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{{Use American English|date = February 2019}}
{{ArticleIcons|ultimate=y}}
{{Short description|Important, well-understood quantum mechanical model}}
{{disambig2|Greninja's appearance in ''Super Smash Bros. Ultimate''|the character in other contexts|Greninja}}
{{Quantum mechanics}}
{{Infobox Character
{{redirect|QHO|text=It is also the [[IATA airport code]] for [[Transportation in Houston#Airports|all airports in the Houston area]]}}
|name = Greninja
|image = [[File:Greninja SSBU.png|250px]]
|game = SSBU
|ssbgame1 = SSB4
|availability = [[Unlockable character|Unlockable]]
|tier = B+
|ranking = 34
}}
<!--
REMINDER: Our Manual of Style states that all Pokémon are to use gender-neutral pronouns.
-->
'''Greninja''' ({{ja|ゲッコウガ|Gekkōga}}, ''Gekkouga'') is a playable character in ''[[Super Smash Bros. Ultimate]]''. It was officially revealed on June 12th, 2018 alongside {{SSBU|Mr. Game & Watch}} and the rest of the returning roster. Greninja is classified as [[Fighter number|Fighter #50]].


[[File:QuantumHarmonicOscillatorAnimation.gif|thumb|300px|right|Some trajectories of a [[harmonic oscillator]] according to [[Newton's laws]] of [[classical mechanics]] (A–B), and according to the [[Schrödinger equation]] of [[quantum mechanics]] (C–H). In A–B, the particle (represented as a ball attached to a [[Hooke's law|spring]]) oscillates back and forth. In C–H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the [[wavefunction]]. C, D, E, F, but not G, H, are [[energy eigenstate]]s. H is a [[Coherent states|coherent state]]—a quantum state that approximates the classical trajectory.]]
Billy Bob Thompson, Yūji Ueda, Frédéric Clou and Benedikt Gutjan's portrayals of Greninja from ''[[Super Smash Bros. 4]]'' were repurposed for the English, Japanese, French and German versions of ''Ultimate'', respectively.


The '''quantum harmonic oscillator''' is the [[quantum mechanics|quantum-mechanical]] analog of the [[harmonic oscillator|classical harmonic oscillator]]. Because an arbitrary smooth [[Potential energy|potential]] can usually be approximated as a [[Harmonic oscillator#Simple harmonic oscillator|harmonic potential]] at the vicinity of a stable [[equilibrium point]], it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, [[List_of_quantum-mechanical_systems_with_analytical_solutions|analytical solution]] is known.<ref>{{Cite book | author=Griffiths, David J. | title=Introduction to Quantum Mechanics | edition=2nd | publisher=Prentice Hall | year=2004 | isbn=978-0-13-805326-0 | author-link=David Griffiths (physicist) | url-access=registration | url=https://archive.org/details/introductiontoel00grif_0 }}</ref><ref>{{Cite book| author=Liboff, Richard L. | title=Introductory Quantum Mechanics | publisher=Addison–Wesley | year=2002 | isbn=978-0-8053-8714-8| author-link=Liboff, Richard L.}}</ref><ref>{{Cite web | last =Rashid | first =Muneer A. | author-link =Munir Ahmad Rashid | title =Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian | website =M.A. Rashid – [[National University of Sciences and Technology, Pakistan|Center for Advanced Mathematics and Physics]] | publisher =[[National Center for Physics]] | year =2006 | url =http://www.ncp.edu.pk/docs/12th_rgdocs/Munir-Rasheed.pdf | format =[[PDF]]-[[Microsoft PowerPoint]] | access-date =19 October 2010 }}</ref>
Greninja is ranked 34th out of 82 on the current [[tier list]], placing it in the B+ tier. This is a marginal rise from its 26th out of 54 placement in ''SSB4''.


==One-dimensional harmonic oscillator==
==How to unlock==
Complete one of the following:
*Play [[VS. match]]es, with Greninja being the 58th character to be unlocked.
*Clear {{SSBU|Classic Mode}} with {{SSBU|Donkey Kong}} or any character in his unlock tree, being the 6th character unlocked after {{SSBU|Sheik}}.
*Have Greninja join the player's party in [[World of Light]].
Greninja must then be defeated on [[Kalos Pokémon League]] (the [[Ω form]] is used in World of Light).


===Hamiltonian and energy eigenstates===
==Attributes==
[[Image:HarmOsziFunktionen.png|thumb|Wavefunction representations for the first eight bound eigenstates, ''n'' = 0 to 7. The horizontal axis shows the position ''x''.]]
<!--
[[Image:Aufenthaltswahrscheinlichkeit harmonischer Oszillator.png|thumb|Corresponding probability densities.]]
REMINDER: Our Manual of Style states that all Pokémon are to use gender-neutral pronouns.
-->
Greninja, true to being a ninja-themed character, has very strong mobility; it has the 4th fastest [[walking speed]], the 8th fastest [[dash|run speed]], the 10th fastest [[air speed]], is tied for the 9th fastest [[fall speed|falling]] and [[fast fall]]ing speed, the 2nd highest [[gravity]], and possesses the 2nd highest [[jump|jump height]] overall. However, unlike most characters who boast similar mobility (such as {{SSBU|Sheik}}), Greninja boasts a surprising amount of KO options, good range on plenty of its attacks, and KO throws.


The [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the particle is:
One of Greninja's most notable traits is its high mobility which complements its grounded moveset. Greninja's dash attack comes out on frame 7 and has very low ending lag, as well as the ability to cross upon shields. Its knockback angle allows for many true follow-ups and strings over a large range of percents. Its neutral jab attack comes out on frame 3, making it a good grounded combo breaker. It can also lock, which gives Greninja access to potent punishes from opponents missing techs. Its down tilt is an excellent combo starter due to its low startup, ending lag and vertical launch angle. Greninja's up tilt is a frame 9 disjointed hitbox that acts well as an anti-air and can also be a combo starter. Its smash attacks are also reliable in their own right; its forward smash is quick for its range and power, down smash is an excellent punishment option for ledge regrabs, as well as sending at a low angle, and up smash is a potent combo finisher.
<math display="block">\hat H = \frac{{\hat p}^2}{2m} + \frac{1}{2} k {\hat x}^2 = \frac{{\hat p}^2}{2m} + \frac{1}{2} m \omega^2 {\hat x}^2 \, ,</math>
where {{mvar|m}} is the particle's mass, {{mvar|k}} is the force constant, <math display="inline">\omega = \sqrt{k / m}</math> is the [[angular frequency]] of the oscillator, <math>\hat{x}</math> is the [[position operator]] (given by {{mvar|x}} in the coordinate basis), and <math>\hat{p}</math>  is the [[momentum operator]] (given by <math>\hat p = -i \hbar \, \partial / \partial x</math> in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in [[Hooke's law]].


One may write the time-independent [[Schrödinger equation]],
Greninja also has a very strong air game due to its aforementioned air speed and jump height. Greninja's aerials are reliable for multiple situations and all have low landing lag (except for down aerial, at 30 frames). Its neutral aerial is a decent low percent combo starter due to it having incredibly low landing lag and a good launch angle. It can also KO at high percentages. Its forward aerial acts as a combo finisher from its combo starters and can KO moderately early. Forward aerial's low landing lag and disjointed nature also make it safe on shield in many situations when spaced correctly. Its up aerial is a great juggling option with low all-around lag and boasting good KO potential near the upper blast line. Greninja can also utilize its multihits to drag down opponents to create tech chase and jab lock situations. Its back aerial is a very fast follow up or offstage edgeguarding option. Down aerial can be used as a mix up to return to the stage from far above, as well as perform surprise combos on hit with both its [[meteor smash]] and sourspot hitbox.
<math display="block"> \hat H \left| \psi \right\rangle = E \left| \psi \right\rangle  ~,</math>
where {{mvar|E}} denotes a to-be-determined real number that will specify a time-independent [[energy level]], or [[eigenvalue]], and the solution  {{math|{{!}}''ψ''⟩}} denotes that level's energy [[eigenstate]].


One may solve the differential equation representing this eigenvalue problem in the coordinate basis, for the [[wave function]] {{math|1=⟨''x''{{!}}''ψ''⟩ = ''ψ''(''x'')}}, using a [[spectral method]]. It turns out that there is a family of solutions. In this basis, they amount to [[Hermite polynomials#Hermite functions| Hermite functions]],
Greninja's grab game is overall very effective, due to its grabs being among the longest ranges of any non-tether grab in the game. Its forward, up, and back throws can KO at high percentages. Down throw acts as a middle percent combo starter, as well as a strong DI mix up, especially at higher percentages at ledge as a 50/50 between DI in and out in conjunction with forward throw. Up throw acts as a versatile combo starter that can lead to juggling situations. Because of this, Greninja has plenty of options off of a grab, as not one of its throws could be considered useless.
<math display="block"> \psi_n(x) = \frac{1}{\sqrt{2^n\,n!}}  \left(\frac{m\omega}{\pi \hbar}\right)^{1/4}  e^{
- \frac{m\omega x^2}{2 \hbar}} H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad n = 0,1,2,\ldots. </math>


The functions ''H<sub>n</sub>'' are the physicists' [[Hermite polynomials]],
Finally, Greninja's special moves are effective in various situations. [[Water Shuriken]] acts as a versatile zoning tool, as well as a high-percentage KO option when fully charged. At low-to-mid percents, it is also a combo starter, allowing Greninja to rush down its opponent and follow up with any aerial attack or an up smash. [[Shadow Sneak]] works as an effective recovery mix up, as well as a potent KO move from a good read or pseudo-combo finisher. Despite lacking an offensive hitbox, [[Hydro Pump]] is a good recovery move for its long distance, and can be used for gimping recoveries due to having windbox properties. [[Substitute]] is a counterattack with the unique attribute of being able to be aimed in one of 8 different directions upon a successful counter. These angled follow ups allow for Greninja to gain pseudo-follow ups as well as KO earlier by picking the optimal angle in regards to stage positioning.
<math display="block">H_n(z)=(-1)^n~ e^{z^2}\frac{d^n}{dz^n}\left(e^{-z^2}\right).</math>


The corresponding energy levels are
However, Greninja is also flawed in many ways. One of Greninja's primary flaws is its inability to break out of disadvantage state. While not as bad as the previous game, Greninja still has difficulties escaping combos due to its fast falling speed and its aerials still having relatively high startup. This can sometimes be alleviated with its back or down aerials, but both are not very effective due to back aerial's almost entirely horizontal hitboxes and down aerial's landing lag. Another option is aerial Water Shuriken, which stalls Greninja in the air and can be used as a landing mix up, as well as Hydro Pump landing mix ups.  
<math display="block"> E_n = \hbar \omega\bigl(n + \tfrac{1}{2}\bigr)=(2 n + 1) {\hbar \over 2} \omega~.</math>


This energy spectrum is noteworthy for three reasons.  First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of {{math|''ħω''}}) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the [[Bohr model]] of the atom, or the [[particle in a box]]. Third, the lowest achievable energy (the energy of the {{math|1=''n'' = 0}} state, called the [[ground state]]) is not equal to the minimum of the potential well, but {{math|''ħω''/2}} above it; this is called [[zero-point energy]]. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the [[Heisenberg uncertainty principle]].
Another notable weaknesses is Greninja's multihit moves, especially back air and up air, being notoriously inconsistent, allowing opponents to escape out of these moves. As both aerials are some of Greninja's best combo tools, the inconsistency greatly affects its combo game.


The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The [[correspondence principle]] is thus satisfied. Moreover, special nondispersive [[wave packet]]s, with minimum uncertainty, called [[Coherent states#The wavefunction of a coherent state|coherent states]] oscillate very much like classical objects, as illustrated in the figure; they are ''not'' eigenstates of the Hamiltonian.
Greninja's biggest weakness, however, is its terrible out of shield game, which is arguably the worst of the entire cast. Because of its high short hop, its aerials slow startup, and lacking a fast grab (although it has good range), Greninja lacks an effective out of shield option faster than frame 14. While its back aerial is fairly quick at frame 5 (making it frame 8 out of shield), it is unable to hit opponents in front of Greninja and is very inconsistent at hitting opponents behind Greninja due to its high short hop. Jumping or rolling out of shield are potential options to reset neutral, but they are very predictable and easily read. Thus, when Greninja is pinned down in shield, it has difficulty escaping the situation without being heavily punished. Combined with its vulnerability to combos, this gives it an atrocious defensive game.  


===Ladder operator method===
Altogether, Greninja's playstyle requires players to think like an actual ninja: utilizing Greninja's superb mobility and fast attacks to rush down opponents, saving the slower attacks for potential mixups, mindgames and surprise KO options, and remaining unpredictable to prevent being trapped into disadvantageous positions.
[[Image:QHarmonicOscillator.png|right|thumb|Probability densities <nowiki>|</nowiki>''ψ<sub>n</sub>''(''x'')<nowiki>|</nowiki><sup>2</sup> <!--or in pseudoTeX: <math>\left |\psi_n(x)\right |^2</math> --> for the bound eigenstates, beginning with the ground state (''n'' = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position {{mvar|x}}, and brighter colors represent higher probability densities.]]


The "[[ladder operator]]" method, developed by [[Paul Dirac]], allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in [[quantum field theory]].  Following this approach, we define the operators {{mvar|a}} and its [[Hermitian adjoint|adjoint]] {{math|''a''<sup>†</sup>}},
==Changes from ''[[Super Smash Bros. 4]]''==
<math display="block">\begin{align}
<!--
          a &=\sqrt{m\omega \over 2\hbar} \left(\hat x + {i \over m \omega} \hat p \right) \\
REMINDER: Our Manual of Style states that all Pokémon are to use gender-neutral pronouns.
  a^\dagger &=\sqrt{m\omega \over 2\hbar} \left(\hat x - {i \over m \omega} \hat p \right)
-->
\end{align}</math>Note these operators classically are exactly the [[Generator (mathematics)|generators]] of normalized rotation in the phase space of <math>x</math> and <math>m\frac{dx}{dt}</math>, ''i.e'' they describe the forwards and backwards evolution in time of a classical harmonic oscillator.
Greninja has been greatly buffed in the transition from ''SSB4'' to ''Ultimate''. An assortment of its traits, such as its strong combo game and great mobility, that made up its playstyle have been further improved, while the general engine changes benefit said playstyle.


These operators lead to the useful representation of <math>\hat{x}</math> and <math>\hat{p}</math>,
Most of the universal changes notably benefit Greninja. As with all other characters in the game, Greninja's already quick mobility is faster like most characters, which benefits its hit-and-run playstyle. The ability to [[dash-canceling|run cancel into any ground move]] allows Greninja to further exploit its amazing ground mobility, allowing for easier setups into its combo starters, such as [[up tilt|up]] and [[down tilt]]s and [[dash attack]]. The universally reduced landing lag on aerial attacks gives Greninja an easier time landing and comboing, while the universal 3-frame jumpsquat improves Greninja's ground-to-air potential. The implementation of [[spot dodge]] canceling and the changes to [[perfect shield]]ing improve its potential punish game, due to its wide variety of combo starters and fast frame data.
<math display="block">\begin{align}
  \hat x &=  \sqrt{\frac{\hbar}{2 m\omega}}(a^\dagger + a) \\
  \hat p &= i\sqrt{\frac{\hbar m \omega}{2}}(a^\dagger - a) ~.
\end{align}</math>


The operator {{mvar|a}} is not [[Hermitian operator|Hermitian]], since itself and its adjoint {{math|''a''<sup>†</sup>}} are not equal. The energy eigenstates {{math|{{ket|''n''}}}} (also known as [[Fock state|Fock states]]), when operated on by these ladder operators, give
Aside from the universal changes, Greninja has also received notable direct buffs. The biggest ones were to its [[grab]] game; its standing grab is faster and its pummel, previously one of the worst in ''SSB4'', deals less damage but is significantly faster, making it better all-around. Greninja's [[forward throw]] has higher knockback, allowing it to KO in an emergency, as with [[up throw]]. Its up and [[down throw]]s also have better combo and juggling potential due to the universal changes to mobility — down throw notably now allows for potential KO confirms into [[forward aerial|forward]] and [[back aerial]]s Outside of its improved grab game, Greninja now has a new down tilt that has lower ending lag and sends at more favorable angles, and its dash attack sends at a higher angle, further improving Greninja's combo game. [[Water Shuriken]] has more range, improving Greninja's camping ability combined with its faster mobility. [[Substitute]] now slows opponents down and offers Greninja intangibility during all of its attack variations, bringing it in line with other counterattacks. Greninja's KO power has also been buffed, with [[forward smash]] and forward aerial receiving higher knockback, [[up smash]] connecting better into its second hit, and [[down smash]] having faster startup.
<math display="block">\begin{align}
  a^\dagger|n\rangle &= \sqrt{n + 1} | n + 1\rangle \\
          a|n\rangle &= \sqrt{n} | n - 1\rangle.
\end{align}</math>


It is then evident that {{math|''a''<sup>†</sup>}}, in essence, appends a single quantum of energy to the oscillator, while {{mvar|a}} removes a quantum. For this reason, they are sometimes referred to as "creation" and "annihilation" operators.
On the other hand, Greninja is not without a few nerfs. Notably, the ability to [[tech]] footstools has made footstool combos much harder to pull off, which hinders Greninja's combo ability (specifically from its [[down aerial]]), though this is alleviated by the various buffs to Greninja's moveset. Substitute's attack variants are all weaker while also being more laggy overall, which compensates for the attack's new intangibility. In exchange for its buffed mobility, Greninja is now lighter, which brings it slightly more in-line with other combo-centric and/or hit-and-run characters, while not compensating much for its vulnerability to combos.


From the relations above, we can also define a number operator {{mvar|N}}, which has the following property:
In terms of universal changes, some of them are as benefitial as they are detrimental to Greninja. The changes to [[air dodge]] mechanics slightly improve its previously below average edgeguarding game, but also make it more vulnerable to combos. The increased [[shieldstun]] makes Greninja's own attacks much safer on shield, but also weakens its already poor [[out of shield]] game, forcing it to abuse the revamped perfect shielding. Lastly, the weakening of [[rage]] makes it slightly harder for Greninja to take advantage of it, despite compensating for its lighter weight.
<math display="block">\begin{align}
                        N &= a^\dagger a \\
  N\left| n \right\rangle &= n\left| n \right\rangle.
\end{align}</math>


The following [[commutator]]s can be easily obtained by substituting the [[canonical commutation relation]],
Collectively, Greninja's strengths and weaknesses are more defined. It now performs much better as a hit-and-run, combo-centric character, with excellent mobility and various KO options and set-ups at a variety of percentages, but with its light weight and attributes making it just as vulnerable to combos while keeping its terrible out of shield game. Despite its nerfs to its previous footstool combos, Greninja has been generally compensated due to the gameplay changes to the engine and the buffs to its moveset giving it a more traditional combo game. That said, various veterans have also been buffed to varying degrees, especially through game updates, while Greninja has comparatively received minor, unimpactful changes from them. Nevertheless, Greninja has improved from ''SSB4'', keeping up with the rest of the cast in ''Ultimate''.
<math display="block">[a, a^\dagger] = 1,\qquad[N, a^\dagger] = a^{\dagger},\qquad[N, a] = -a, </math>


And the Hamilton operator can be expressed as
{{SSB4 to SSBU changelist|char=Greninja}}
<math display="block">\hat H = \hbar\omega\left(N + \frac{1}{2}\right),</math>


so the eigenstate of {{mvar|N}} is also the eigenstate of energy.
==Update history==
Greninja has received a mix of minor buffs and nerfs and glitch fixes via game updates. Some important nerfs include removing the ability to cancel endlag with Hydro Pump and the universal projectile shield damage nerf affected Greninja's safety and neutral game making actions in general less safe. To partially compensate, other aspects like down smash and pummel were made more reliable and increased shield size improve the potency of rushdown.


The commutation property yields
These changes did not affect Greninja in significant ways, and thus Greninja's competitive standing has remained virtually the same since the game's launch.
<math display="block">\begin{align}
  Na^{\dagger}|n\rangle &= \left(a^\dagger N + [N, a^\dagger]\right)|n\rangle \\
                        &= \left(a^\dagger N + a^\dagger\right)|n\rangle \\
                        &= (n + 1)a^\dagger|n\rangle,
\end{align} </math>


and similarly,
'''{{GameIcon|ssbu}} {{SSBU|1.2.0}}'''
<math display="block">Na|n\rangle = (n - 1)a | n \rangle.</math>
{{UpdateList (SSBU)/1.2.0|char=Greninja}}


This means that {{mvar|a}} acts on  {{math|{{!}}''n''⟩}}  to produce, up to a multiplicative constant,  {{math|{{!}}''n''–1⟩}}, and {{math|''a''<sup>†</sup>}} acts on  {{math|{{!}}''n''⟩}} to produce {{math|{{!}}''n''+1⟩}}. For this reason, {{mvar|a}} is called a '''annihilation operator''' ("lowering operator"), and {{math|''a''<sup>†</sup>}} a '''creation operator''' ("raising operator"). The two operators together are called [[ladder operator]]s. In quantum field theory, {{mvar|a}} and {{math|''a''<sup>†</sup>}} are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.
'''{{GameIcon|ssbu}} {{SSBU|2.0.0}}'''
{{UpdateList (SSBU)/2.0.0|char=Greninja}}


Given any energy eigenstate, we can act on it with the lowering operator, {{mvar|a}}, to produce another eigenstate with {{math|''ħω''}} less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to {{math|1=''E'' = −∞}}. However, since
'''{{GameIcon|ssbu}} {{SSBU|3.0.0}}'''
<math display="block">n = \langle n | N | n \rangle = \langle n | a^\dagger a | n \rangle = \Bigl(a | n \rangle \Bigr)^\dagger a | n \rangle \geqslant 0,</math>
{{UpdateList (SSBU)/3.0.0|char=Greninja}}


the smallest eigen-number is 0, and
'''{{GameIcon|ssbu}} {{SSBU|3.1.0}}'''
<math display="block">a \left| 0 \right\rangle = 0. </math>
{{UpdateList (SSBU)/3.1.0|char=Greninja}}


In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstates. Furthermore, we have shown above that
'''{{GameIcon|ssbu}} {{SSBU|4.0.0}}'''
<math display="block">\hat H \left|0\right\rangle = \frac{\hbar\omega}{2} \left|0\right\rangle</math>
{{UpdateList (SSBU)/4.0.0|char=Greninja}}


Finally, by acting on  |0⟩ with the raising operator and multiplying by suitable [[Wave function#Normalization condition|normalization factors]], we can produce an infinite set of energy eigenstates
'''{{GameIcon|ssbu}} {{SSBU|7.0.0}}'''
<math display="block">\left\{\left| 0 \right\rangle, \left| 1 \right\rangle, \left| 2 \right\rangle, \ldots , \left| n \right\rangle, \ldots\right\},</math>
{{UpdateList (SSBU)/7.0.0|char=Greninja}}


such that
'''{{GameIcon|ssbu}} {{SSBU|8.0.0}}'''
<math display="block">\hat H \left| n \right\rangle = \hbar\omega \left( n + \frac{1}{2} \right) \left| n \right\rangle, </math>
{{UpdateList (SSBU)/8.0.0|char=Greninja}}
which matches the energy spectrum given in the preceding section.


Arbitrary eigenstates can be expressed in terms of |0⟩,
==Moveset==
<math display="block">|n\rangle = \frac{(a^\dagger)^n}{\sqrt{n!}} |0\rangle. </math>
* Greninja can [[Crawling|crawl]], [[wall cling]], and [[wall jump]].
{{math proof|<math display="block">\begin{align}
''For a gallery of Greninja's hitboxes, see [[Greninja (SSBU)/Hitboxes|here]].''
  \langle n | aa^\dagger | n \rangle &= \langle n|\left([a, a^\dagger] + a^\dagger a\right)| n \rangle = \langle n|(N + 1)|n\rangle = n + 1 \\
    \Rightarrow a^\dagger | n\rangle &= \sqrt{n + 1} | n + 1\rangle \\
                    \Rightarrow|n\rangle &= \frac{a^\dagger}{\sqrt{n}} | n - 1 \rangle = \frac{(a^\dagger)^2}{\sqrt{n(n - 1)}} | n - 2 \rangle = \cdots = \frac{(a^\dagger)^n}{\sqrt{n!}}|0\rangle.
\end{align}</math>}}


====Analytical questions====
{{MovesetTable
 
|neutralname=Single Hit ({{ja|たんだ|Tanda}}) / Palm Strike ({{ja|しょうだ|Shōda}}) / Water Palm Heel ({{ja|しょうてい|Mizu Shōtei}}) / Water Cutter ({{ja|みずカッター|Mizu Kattā}}) / Cutter Finish ({{ja|カッターフィニッシュ|Kattā Finisshu}})
The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation <math>a\psi_0 = 0</math>. In the position representation, this is the first-order differential equation
|neutralcount=3
<math display="block">\left(x+\frac{\hbar}{m\omega}\frac{d}{dx}\right)\psi_0 = 0,</math>
|neutralinf=y
whose solution is easily found to be the Gaussian<ref>The normalization constant is <math>C = \left(\frac{m\omega}{\pi \hbar}\right)^{\frac{1}{4}}</math>, and satisfies the normalization condition <math>\int_{-\infty}^{\infty}\psi_0(x)^{*}\psi_0(x)dx = 1</math>.</ref>
|neutral1dmg=2%
<math display="block">\psi_0(x)=Ce^{-\frac{m\omega x^2}{2\hbar}}.</math>
|neutral2dmg=2%
Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the harmonic oscillator. Once the ground state is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. One can also prove that, as expected from the uniqueness of the ground state, the Hermite functions energy eigenstates <math>\psi_n</math> constructed by the ladder method form a ''complete'' orthonormal set of functions.<ref>See Theorem 11.4 in {{citation|first=Brian C.|last=Hall|title=Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267|isbn=978-1461471158 |publisher=Springer|year=2013}}</ref>
|neutral3dmg=3%
 
|neutralinfdmg=0.5% (loop), 2% (last)
Explicitly connecting with the previous section, the ground state  |0⟩  in the position representation is determined by <math> a| 0\rangle =0</math>,
|neutraldesc=Two alternating palm thrusts followed by a double palm thrust that emits a small blast of water. If button mashed, it is instead followed by a series of knifehand strikes that emit blade-shaped water blasts that concludes with an outward knifehand strike that emits a wide blast of water. It can also be jab canceled, such as into forward tilt, down tilt and forward smash.
<math display="block">  \left\langle x \mid a \mid 0 \right\rangle = 0 \qquad
|ftiltname=Rolling Kick ({{ja|まわしげり|Mawashigeri}}, ''Roundhouse Kick'')
  \Rightarrow \left(x + \frac{\hbar}{m\omega}\frac{d}{dx}\right)\left\langle x\mid 0\right\rangle = 0 \qquad
|ftiltdmg=7.3%
  \Rightarrow          </math>
|ftiltdesc=A hook kick which stops half way. It can be angled and can [[lock]] opponents.  
<math display="block"> \left\langle x\mid 0\right\rangle = \left(\frac{m\omega}{\pi\hbar}\right)^\frac{1}{4} \exp\left( -\frac{m\omega}{2\hbar}x^2 \right)  
|utiltname=Anti-Air Tongue ({{ja|ぜつたいくう|Zetsu Taikū}})
    = \psi_0  ~,</math>
|utiltdmg=4.5%
hence
|utiltdesc=Swings its tongue upwards. A good aerial combo starter and juggling tool due to its low knockback and somewhat disjointed hitbox. It can combo into itself at low percents and is a reliable way to connect into up aerial.
<math display="block"> \langle x \mid a^\dagger \mid 0 \rangle = \psi_1 (x) ~,</math>
|dtiltname=Shin Slash ({{ja|すねぎり|Sune Giri}})
so that <math>\psi_1(x,t)=\langle x \mid e^{-3i\omega t/2} a^\dagger \mid 0 \rangle </math>, and so on.
|dtiltdmg=4%
 
|dtiltdesc=Does a downward hand sweep. It sends opponents at an upward angle, making it a versatile combo starter. Notably, it can confirm a KO into an up smash rather easily.
===Natural length and energy scales===
|dashname=Surface Kick ({{ja|すいめんげり|Suimen Geri}})
The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by [[nondimensionalization#Quantum harmonic oscillator|nondimensionalization]].
|dashdmg=8%
 
|dashdesc=Does a sweep kick. Is arguably one of the best dash attacks in the game, as it launches opponents at an excellent angle for combos, making it one of Greninja's best combo starters. Reliably combos into back aerial at virtually any percent, and can set up up aerial strings or drag-down combos with up aerial.
The result is that, if  ''energy'' is measured in units of  {{math|''ħω''}} and ''distance'' in units of {{math|{{sqrt|''ħ''/(''mω'')}}}}, then the Hamiltonian simplifies to
|fsmashname=Water Kunai ({{ja|みずくない|Mizu Kunai}})
<math display="block"> H = -\frac{1}{2} {d^2 \over dx^2} +\frac{1}{2}  x^2 ,</math>
|fsmashdmg=14%
while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half,
|fsmashdesc=An inward slash with a water kunai. Deals good knockback and has good range. However, it has notable ending lag.
<math display="block">\psi_n(x)= \left\langle x \mid n \right\rangle = {1 \over \sqrt{2^n n!}}~ \pi^{-1/4} \exp(-x^2 / 2)~ H_n(x),</math>
|usmashname=Double Water Kunai ({{ja|ダブルみずくない|Daburu Mizu Kunai}})
<math display="block">E_n = n + \tfrac{1}{2} ~,</math>
|usmashdmg=5% (hit 1), 14% (hit 2 clean center), 11% (hit 2 clean sides), 10% (hit 2 late)
where {{math|''H''<sub>''n''</sub>(''x'')}} are the [[Hermite polynomials]].
|usmashdesc=Two reverse gripped inward slashes with water kunai, similar to {{SSBU|Sheik}}'s up smash. Greninja's strongest finisher, especially when hit clean. Can be combo'd into from a down-tilt at specific percentages for a KO.
 
|dsmashname=Water Kunai Shake Off ({{ja|ダブルふりはらいみずくない|Daburu Furiharai Mizu Kunai}})
To avoid confusion, these "natural units"  will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.
|dsmashdmg=13% (kunai), 11% (arms)
 
|dsmashdesc=Hits both sides with water kunai. Due to it sending opponents at an [[semi-spike]] angle and coming out on frame 11, it is a quick and effective way to set up an edge-guard situation.
For example, the [[fundamental solution]] ([[Propagator#Basic_examples:_propagator_of_free_particle_and_harmonic_oscillator|propagator]]) of {{math|''H'' − ''i∂<sub>t</sub>''}}, the time-dependent Schrödinger operator for this oscillator, simply boils down to the [[Mehler kernel]],<ref>[[Wolfgang Pauli|Pauli, W.]] (2000), ''Wave Mechanics: Volume 5 of Pauli Lectures on Physics'' (Dover Books on Physics). {{ISBN|978-0486414621}} ; Section 44.</ref><ref>[[Edward Condon|Condon, E. U.]]  (1937). "Immersion of the Fourier transform in a continuous group of functional transformations", ''Proc. Natl. Acad. Sci. USA''  '''23''', 158–164. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf online]</ref>
|nairname=Water Burst ({{ja|みずはれつ|Mizu Haretsu}})
<math display="block">\langle x \mid \exp (-itH) \mid y \rangle \equiv K(x,y;t)= \frac{1}{\sqrt{2\pi i \sin t}} \exp \left(\frac{i}{2\sin t}\left ((x^2+y^2)\cos t - 2xy\right )\right )~,</math>
|nairdmg=11% (clean), 6% (late)
where {{math|1= ''K''(''x'',''y'';0) = ''δ''(''x'' − ''y'')}}. The most general solution for a given initial configuration {{math|''ψ''(''x'',0)}} then is simply
|nairdesc=Strikes a ninjutsu pose while emitting an exploding water bubble. Despite noticeable start-up lag for a neutral aerial(frame 12), it boasts great combo potential at low to mid percentages with its strong and weak hits and can KO confirm into up-smash at high percents with the weak hit. It can also KO at very high percents with its strong hit.
<math display="block">\psi(x,t)=\int dy~ K(x,y;t) \psi(y,0) \,.</math>
|fairname=Aerial Water Kunai ({{ja|くうちゅうみずくない|Kūchū Mizu Kunai}})
{{see also|Path integral formulation#Simple harmonic oscillator}}
|fairdmg=14%
 
|fairdesc=Slashes with a water kunai. It has some start-up and suffers from high end lag, but it is a great tool in the neutral for spacing due to its disjointed hitbox and can be used for KOing. It is also safe on shield if spaced correctly.
===Coherent states===
|bairname=Triple Kick ({{ja|さんてんげり|Santen Geri}})
 
|bairdmg= 3% (hit 1), 2.5% (hit 2), 6% (hit 3)
{{main|Coherent state}}
|bairdesc=Kicks backwards three times. It is Greninja's fastest aerial attack, although it is also one of the weakest aerials of its kind. However, this makes it a useful combo tool in return, as it can be followed up from a down throw, down tilt, as well as a dash attack. It can be a situational out-of-shield option if the opponent crosses-up on its shield.
 
|uairname=Spiral Kick ({{ja|らせんげり|Rasen Geri}})
[[File:QHO-coherentstate3-animation-color.gif|thumb|Time evolution of the probability distribution (and phase, shown as color) of a coherent state with &#124;''α''&#124;=3.]]
|uairdmg=1.3% (hits 1-5), 3% (hit 6)
 
|uairdesc=Does a upward corkscrew kick, similar to both {{SSBU|Sheik}}'s and {{SSBU|Joker}}'s up aerials. One of Greninja's best combo and KO tools, as it can juggle and KO effectively due to its great jumping prowess. It also allows for drag-down combos because it's multi-hit properties, although this requires precise timing to land it at the right time, as its final launching hit comes out too fast to set up drag-down combos otherwise.
The [[Coherent states#The wavefunction of a coherent state|coherent states]] (also known as Glauber states) of the harmonic oscillator are special nondispersive [[wave packet]]s, with minimum uncertainty {{math|1=''σ<sub>x</sub>'' ''σ<sub>p</sub>'' = {{frac|''ℏ''|2}}}}, whose [[observable]]s' [[Expectation value (quantum mechanics)|expectation values]] evolve like a classical system.  They are eigenvectors of the annihilation operator, ''not'' the Hamiltonian, and form an [[Overcompleteness|overcomplete]] basis which consequentially lacks orthogonality.
|dairname=Aerial Kick ({{ja|きゅうしゅうげり|Kūchū Geri}})
 
|dairdmg=8%
The coherent states are indexed by {{math|''α'' ∈ '''C'''}} and expressed in the {{math|{{braket|ket|''n''}}}} basis as
|dairdesc=A diving double foot stomp. It acts as a [[stall-then-fall]] and bounces off opponents. The clean hit [[meteor smash]]es opponents while the late hit sends the opponent upwards, allowing for some situational combos.
 
|grabname=Grab ({{ja|つかみ|Tsukami}})
<math display="block">|\alpha\rangle = \sum_{n=0}^\infty |n\rangle \langle n | \alpha \rangle = e^{-\frac{1}{2} |\alpha|^2} \sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}} |n\rangle = e^{-\frac{1}{2} |\alpha|^2} e^{\alpha a^\dagger} e^{-{\alpha^* a}} |0\rangle.</math>
|grabdesc=Grabs with a whirlpool. While its standing grab is slow, it is among the longest-reaching grabs in the game.
 
|pummelname=Grab Water Bind ({{ja|つかみみずしばり|Tsukami Mizu Shibari}})
Because <math>a \left| 0 \right\rangle = 0 </math> and via the Kermack-McCrae identity, the last form is equivalent to a [[Unitary operator|unitary]] [[displacement operator]] acting on the ground state: <math>|\alpha\rangle=e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle  = D(\alpha)|0\rangle</math>. The [[position space]] wave functions are
|pummeldmg=1%
|pummeldesc=Compresses target with water. Decent speed.
|fthrowname=Push Away ({{ja|おしのけ|Oshinoke}})
|fthrowdmg=3.5% (hit 1), 4.5% (throw)
|fthrowdesc=Shoves the opponent forward. Can KO at high percentages near the edge.
|bthrowname=Send-off　({{ja|さかおくり|Sakaokuri}})
|bthrowdmg=9%
|bthrowdesc=Greninja leans forward and flings the opponent backwards. Like forward throw, its decent knockback gives it good edgeguarding potential at high percentages.
|uthrowname=Toss Up ({{ja|なげあげ|Nage Age}})
|uthrowdmg=5%
|uthrowdesc=Tosses the opponent upwards. One of Greninja's most useful throws, as it can combo into a up tilt or up aerial at low and medium percents and can even KO at later percents.
|dthrowname=Slam ({{ja|たたきつけ|Tatakitsuke}})
|dthrowdmg=5%
|dthrowdesc=Slams the opponent onto the ground. It can combo into a down tilt, forward tilt, and dash attack at low percentages. It can combo into back aerial at medium percents and later into forward aerial as well at higher percentages with good timing.
|floorfname=&nbsp;
|floorfdmg=8%
|floorfdesc=Sweep kicks around itself while getting up.
|floorbname=&nbsp;
|floorbdmg=8%
|floorbdesc=Sweep kicks around itself while getting up.
|floortname=&nbsp;
|floortdmg=8%
|floortdesc=Sweep kicks around itself while getting up.
|edgename=&nbsp;
|edgedmg=9%
|edgedesc=Performs a roundhouse kick while climbing up.
|nsname=Water Shuriken
|nsdmg=3%-10.8% (uncharged), 1.0% (fully charged looping hits), 9% (fully charged final hit)
|nsdesc=Uses Water Shuriken, which can be charged. Depending on how long the move is charged, the shuriken will be larger and do more damage, however the speed and distance will decrease as a result. At full charge, it hits multiple times and can kill with a Shadow Sneak follow-up at low and medium percents. It is also a decent KOing option at higher percents. A fully charged water shuriken that is reflected at a higher speed will have trouble landing the final hit on Greninja because of the looping hits not moving Greninja far enough for it.
|ssname=Shadow Sneak
|ssdmg=10% (normal), 12% (reverse)
|ssdesc=Greninja moves its shadow along the ground; the way the shadow moves is relative to Greninja's position, and it can be moved further or closer to change how far Greninja teleports. After a short time or when the special button is released, Greninja disappears briefly, then reappears at its shadow while performing either a flip kick forwards or a stronger kick backwards (depending on control stick direction) to attack. While performing Shadow Sneak, Greninja cannot run, attack, grab, dodge or shield. However, it can walk slowly, jump and taunt. Has strong knockback and base knockback, which allows it to kill notably early when hitting the move offstage.
|usname=Hydro Pump
|usdmg=2% (per shot)
|usdesc=Uses Hydro Pump to propel itself in the inputted direction; it can be used twice. Each shot has a [[windbox]] effect that pushes opponents, making it a decent option for gimping predictable recoveries.
|dsname=Substitute
|dsdmg=13% (up or down), 12% (diagonally), 11% (left or right)
|dsdesc=Does a pose, and if anyone hits it while posing, Greninja will temporarily disappear, get replaced by a wooden log or a Substitute doll, and then appear behind the opponent and strike them. It deviates noticeably from other counters, as the attack itself can be angled in an inputted direction and launching the opponent in that direction (with the down input meteor smashing opponents, which makes it a good punish against reckless edgeguarders).
|fsname=Secret Ninja Attack
|fsdmg=6% (the mat), 50% (Total in the attack after mat)
|fsdesc=Turns into Ash-Greninja and attacks the opponent with a mat. If an opponent is caught in the mat, Greninja will send them up into the air and strike the opponent repeatedly in midair with its kunai, before slamming them down with a final hit.
|game=SSBU
|dtauntname=
|dtauntdmg=0.5%
|dtauntdesc=While standing on one foot, Greninja holds out its hands, faces the screen, and summons small sprays of water. The sprays produce some knockback, though they're only able to KO Mario at 599% percent. A video showing the exact KO percentages at which each character can be KO'd can be found [https://www.youtube.com/watch?v=cH7IDQ_8vic here]. Unlike the rest of Greninja's taunts, this one cannot be cancelled (unless Shadow Sneak is being charged before using it).
}}


<math display="block">\psi_\alpha(x')= \left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}} e^{\frac{i}{\hbar} \langle\hat{p}\rangle_\alpha x' - \frac{m\omega}{2\hbar}(x' - \langle\hat{x}\rangle_\alpha)^2} .</math>
===Stats===


Since coherent states are not energy eigenstates, their time evolution is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter {{mvar|α}} instead: <math>\alpha(t) = \alpha(0) e^{-i\omega t}</math>.
{{Attributes
 
| cast=89
===Highly excited states===
| weight=88 | rweight=66-68
{{multiple image
| dash=2.178 | rdash=14-15
| width = 320
| run=2.288 | rrun=8
| direction = vertical
| walk=1.502 | rwalk=4
| image1 = Excited_state_for_quantum_harmonic_oscillator.svg
| trac=0.087 | rtrac=75-76
| image2 = QHOn30pdf.svg
| airfric=0.015 | rairfric=9-30
| footer = Wavefunction (top) and probability density (bottom) for the {{math|1=''n'' = 30}} excited state of the quantum harmonic oscillator. Vertical dashed lines indicate the classical turning points, while the dotted line represents the classical probability density.
| air=1.239 | rair=10
| baseaccel=0.01 | rbaseaccel=15-87
| addaccel=0.07 | raddaccel=26-38
| gravity=0.18 | rgravity=2
| fall=1.85 | rfall=9-11
| ff=2.96 | rff=9-11
| jumpsquat=3 | rjumpsquat=1-88
| jumpheight=46 | rjumpheight=2
| shorthop=22.11 | rshorthop=1
| djump=46 | rdjump=5
}}
}}
When {{mvar|n}} is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy {{math|''E''<sub>''n''</sub>}} can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through [[Hermite_polynomials#Asymptotic_expansion|asymptotics of the Hermite polynomials]],  and also through the [[WKB approximation]].
The frequency of oscillation at {{mvar|x}} is proportional to the momentum {{math|''p''(''x'')}} of a classical particle of energy {{math|''E''<sub>''n''</sub>}}  and position {{mvar|x}}.  Furthermore, the square of the amplitude (determining the probability density) is ''inversely'' proportional to  {{math|''p''(''x'')}},  reflecting the length of time the classical particle spends near {{mvar|x}}. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an [[Airy function]]. Using properties of the Airy function,  one may estimate the probability of finding the particle outside the classically allowed region,  to be approximately
<math display="block">\frac{2}{n^{1/3}3^{2/3}\Gamma^2(\tfrac{1}{3})}=\frac{1}{n^{1/3}\cdot  7.46408092658...}</math>
This is also given, asymptotically, by the integral
<math display="block">\frac{1}{2\pi}\int_{0}^{\infty}e^{(2n+1)\left (x-\tfrac{1}{2}\sinh(2x)  \right )}dx ~.</math>
===Phase space solutions===
In the [[phase space formulation]] of quantum mechanics, eigenstates of the quantum harmonic oscillator in [[quasiprobability distribution#Fock state|several different representations]] of the [[quasiprobability distribution]] can be written in closed form. The most widely used of these is for the [[Wigner quasiprobability distribution]].
The Wigner quasiprobability distribution for the energy eigenstate {{math|{{!}}''n''⟩}} is, in the natural units described above,{{citation needed|date=July 2020}}
<math display="block">F_n(x, p) = \frac{(-1)^n}{\pi \hbar} L_n\left(2(x^2 + p^2)\right) e^{-(x^2 + p^2)} \,,</math>
where ''L<sub>n</sub>'' are the [[Laguerre polynomials]]. This example illustrates how the Hermite and Laguerre polynomials are [[Hermite polynomials#Wigner distributions of Hermite functions|linked]] through the [[Wigner–Weyl transform|Wigner map]].
Meanwhile, the [[Husimi_Q_representation|Husimi Q function]] of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we have
<math display="block">Q_n(x,p)=\frac{(x^2+p^2)^n}{n!}\frac{e^{-(x^2+p^2)}}{\pi}</math>
This claim can be verified using the [[Segal–Bargmann_space#The Segal.E2.80.93Bargmann transform|Segal–Bargmann transform]]. Specifically, since the [[Segal–Bargmann space#The canonical commutation relations|raising operator in the Segal–Bargmann representation]] is simply multiplication by <math>z=x+ip</math> and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simply <math>z^n/\sqrt{n!}</math> . At this point, we can appeal to the formula for the Husimi Q function in terms of the Segal–Bargmann transform.
==''N''-dimensional isotropic harmonic oscillator==
The one-dimensional harmonic oscillator is readily generalizable to {{math|''N''}} dimensions, where {{math|1=''N'' = 1, 2, 3, …}}. In one dimension, the position of the particle was specified by a single [[coordinate system|coordinate]], {{math|''x''}}. In {{math|''N''}} dimensions, this is replaced by {{math|''N''}} position coordinates, which we label {{math|''x''<sub>1</sub>, …, ''x''<sub>''N''</sub>}}. Corresponding to each position coordinate is a momentum; we label these {{math|''p''<sub>1</sub>, …, ''p''<sub>''N''</sub>}}. The [[canonical commutation relations]] between these operators are
<math display="block">\begin{align}
{[}x_i , p_j{]} &= i\hbar\delta_{i,j} \\
{[}x_i , x_j{]} &= 0                  \\
{[}p_i , p_j{]} &= 0
\end{align}</math>
The Hamiltonian for this system is
<math display="block"> H = \sum_{i=1}^N \left( {p_i^2 \over 2m} + {1\over 2} m \omega^2 x_i^2 \right).</math>
As the form of this Hamiltonian makes clear, the {{math|''N''}}-dimensional harmonic oscillator is exactly analogous to {{math|''N''}} independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities {{math|''x''<sub>1</sub>, ..., ''x''<sub>''N''</sub>}} would refer to the positions of each of the {{math|''N''}} particles. This is a convenient property of the {{math|''r''<sup>2</sup>}} potential, which allows the potential energy to be separated into terms depending on one coordinate each.
This observation makes the solution straightforward. For a particular set of quantum numbers <math>\{n\}\equiv
\{n_1, n_2, \dots, n_N\}</math> the energy eigenfunctions for the {{math|''N''}}-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:
<math display="block">\langle \mathbf{x}|\psi_{\{n\}}\rangle = \prod_{i=1}^N\langle x_i\mid \psi_{n_i}\rangle</math>
In the ladder operator method, we define {{math|''N''}} sets of ladder operators,
<math display="block">\begin{align}
a_i &= \sqrt{m\omega \over 2\hbar} \left(x_i + {i \over m \omega} p_i \right), \\
a^{\dagger}_i &= \sqrt{m \omega \over 2\hbar} \left( x_i - {i \over m \omega} p_i \right).
\end{align}</math>
By an analogous procedure to the one-dimensional case, we can then show that each of the {{math|''a<sub>i</sub>''}} and {{math|''a''<sup>†</sup><sub>''i''</sub>}} operators lower and raise the energy by {{math|''ℏω''}} respectively. The Hamiltonian is
<math display="block">H = \hbar \omega \, \sum_{i=1}^N \left(a_i^\dagger \,a_i + \frac{1}{2}\right).</math>
This Hamiltonian is invariant under the dynamic symmetry group {{math|''U''(''N'')}} (the unitary group in {{math|''N''}} dimensions), defined by
<math display="block">
U\, a_i^\dagger \,U^\dagger = \sum_{j=1}^N  a_j^\dagger\,U_{ji}\quad\text{for all}\quad
U \in U(N),</math>
where <math>U_{ji}</math> is an element in the defining matrix representation of {{math|''U''(''N'')}}.
The energy levels of the system are
<math display="block"> E = \hbar \omega \left[(n_1 + \cdots + n_N) + {N\over 2}\right].</math>
<math display="block">n_i = 0, 1, 2, \dots \quad (\text{the energy level in dimension } i).</math>
As in the one-dimensional case, the energy is quantized. The ground state energy is {{math|''N''}} times the one-dimensional ground energy, as we would expect using the analogy to {{math|''N''}} independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In {{math|''N''}}-dimensions, except for the ground state, the energy levels are ''degenerate'', meaning there are several states with the same energy.
The degeneracy can be calculated relatively easily.  As an example, consider the 3-dimensional case: Define {{math|1=''n'' = ''n''<sub>1</sub> + ''n''<sub>2</sub> + ''n''<sub>3</sub>}}. All states with the same {{math|''n''}} will have the same energy.  For a given {{math|''n''}}, we choose a particular {{math|''n''<sub>1</sub>}}. Then {{math|1=''n''<sub>2</sub> + ''n''<sub>3</sub> = ''n'' − ''n''<sub>1</sub>}}. There are {{math|''n'' − ''n''<sub>1</sub> + 1}} possible pairs {{math|{{mset|''n''<sub>2</sub>, ''n''<sub>3</sub>}}}}.  {{math|''n''<sub>2</sub>}} can take on the values {{math|0}} to {{math|''n'' − ''n''<sub>1</sub>}}, and for each {{math|''n''<sub>2</sub>}} the value of {{math|''n''<sub>3</sub>}} is fixed. The degree of degeneracy therefore is:
<math display="block">g_n = \sum_{n_1=0}^n n - n_1 + 1 = \frac{(n+1)(n+2)}{2}</math>
Formula for general {{math|''N''}} and {{math|''n''}} [{{math|''g''<sub>''n''</sub>}} being the dimension of the symmetric irreducible {{math|''n''}}-th power representation of the unitary group {{math|''U''(''N'')}}]:
<math display="block">g_n = \binom{N+n-1}{n}</math>
The special case {{math|''N''}} = 3, given above, follows directly from this general equation.  This is however, only true for distinguishable particles, or one particle in {{math|''N''}} dimensions (as dimensions are distinguishable). For the case of {{math|''N''}} bosons in a one-dimension harmonic trap, the degeneracy scales as the number of ways to partition an integer {{math|''n''}} using integers less than or equal to {{math|''N''}}.


<math display="block">g_n = p(N_{-},n)</math>
===[[Announcer]] call===
{{audio|Needs announcer calls from other languages.}}
<gallery>
Greninja English Announcer SSB4-SSBU.wav|English
Greninja Japanese Announcer SSB4-SSBU.wav|Japanese/Chinese
Greninja Korean Announcer SSB4-SSBU.wav|Korean
Greninja French Announcer SSBU.wav|French
Greninja Russian Announcer SSBU.wav|Russian
</gallery>


This arises due to the constraint of putting {{math|''N''}} quanta into a state ket where <math display="inline">\sum_{k=0}^\infty k n_k = n  </math>  and <math display="inline"> \sum_{k=0}^\infty  n_k = N </math>, which are the same constraints as in integer partition.
===[[On-screen appearance]]===
*Emerges from a [[Poké Ball]], then performs a ninjutsu hand sign that emits a small burst of water from its hands.
<gallery>
GreninjaOnScreenAppearanceSSBU.gif|Greninja's on-screen appearance
</gallery>


===Example: 3D isotropic harmonic oscillator===
===[[Taunt]]s===
[[File:2D_Spherical_Harmonic_Orbitals.png|thumb|300px|right|Schrödinger 3D spherical harmonic orbital solutions in 2D density plots; the [[Mathematica]] source code that used for generating the plots is at the top]]
*'''Up Taunt''': Stands upright, clasping its hands together before assuming a ninjutsu stance. The stance resembles one of its attack animations from the ''Pokémon'' series.
The Schrödinger equation for a particle in a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables; see [[Particle in a spherically symmetric potential#3D isotropic harmonic oscillator|this article]] for the present case. This procedure is analogous to the separation performed in the [[Hydrogen-like atom#Schrödinger equation in a spherically symmetric potential|hydrogen-like atom]] problem, but with a different [[Particle in a spherically symmetric potential|spherically symmetric potential]]
*'''Side Taunt''': Shakes head from side to side, causing its tongue to whip out in the same directions. Particles of saliva fly off with each whip.
<math display="block">V(r) = {1\over 2} \mu \omega^2 r^2,</math>
*'''Down Taunt''': Poses with arms out and palms upward, and summons small sprays of water from them, which deal 1% or 2% damage and produce some knockback, though they're able to KO only at above 420%. A video showing the exact KO percentages at which each character can be KO'd can be found [https://www.youtube.com/watch?v=cH7IDQ_8vic here].
where {{mvar|μ}} is the mass of the particle. Because {{mvar|m}} will be used below for the magnetic quantum number, mass is indicated by  {{mvar|μ}}, instead of {{mvar|m}}, as earlier in this article.
<gallery>
SSBUGreninjaTaunt1.gif|Greninja's up taunt.
SSBUGreninjaTaunt2.gif|Greninja's side taunt.
SSBUGreninjaTaunt3.gif|Greninja's down taunt.
</gallery>


The solution reads<ref>[[Albert Messiah]], ''Quantum Mechanics'', 1967, North-Holland, Ch XII,  § 15, p 456.[https://archive.org/details/QuantumMechanicsVolumeI/page/n239 online]</ref>
===[[Idle Pose]]s===
<math display="block">\psi_{klm}(r,\theta,\phi) = N_{kl} r^{l}e^{-\nu r^2}L_k^{\left(l+{1\over 2}\right)}(2\nu r^2) Y_{lm}(\theta,\phi)</math>
*Crosses arms over its body, then separates them with a flourish.
where
*Hunches over and assumes a ninjutsu stance.
:<math>N_{kl}=\sqrt{\sqrt{\frac{2\nu^3}{\pi }}\frac{2^{k+2l+3}\;k!\;\nu^l}{(2k+2l+1)!!}}~~</math> is a normalization constant; <math>\nu \equiv {\mu \omega \over 2 \hbar}~</math>;
<gallery>
:<math>{L_k}^{(l+{1\over 2})}(2\nu r^2)</math>
SSBUGreninjaIdle1.gif|Greninja's first idle pose.
are [[Laguerre polynomials#Generalized Laguerre polynomials|generalized Laguerre polynomials]]; The order {{mvar|k}}  of the polynomial is a non-negative integer;
SSBUGreninjaIdle2.gif|Greninja's second idle pose.
*<math>Y_{lm}(\theta,\phi)\,</math> is a [[spherical harmonics|spherical harmonic function]];
</gallery>
*{{mvar|ħ}} is the reduced [[Planck constant]]: <math>\hbar\equiv\frac{h}{2\pi}~.</math>


The energy eigenvalue is
===[[Crowd cheer]]===
<math display="block">E=\hbar \omega \left(2k + l + \frac{3}{2}\right) .</math>
<div class="tabber">
The energy is usually described by the single [[quantum number]]
<div class="tabbertab" title="English, Japanese/Chinese, Italian, Dutch, French">
<math display="block">n\equiv 2k+l  \,.</math>
{| class="wikitable" border="1" cellpadding="4" cellspacing="1"
|-
!{{{name|}}}
!Cheer (English)
!Cheer (Japanese/Chinese)
!Cheer (Italian)
!Cheer (Dutch)
!Cheer (French)
|-
! scope="row"|Cheer
|[[File:Greninja Cheer English SSBU.ogg|center]]||[[File:Greninja Cheer Japanese SSBU.ogg|center]]||[[File:Greninja Cheer Italian SSBU.ogg|center]]||[[File:Greninja Cheer Dutch SSBU.ogg|center]]||{{NTSC}} [[File:Greninja Cheer French NTSC SSBU.ogg|center]] <br> {{PAL}} [[File:Greninja Cheer French PAL SSBU.ogg|center]]
|-
! scope="row"|Description
|Gre - ninja! || Ge - kkou -ga! || Gre - nin - ja! || Gre - ninja! || Am - phi - no - bi!
|}
</div>
<div class="tabbertab" title="German, Spanish, Russian, Korean">
{| class="wikitable" border="1" cellpadding="4" cellspacing="1"
|-
!{{{name|}}}
!Cheer (German)
!Cheer (Spanish)
!Cheer (Russian)
!Cheer (Korean)
|-
! scope="row"|Cheer
|[[File:Greninja Cheer German SSBU.ogg|center]]||{{NTSC}} [[File:Greninja Cheer Spanish NTSC SSBU.ogg|center]] <br> {{PAL}} [[File:Greninja Cheer Spanish PAL SSBU.ogg|center]]||[[File:Greninja Cheer Russian SSBU.ogg|center]]||[[File:Greninja Cheer Korean SSBU.ogg|center]]
|-
! scope="row"|Description
|Quaaaaa - jutsu! || Greninja! Greninja! Ya ya ya! || Gre - ninja! || Gae - gul - nin - ja!
|}
</div>
</div>


Because {{mvar|k}} is a non-negative integer, for every even {{mvar|n}} we have {{math|1=''ℓ'' = 0, 2, …, ''n'' − 2, ''n''}} and for every odd {{mvar|n}}  we have {{math|1=''ℓ'' = 1, 3, …, ''n'' − 2, ''n''}} . The magnetic quantum number {{mvar|m}} is an integer satisfying {{math|−''ℓ'' ≤ ''m'' ≤ ''ℓ''}}, so for every {{mvar|n}} and ''ℓ'' there are 2''ℓ''&nbsp;+&nbsp;1 different [[quantum state]]s, labeled by {{mvar|m}} . Thus, the degeneracy at level {{mvar|n}} is
===[[Victory pose]]s===
<math display="block">\sum_{l=\ldots,n-2,n} (2l+1) = {(n+1)(n+2)\over 2} \,,</math>
*'''Left:''' Does a few hand seals with splashing water, and then a ninja pose. It resembles one of its attack animations in [[bulbapedia:Pokémon X and Y|''Pokémon X'' and ''Y'']].
where the sum starts from 0 or 1, according to whether {{mvar|n}} is even or odd.
*'''Up:''' Performs Double Team to briefly create three afterimages of itself.
This result is in accordance with the dimension formula above, and amounts to the dimensionality of a symmetric representation of {{math|SU(3)}},<ref>Fradkin, D. M. "Three-dimensional isotropic harmonic oscillator and SU3." ''American Journal of Physics'' '''33''' (3) (1965) 207–211.</ref> the relevant degeneracy group.
*'''Right:''' Does a flip, lands in a spinning pose, and crosses its arms.
[[File:PokemonSeriesVictoryThemeUltimate.ogg|thumb|A small excerpt of the title theme of ''Pokémon Red, Blue, Yellow, and Green Versions'', a track which would go on to become the ''Pokémon'' main theme and the title theme for the entire series.]]
<gallery>
GreninjaVictoryPose1SSBU.gif
GreninjaVictoryPose2SSBU.gif
GreninjaVictoryPose3SSBU.gif
</gallery>


==Applications==
==In [[competitive play]]==
===Harmonic oscillators lattice: phonons===
===Most historically significant players===
{{see also|Canonical quantization}}
<!--This character has a ten player limit for this section. Before adding and/or removing a player, read these guidelines: https://www.ssbwiki.com/SmashWiki:Notability#%22Most_historically_significant_players%22_guidelines -->


We can extend the notion of a harmonic oscillator to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical ''harmonic chain'' of ''N'' identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how [[phonon]]s arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.
''See also: [[:Category:Greninja players (SSBU)]]''


As in the previous section, we denote the positions of the masses by  {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, …}}, as measured from their equilibrium positions (i.e. {{math|1=''x<sub>i</sub>'' = 0}} if the particle {{mvar|i}} is at its equilibrium position). In two or more dimensions, the {{math|''x<sub>i</sub>''}} are vector quantities. The [[Hamiltonian (quantum mechanics)|Hamiltonian]] for this system is
*{{Sm|iStudying|Netherlands}} - One of the best Greninja players in Europe in the early metagame, with his best results being placing 7th at {{Trn|Tech Republic IV}} defeating {{Sm|quiK}}, placing 9th at {{Trn|Ultimate Fighting Arena 2019}}, and defeating {{Sm|Stroder}} at {{Trn|EVO 2019}}. He remained the second-best European Greninja player in the post-online metagame, most notably placing 9th at {{Trn|Temple: Hermès Edition}} defeating {{Sm|Tilde}}, but became inactive by the end of 2022.
*{{Sm|JW|Canada}} - The second-best Greninja player in North America in the early metagame and the best player from Canada for the first half of 2019. He is best known for defeating {{Sm|MkLeo}} at {{Trn|Pound 2019}}, ultimately placing 9th, and he has also placed 13th at {{Trn|Shine 2019}} defeating {{Sm|Goblin}} and 17th at {{Trn|The Big House 9}} defeating {{Sm|Kola}} and {{Sm|ESAM}}. Although less active in the post-online metagame, he remained Canada's best Greninja player, notably placing 4th at {{Trn|Pinnacle 2021}} defeating {{Sm|Ouch!?}}, before becoming inactive in 2023.
*{{Sm|Lea|Japan}} - The best Greninja player of all-time who spearheaded Greninja's early metagame, ranking 21st on the [[Fall 2019 PGRU]] and placing 5th at several {{Trn|Umebura|series}} majors as well as 7th at {{Trn|2GG: Kongo Saga}} defeating {{Sm|Shuton}} and {{Sm|Dabuz}}. He remained the best Greninja player in the world until his hiatus in mid-2022, notably placing 7th at {{Trn|Kagaribi 7}} during this time, and has sporadically competed since with mixed results.
*{{Sm|Somé|Japan}} - The second-best Greninja player in Japan's early metagame, winning {{Trn|Karisuma SP 4}} and {{Trn|TSC 11}}, and also placing 7th at {{Trn|Sumabato SP 4}} defeating {{Sm|Etsuji}}, 9th at {{Trn|Umebura SP 6}} defeating {{Sm|HIKARU}}, and 13th at {{Trn|Umebura SP 4}}. He took a hiatus from offline competitive play due to the pandemic and did not return until 2023.
*{{Sm|Stroder Ame|USA}} - Plays a variety of characters and primarily played Greninja in the early metagame. He was the second-best Greninja player in the world during this time, ranking 29th on the [[Spring 2019 PGRU]]. He is best known for winning {{Trn|Crown}} over {{Sm|Kurama}} and has also placed 5th at {{Trn|Ultimate Nimbus}} defeating {{Sm|Tweek}}, 5th at {{Trn|Glitch 8 - Missingno}} defeating {{Sm|Cosmos}} and {{Sm|Maister}}, and 13th at {{Trn|2GG: Prime Saga}} defeating {{Sm|ESAM}}. He has since switched to other characters but continues to use Greninja as an occasional secondary.
*{{Sm|Tarik|Germany}} - The second-best Greninja player of all-time and the best Greninja player after Lea's hiatus, ranking 55th on the [[LumiRank 2023]]. He is one of the best players in Europe in the post-online metagame, having placed 7th at several European majors such as {{Trn|Ultimate Fighting Arena 2023}} and {{Trn|King Con}} and having multiple wins over {{Sm|Bloom4Eva}} at ranked events. Internationally, he is known for placing 5th at {{Trn|Rise 'N Grind 2023}} defeating {{Sm|SHADIC}} and defeating {{Sm|Shuton}} at {{Trn|Port Priority 8}}, {{Sm|Onin}} at {{Trn|GENESIS X}}, and {{Sm|ShinyMark}} at {{Trn|Battle of BC 5}}.
*{{Sm|Tsuna|Japan}} - A known Greninja player in Japan since the early metagame, having placing 2nd at {{Trn|Karisuma SP 6}}, 5th at {{Trn|Sumabato SP 10}}, and 9th at {{Trn|Sumabato SP 11}} during this time. He has since become the best Greninja player in Japan following Lea's hiatus, frequently placing at or near the top 8 at {{Trn|Sumabato}} events, including 5th at {{Trn|Sumabato SP 46}} over {{Sm|Asimo}} and {{Sm|Jogibu}}.


<math display="block">\mathbf{H} = \sum_{i=1}^N {p_i^2 \over 2m} + {1\over 2} m \omega^2 \sum_{\{ij\} (nn)} (x_i - x_j)^2 \,,</math>
===Tier placement and history===
where {{mvar|m}} is the (assumed uniform) mass of each atom, and  {{math|''x<sub>i</sub>''}} and {{math|''p<sub>i</sub>''}} are the position and [[momentum]] operators for the ''i'' th atom and the sum is made over the nearest neighbors (nn).  However, it is customary to rewrite the Hamiltonian in terms of the [[normal modes]] of the [[wavevector]] rather than in terms of the particle coordinates so that one can work in the more convenient [[Fourier space]].
In the early metagame, players quickly noticed that Greninja had been buffed from ''Smash 4'', with improved versatility and speed, and despite losing its [[footstool]] combos, it gained a stronger combo game thanks to improved frame data on moves such as its dash attack, up throw, down throw, and neutral air. Greninja's high learning curve did not deter players such as {{Sm|Stroder}}, {{Sm|Jw}}, and {{Sm|Lea}} from seeing success with the character, and Greninja became a popular pick in competitive play, culminating in the character peaking at 10th in terms of representation between July and December 2019. Due to Greninja's buffs and its strong representation, it was considered an upper-high tier character in the early metagame.


We introduce, then, a set of {{mvar|N}} "normal coordinates" {{math|''Q<sub>k</sub>''}}, defined as the [[discrete Fourier transform]]s of the {{mvar|x}}s, and {{mvar|N}} "conjugate momenta"  {{mvar|Π}} defined as the Fourier transforms of the {{mvar|p}}s,
This perception shifted following the end of the online metagame, especially as Greninja players began pointing out frustrating traits in the character's moveset, most notably inconsistencies in Greninja's multihit moves. Furthermore, Greninja's best players from before the pandemic either became less active, changed mains, or had worse performance: for example, Lea would go on hiatus in 2022, with mixed results in the events he has attended since, while Stroder would begin playing other characters over Greninja. Although Greninja still saw strong results from players such as {{Sm|Tarik}}, Greninja's overall representation would drop, bottoming out at 45th between January and June 2022. Greninja's lower representation and more apparent flaws led to a colder reception than in the early metagame, and as such, Greninja was ranked 33rd on the first tier list, towards the lower end of high tier.  
<math display="block">Q_k = {1\over\sqrt{N}} \sum_{l} e^{ikal} x_l</math>
<math display="block">\Pi_{k} = {1\over\sqrt{N}} \sum_{l}  e^{-ikal} p_l \,.</math>


The quantity {{math|''k<sub>n</sub>''}} will turn out to be the [[Wavenumber|wave number]] of the phonon, i.e. 2''π''  divided by the [[wavelength]]. It takes on quantized values, because the number of atoms is finite.
Following the release of the first list, Greninja's results would noticeably improve, thanks to Tarik's results remaining strong and the rise of many other Greninja players, including {{Sm|Tsuna}} and {{Sm|Anarchy}}, in their respective scenes. Continued efforts from dedicated mains prevented Greninja from dropping further, and he only slid down a spot to 34th on the second list, ranking towards the middle of B+ tier, with this drop mainly being the result of the meteoric rise of {{SSBU|Bayonetta}} and {{SSBU|Corrin}} from the first tier list. Overall, although less common than the early metagame, Greninja remains a rather viable pick in the post-online metagame.


This preserves the desired commutation relations in either real space or wave vector space
=={{SSBU|Classic Mode}}: Your Turn, Greninja!==
<math display="block"> \begin{align}  
[[File:SSBU Congratulations Greninja.png|thumb|Greninja's congratulations screen.]]
\left[x_l , p_m \right]&=i\hbar\delta_{l,m} \\
Greninja battles characters that represent different {{iw|bulbapedia|type}}s from the ''Pokémon'' games: for example, Charizard and Bowser represent the Fire type, while Mewtwo, Ness and Lucas represent the Psychic type. Interestingly, the types represented correspond to those initially available in the {{iw|bulbapedia|Pokémon Trading Card Game}}.
\left[ Q_k , \Pi_{k'} \right] &={1\over N} \sum_{l,m} e^{ikal} e^{-ik'am} [x_l , p_m ] \\
&= {i \hbar\over N} \sum_{m} e^{iam(k-k')} = i\hbar\delta_{k,k'} \\
\left[ Q_k , Q_{k'} \right] &= \left[ \Pi_k , \Pi_{k'} \right] = 0 ~.
\end{align}</math>


From the general result
{|class="wikitable" style="text-align:center"
<math display="block"> \begin{align}
!Round!!Opponent!!Stage!!Music!!Notes
\sum_{l}x_l x_{l+m}&={1\over N}\sum_{kk'}Q_k Q_{k'}\sum_{l} e^{ial\left(k+k'\right)}e^{iamk'}= \sum_{k}Q_k Q_{-k}e^{iamk} \\
|-
\sum_{l}{p_l}^2 &= \sum_{k}\Pi_k \Pi_{-k}   ~,
|1||{{CharHead|Charizard|SSBU|hsize=20px}} and {{CharHead|Bowser|SSBU|hsize=20px}}||[[Pokémon Stadium]]||''{{SSBUMusicLink|Pokémon|Battle! (Elite Four) / Battle! (Solgaleo/Lunala)}}''||Represents {{iw|bulbapedia|Fire}}-type. Charizard's {{SSBU|Pokémon Trainer}} is absent.
\end{align}</math>
|-
it is easy to show, through elementary trigonometry, that the potential energy term is
|2||{{CharHead|Pikachu|SSBU|hsize=20px}}, {{CharHead|Pichu|SSBU|hsize=20px}}, and {{CharHead|Zero Suit Samus|SSBU|hsize=20px}}||[[Pokémon Stadium 2]]||''{{SSBUMusicLink|Pokémon|Battle! (Steven)}}''||Represents {{iw|bulbapedia|Electric}}-type.
<math display="block"> {1\over 2} m \omega^2 \sum_{j} (x_j - x_{j+1})^2= {1\over 2} m \omega^2\sum_{k}Q_k Q_{-k}(2-e^{ika}-e^{-ika})= {1\over 2} m \sum_{k}{\omega_k}^2Q_k Q_{-k} ~ ,</math>
|-
where
|3||{{CharHead|Lucario|SSBU|hsize=20px}}, {{CharHead|Ryu|SSBU|hsize=20px}}, and {{CharHead|Ken|SSBU|hsize=20px}}||Pokémon Stadium||''{{SSBUMusicLink|Pokémon|Battle! (Reshiram / Zekrom)}}''||Represents {{iw|bulbapedia|Fighting}}-type.
<math display="block">\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))} ~.</math>
|-
|4||{{CharHead|Ivysaur|SSBU|hsize=20px}}||Pokémon Stadium||''{{SSBUMusicLink|Pokémon|Battle! (Gladion)}}''||Represents {{iw|bulbapedia|Grass}}-type. Ivysaur's Pokémon Trainer is absent.
|-
|5||{{CharHead|Mewtwo|SSBU|hsize=20px}}, {{CharHead|Ness|SSBU|hsize=20px}}, and {{CharHead|Lucas|SSBU|hsize=20px}}||Pokémon Stadium 2||''{{SSBUMusicLink|Pokémon|Battle! (Dialga/Palkia) / Spear Pillar}}''||Represents {{iw|bulbapedia|Psychic}}-type.
|-
|6||{{CharHead|Squirtle|SSBU|hsize=20px}} and {{CharHead|Greninja|SSBU|hsize=20px|color=Black}}||[[Kalos Pokémon League]]||''{{SSBUMusicLink|Pokémon|Battle! (Champion) - Pokémon X / Pokémon Y}}''||Represents {{iw|bulbapedia|Water}}-type. Squirtle's Pokémon Trainer is absent.  The CPU will be the {{Head|Greninja|g=SSBU|s=20px}} default Greninja if the player chooses the black costume.
|-
|colspan="5"|[[Bonus Stage]]
|-
|Final||{{SSBU|Master Hand}}||{{SSBU|Final Destination}}||''{{SSBUMusicLink|Super Smash Bros.|Master Hand}}'' <small>(Less than 7.0 intensity)</small><br>''{{SSBUMusicLink|Super Smash Bros.|Master Hand / Crazy Hand}}'' <small>(Intensity 7.0 or higher)</small>||On intensity 7.0 and higher, {{SSBU|Crazy Hand}} fights alongside Master Hand.
|}


The Hamiltonian may be written in wave vector space as
Note: All rounds except the sixth round take place on Pokémon Stadium and Pokémon Stadium 2. If applicable, each stage will also shift to their appropriately-typed form at the earliest possible opportunity. (The stages remain in their default form in rounds 3 and 5, as none of the stages have Psychic or Fighting-themed forms.)
<math display="block">\mathbf{H} = {1\over {2m}}\sum_k \left(
{ \Pi_k\Pi_{-k} } + m^2 \omega_k^2 Q_k Q_{-k}
\right) ~.</math>


Note that the couplings between the position variables have been transformed away; if the {{mvar|Q}}s and {{mvar| Π}}s were [[Hermitian operator|hermitian]] (which they are not), the transformed Hamiltonian would describe {{mvar|N}} ''uncoupled'' harmonic oscillators.
[[Credits]] roll after completing Classic Mode. Completing it as Greninja has ''{{SSBUMusicLink|Pokémon|Battle! (Trainer Battle) - Pokémon X / Pokémon Y}}'' accompany the credits.
{{clr}}


The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose ''periodic'' boundary conditions, defining the {{math|(''N'' + 1)}}-th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
==Role in [[World of Light]]==
[[File:WoL-50Greninja.jpg|thumb|Finding Greninja in World of Light|left]]
Greninja was among the fighters that were summoned to fight the army of [[Master Hand]]s.


<math display="block">k=k_n = {2n\pi \over Na}
During the opening cutscene, Greninja was present on the cliffside when [[Galeem]] unleashed his beams of light. Greninja leaped into the air to avoid one of the beams, which hit {{SSBU|Lucario}} instead. Greninja was hit shortly after and vaporized, getting imprisoned by Galeem afterward along with the other fighters, sans {{SSBU|Kirby}}. A puppet fighter cloned from Greninja is later seen alongside ones cloned from {{SSBU|Fox}}, {{SSBU|Samus}}, {{SSBU|Link}} and other fighters.
\quad \hbox{for}\ n = 0, \pm1, \pm2, \ldots , \pm {N \over 2}. </math>


The upper bound to {{mvar|n}} comes from the minimum wavelength, which is twice the lattice spacing {{mvar|a}}, as discussed above.
Greninja was one of the many fighters that fell under [[Dharkon]]'s control upon Galeem's first defeat, and it can be found in the [[Mysterious Dimension]] at [[The Dark Realm]]. It can be seen impeding the path, making it an obligatory unlock.


The harmonic oscillator eigenvalues or energy levels for the mode {{math|''ω<sub>k</sub>''}} are
Greninja is later seen among several other fighters, making their last stand against Galeem and Dharkon. It also shows up in the bad ending where Galeem emerges victorious against Dharkon, witnessing Galeem engulf the world in light.
<math display="block">E_n = \left({1\over2}+n\right)\hbar\omega_k  \quad\hbox{for}\quad n=0,1,2,3,\ldots</math>
{{clrl}}


If we ignore the [[zero-point energy]] then the levels are evenly spaced at
===Fighter Battle===
<math display="block">\hbar\omega,\, 2\hbar\omega,\, 3\hbar\omega,\, \ldots  </math>
{|class="wikitable" style="width:100%;"
|-
!style="width:5%;"|No.
!style="width:5%;"|Image
!Name
!Type
!Power
!Stage
!Music
|-
|50
|[[File:Greninja SSBU.png|center|64x64px]]
|Greninja
|{{SpiritType|Shield}} <center>{{color|#18aef5|Shield}}</center>
|10,600
|[[Kalos Pokémon League]] ([[Ω form]])
|''{{SSBUMusicLink|Pokémon|Battle! (Trainer Battle) - Pokémon X / Pokémon Y}}''
|}
{{clr}}


So an '''exact''' amount of [[energy]] {{math|''ħω''}}, must be supplied to the harmonic oscillator lattice to push it to the next energy level. In analogy to the [[photon]] case when the [[electromagnetic field]] is quantised, the quantum of vibrational energy is called a [[phonon]].
==[[Spirit]]==
Greninja's fighter spirit can be obtained by completing {{SSBU|Classic Mode}}. It is also available periodically for purchase in the shop for 300 Gold, but only after Greninja has been unlocked. Unlocking Greninja in World of Light allows the player to preview the spirit below in the Spirit List under the name "???". As a fighter spirit, it cannot be used in Spirit Battles and is purely aesthetic. Its fighter spirit has an alternate version that replaces it with its artwork in ''Ultimate''.


All quantum systems show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods of [[second quantization]] and operator techniques described elsewhere.<ref name="Mahan">{{cite book| last=Mahan |first=GD |title=Many particle physics|publisher= Springer|location=New York | isbn=978-0306463389 |year=1981}}</ref>
<center>
<gallery>
SSBU spirit Greninja.png|418. '''''Greninja'''''
</gallery>
</center>


In the continuum limit, ''a''→0, ''N''→∞, while ''Na'' is held fixed. The canonical coordinates ''Q<sub>k</sub>'' devolve to the decoupled momentum modes of a scalar field, <math>\phi_k</math>, whilst the location index {{mvar|i}} (''not the displacement dynamical variable'') becomes the ''parameter {{mvar|x}} argument of the scalar field, <math>\phi (x,t)</math>.
==In Spirit Battles==
===As the main opponent===
{|class="wikitable sortable" style="width:100%;"
! colspan=4|Spirit
! colspan=7|Battle parameters
! colspan=1|Inspiration
|-
! style="width:5%;"|No.
! style="width:5%;"|Image
! Name
! Series
! Enemy Fighter(s)
! style="width:5%;"|Type
! style="width:5%;"|Power
! Stage
! Rules
! Conditions
! Music
! Character
|-
|154
|{{SpiritTableName|Winky|size=64}}
|''Donkey Kong'' Series
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Green}}
|{{SpiritType|Attack}}
|1,700
|[[Mushroom Kingdom U]]
|N/A
|•The enemy deals damage when falling<br>•The enemy has increased jump power
|{{SSBUMusicLink|Donkey Kong|Jungle Level (Brawl)}}
|
|-
|199
|{{SpiritTableName|Zora|size=64}}
|''The Legend of Zelda'' Series
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Violet}}
|{{SpiritType|Shield}}
|1,800
|[[Great Bay]]
|N/A
|•The enemy's neutral special has increased power
|{{SSBUMusicLink|The Legend of Zelda|Ocarina of Time Medley}}
|
|-
|385
|{{SpiritTableName|Slippy Toad|link=y|size=64}}
|''Star Fox'' Series
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Green}}<br>•{{SSBU|Fox}} {{Head|Fox|g=SSBU|s=20px|cl=Black}}
|{{SpiritType|Shield}}
|9,600
|[[Frigate Orpheon]] (hazards off)
|N/A
|•Defeat the main fighter to win<br>•Timed battle (1:30)<br>•The enemy tends to avoid conflict
|{{SSBUMusicLink|Star Fox|Corneria - Star Fox}}
|
|-
|482
|{{SpiritTableName|Raikou, Entei, & Suicune|customname=[[Raikou]], [[Entei]], & [[Suicune]]|size=64}}
|''Pokémon'' Series
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Violet}}<br>•{{SSBU|Incineroar}} {{Head|Incineroar|g=SSBU|s=20px|cl=White}}<br>•{{SSBU|Pikachu}} {{Head|Pikachu|g=SSBU|s=20px|cl=Libre}}
|{{SpiritType|Shield}}
|9,900
|[[Suzaku Castle]]
|•Hazard: Lava Floor
|•The floor is lava
|{{SSBUMusicLink|Pokémon|Pokémon Red / Pokémon Blue Medley}}
|Suicune
|-
|516
|{{SpiritTableName|Darkrai|link=y|size=64}}
|''Pokémon'' Series
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Black}}
|{{SpiritType|Shield}}
|9,900
|[[Luigi's Mansion]] ([[Ω form]])
|•Item: [[Black Hole]]<br>•Hazard: Slumber Floor
|•The floor is sleep-inducing<br>•Only certain Pokémon will emerge from Poké Balls (Darkrai)
|{{SSBUMusicLink|Pokémon|Battle! (Team Galactic)}}
|
|-
|770
|{{SpiritTableName|Metal Gear RAY|size=64}}
|''Metal Gear Solid'' Series
|•Metal {{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Grey}} (140 HP)
|{{SpiritType|Grab}}
|4,200
|[[Shadow Moses Island]]
|•Item: Exploding Types
|•[[Stamina battle]]<br>•Explosion attacks aren't as effective against the enemy<br>•The enemy is metal
|{{SSBUMusicLink|Metal Gear|Yell "Dead Cell"}}
|
|-
|893
|{{SpiritTableName|Shadow Man|size=64}}
|''Mega Man'' Series
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Purple}}×3 (60 HP)
|{{SpiritType|Shield}}
|3,500
|[[Norfair]] ([[Battlefield form]])
|N/A
|•The enemy's neutral special has increased power<br>•[[Stamina battle]]<br>•The enemy favors neutral specials
|{{SSBUMusicLink|Mega Man|Shadow Man Stage}}
|
|-
|1,014
|{{SpiritTableName|Luka|size=64}}
|''Bayonetta'' Series
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Black}}
|{{SpiritType|Shield}}
|3,800
|[[New Donk City Hall]]
|•Temporary Invincibility
|•The enemy becomes temporarily invincible when badly damaged
|{{SSBUMusicLink|Bayonetta|Riders Of The Light}}
|
|-
|1,048
|{{SpiritTableName|Octoling Octopus|size=64}}
|''Splatoon'' Series
|•{{SSBU|Greninja}} Team {{Head|Greninja|g=SSBU|s=20px|cl=Pink}}×4
|{{SpiritType|Shield}}
|3,900
|[[Moray Towers]]
|N/A
|•Timed battle (2:00)
|{{SSBUMusicLink|Splatoon|Octoweaponry}}
|
|-
|1,143
|{{SpiritTableName|Frog & Snake|customname=[[Sablé Prince|Frog & Snake]]|size=64}}
|''Kaeru no Tame ni Kane wa Naru''
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Green}}<br>•{{SSBU|King K. Rool}} {{Head|King K. Rool|g=SSBU|s=20px|cl=Blue}}
|{{SpiritType|Shield}}
|3,600
|[[Dream Land GB]] (Castle Lololo interior)
|•Assist Trophy Enemies ([[Sablé Prince]])
|•Hostile assist trophies will appear
|{{SSBUMusicLink|Kirby|Kirby Retro Medley}} (Castle Lololo)
|Frog
|-
|rowspan="2"|1,291
|{{SpiritTableName|Ninjara|link=y|size=64|dlcalt=y}}
|rowspan="2"|''ARMS''
|•{{SSBU|Greninja}} {{Head|Greninja|g=SSBU|s=20px|cl=Green}}
|rowspan="2"|{{SpiritType|Grab}}
|rowspan="2"|3,600
|rowspan="2"|[[Suzaku Castle]]
|rowspan="2"|•Item: [[Boomerang]]
|rowspan="2"|•The enemy has increased move speed
|rowspan="2"|{{SSBUMusicLink|ARMS|Ninja College}}
|rowspan="2"|
|-
|style="background-color:#EEE;"|•{{SSBU|Mii Brawler}} {{Head|Mii Brawler|g=SSBU|s=20px}} (Moveset [[Flashing Mach Punch|2]][[Suplex|3]][[Soaring Axe Kick|1]][[Counter Throw|3]], Ninjara Wig, Ninjara Outfit)<ref group="SB" name="DLC"/>
|}


===Molecular vibrations===
<references group="SB">
{{main|Molecular vibration}}
<ref name="DLC">This alternative occurs when the corresponding DLC has been purchased and downloaded.</ref>
* The vibrations of a [[diatomic molecule]] are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by <math display="block">\omega = \sqrt{\frac{k}{\mu}} </math> where <math>\mu = \frac{m_1 m_2}{m_1 + m_2}</math> is the [[reduced mass]] and <math>m_1</math> and <math>m_2</math> are the masses of the two atoms.<ref>{{Cite web | title=Quantum Harmonic Oscillator | website=Hyperphysics | access-date=24 September 2009 | url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html}}</ref>
</references>
* The [[Hooke's atom]] is a simple model of the [[helium]] atom using the quantum harmonic oscillator.
* Modelling phonons, as discussed above.
* A charge <math>q</math> with mass <math>m</math> in a uniform magnetic field <math>\mathbf{B}</math> is an example of a one-dimensional quantum harmonic oscillator: [[Landau quantization]].


==See also==
==[[Alternate costume (SSBU)#Greninja|Alternate costumes]]==
{{Div col}}
{|style="margin:1em auto 1em auto;text-align:center"
*[[Quantum pendulum]]
|-
*[[Quantum machine]]
|colspan=8|[[File:Greninja Palette (SSBU).png|link=Alternate costume (SSBU)#Greninja|1000px]]
*[[Gas in a harmonic trap]]
|-
*[[Creation and annihilation operators]]
|{{Head|Greninja|g=SSBU|s=50px}}
*[[Coherent state]]
|{{Head|Greninja|g=SSBU|s=50px|cl=Red}}
*[[Morse potential]]
|{{Head|Greninja|g=SSBU|s=50px|cl=Pink}}
*[[Bertrand's theorem]]
|{{Head|Greninja|g=SSBU|s=50px|cl=Black}}
*[[Mehler kernel]]
|{{Head|Greninja|g=SSBU|s=50px|cl=Violet}}
*[[Molecular vibration#Quantum mechanics|Molecular vibration]]
|{{Head|Greninja|g=SSBU|s=50px|cl=Green}}
{{Div col end}}
|{{Head|Greninja|g=SSBU|s=50px|cl=Grey}}
|{{Head|Greninja|g=SSBU|s=50px|cl=Purple}}
|}


==References==
==Gallery==
{{Reflist}}
<gallery>
Pokémon Smash Bros.png|Artwork of all playable Pokémon characters and Poké Ball Pokémon, as posted by the official Pokémon Twitter account.
SSBU Greninja Number.png|Greninja's fighter card.
Greninja unlock notice SSBU.jpg|Greninja's unlock notice.
SSBUWebsiteGreninja1.jpg|Greninja coming to a halt on [[Kalos Pokémon League]].
SSBUWebsiteGreninja2.jpg|Charging [[Water Shuriken]] next to {{SSBU|Ryu}} on [[Suzaku Castle]].
SSBUWebsiteGreninja3.jpg|Jumping on {{SSBU|Battlefield}}.
SSBUWebsiteGreninja4.jpg|Performing its neutral aerial with [[Leviathan]] on [[Midgar]].
SSBUWebsiteGreninja5.jpg|With its [[Substitute|Substitute Doll]] on [[Wuhu Island]] after [[tripping]].
SSBUWebsiteGreninja6.jpg|Performing [[Shadow Sneak]] on [[Luigi's Mansion]].
SSBUWebsiteLucario3.jpg|Struck by {{SSBU|Lucario}} on [[Pokémon Stadium 2]].
The Show's Over.png|Greninja being hit with Joker's [[All-Out Attack]].
</gallery>


==External links==
===Fighter Showcase Video===
*[http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html Quantum Harmonic Oscillator]
{{#widget:YouTube|id=rMCn8NuATaE}}
*[http://behindtheguesses.blogspot.com/2009/03/quantum-harmonic-oscillator-ladder.html Rationale for choosing the ladder operators]
*[http://www.brummerblogs.com/curvature/3d-harmonic-oscillator-eigenfunctions/  Live 3D intensity plots of quantum harmonic oscillator]
*[http://ltl.tkk.fi/~ethuneb/courses/monqo.pdf Driven and damped quantum harmonic oscillator (lecture notes of course "quantum optics in electric circuits")]


{{Use dmy dates|date=August 2019}}
==Trivia==
*In the ''Pokémon'' series, Ash-Greninja is only obtainable as a male. The fact that Greninja transforms into Ash-Greninja for its [[Final Smash]], [[Secret Ninja Attack]], implies that Greninja is a male in ''Ultimate''.
*Greninja's new character portrait resembles its [[air dodge]] animation.
**It also resembles {{SSB4|Fox}}'s character portrait from ''Super Smash Bros. 4'' but with the arm and leg positions mirrored.
*Greninja's down taunt has a unique property where it will seemingly be absorbed by counters without triggering the counterattack, even against grounded opponents despite the grounded hitboxes being standard hitboxes. This is because despite the grounded hitboxes being standard hitboxes that are counterable like any other hitbox, the aerial hitboxes actually take priority over the grounded hitboxes, and the aerial hitboxes are windboxes, which cannot trigger counters but will still be negated by them. Since counters respond to both grounded and aerial hitboxes, the game treats this interaction as a windbox connecting with a counter and completely ignores the regular hitboxes that hit grounded opponents.
**As Greninja's down taunt is the only attack in the game with direct windboxes that trigger hitlag for the user, this gives the unique appearance of the attack striking the counter but being negated.
*Greninja's fighter number, 50, is the same as the number of its [[mariowiki:Costume Mario|costume]] in ''Super Mario Maker''. It shares this distinction with {{SSBU|Inkling}}.
*Greninja, {{SSBU|Ivysaur}}, {{SSBU|Olimar}}, {{SSBU|Little Mac}}, {{SSBU|Ryu}} and {{SSBU|Ken}} are the only base game characters to never appear as minions in any Spirit battles.
*Alongside {{SSBU|Luigi}}, Greninja is one of two characters in ''Ultimate'' with a taunt that cannot be canceled, due to the fact that their non-cancelable taunts have hitboxes.
**Strangely, this does not apply for {{SSBU|Snake}} and {{SSBU|Kazuya}}'s taunts that have damaging hitboxes
**Greninja and Luigi are also the only two characters whose Classic Mode titles feature their names.
*Greninja can also be unlocked immediately after clearing Classic Mode as Sheik, referencing their ninja-like traits and movements.
*Greninja appears slightly tilted in its [[damage meter]] compared to its character artwork. This distinction is shared with fellow ''Pokémon'' series character {{SSBU|Incineroar}}.
**Coincidentally, both are final evolutions of starter Pokémon and both have Dark as their secondary type.
**Both are also found and unlocked in the Dark Realm in World of Light.
*Incineroar and {{SSBU|Jigglypuff}} are the only Pokémon that are not encountered in Greninja's Classic Mode route.
*Oddly, Greninja does not vanish when performing a directional [[air dodge]] despite the sound effects playing. It shares this oddity with {{SSBU|Rosalina & Luma}} and {{SSBU|Palutena}}.
*In ''Ultimate'', Greninja has a weight of 88, which almost matches its weight in ''Pokémon'' (in pounds), being 88.2 lbs.


{{DEFAULTSORT:Quantum Harmonic Oscillator}}
{{SSBUCharacters}}
[[Category:Quantum models]]
{{Pokémon universe}}
[[Category:Quantum mechanics]]
[[Category:Greninja (SSBU)| ]]
[[Category:Oscillators]]
[[Category:Pokémon (SSBU)]]
[[Category:Spirits]]
[[es:Greninja (SSBU)]]

Latest revision as of 01:43, May 29, 2024

This article is about Greninja's appearance in Super Smash Bros. Ultimate. For the character in other contexts, see Greninja.
Greninja
in Super Smash Bros. Ultimate
Greninja SSBU.png
PokemonSymbol.svg
Universe Pokémon
Other playable appearance in SSB4


Availability Unlockable
Final Smash Secret Ninja Attack
Tier B+ (34)
GreninjaHeadSSBU.png

Greninja (ゲッコウガ, Gekkouga) is a playable character in Super Smash Bros. Ultimate. It was officially revealed on June 12th, 2018 alongside Mr. Game & Watch and the rest of the returning roster. Greninja is classified as Fighter #50.

Billy Bob Thompson, Yūji Ueda, Frédéric Clou and Benedikt Gutjan's portrayals of Greninja from Super Smash Bros. 4 were repurposed for the English, Japanese, French and German versions of Ultimate, respectively.

Greninja is ranked 34th out of 82 on the current tier list, placing it in the B+ tier. This is a marginal rise from its 26th out of 54 placement in SSB4.

How to unlockEdit

Complete one of the following:

Greninja must then be defeated on Kalos Pokémon League (the Ω form is used in World of Light).

AttributesEdit

Greninja, true to being a ninja-themed character, has very strong mobility; it has the 4th fastest walking speed, the 8th fastest run speed, the 10th fastest air speed, is tied for the 9th fastest falling and fast falling speed, the 2nd highest gravity, and possesses the 2nd highest jump height overall. However, unlike most characters who boast similar mobility (such as Sheik), Greninja boasts a surprising amount of KO options, good range on plenty of its attacks, and KO throws.

One of Greninja's most notable traits is its high mobility which complements its grounded moveset. Greninja's dash attack comes out on frame 7 and has very low ending lag, as well as the ability to cross upon shields. Its knockback angle allows for many true follow-ups and strings over a large range of percents. Its neutral jab attack comes out on frame 3, making it a good grounded combo breaker. It can also lock, which gives Greninja access to potent punishes from opponents missing techs. Its down tilt is an excellent combo starter due to its low startup, ending lag and vertical launch angle. Greninja's up tilt is a frame 9 disjointed hitbox that acts well as an anti-air and can also be a combo starter. Its smash attacks are also reliable in their own right; its forward smash is quick for its range and power, down smash is an excellent punishment option for ledge regrabs, as well as sending at a low angle, and up smash is a potent combo finisher.

Greninja also has a very strong air game due to its aforementioned air speed and jump height. Greninja's aerials are reliable for multiple situations and all have low landing lag (except for down aerial, at 30 frames). Its neutral aerial is a decent low percent combo starter due to it having incredibly low landing lag and a good launch angle. It can also KO at high percentages. Its forward aerial acts as a combo finisher from its combo starters and can KO moderately early. Forward aerial's low landing lag and disjointed nature also make it safe on shield in many situations when spaced correctly. Its up aerial is a great juggling option with low all-around lag and boasting good KO potential near the upper blast line. Greninja can also utilize its multihits to drag down opponents to create tech chase and jab lock situations. Its back aerial is a very fast follow up or offstage edgeguarding option. Down aerial can be used as a mix up to return to the stage from far above, as well as perform surprise combos on hit with both its meteor smash and sourspot hitbox.

Greninja's grab game is overall very effective, due to its grabs being among the longest ranges of any non-tether grab in the game. Its forward, up, and back throws can KO at high percentages. Down throw acts as a middle percent combo starter, as well as a strong DI mix up, especially at higher percentages at ledge as a 50/50 between DI in and out in conjunction with forward throw. Up throw acts as a versatile combo starter that can lead to juggling situations. Because of this, Greninja has plenty of options off of a grab, as not one of its throws could be considered useless.

Finally, Greninja's special moves are effective in various situations. Water Shuriken acts as a versatile zoning tool, as well as a high-percentage KO option when fully charged. At low-to-mid percents, it is also a combo starter, allowing Greninja to rush down its opponent and follow up with any aerial attack or an up smash. Shadow Sneak works as an effective recovery mix up, as well as a potent KO move from a good read or pseudo-combo finisher. Despite lacking an offensive hitbox, Hydro Pump is a good recovery move for its long distance, and can be used for gimping recoveries due to having windbox properties. Substitute is a counterattack with the unique attribute of being able to be aimed in one of 8 different directions upon a successful counter. These angled follow ups allow for Greninja to gain pseudo-follow ups as well as KO earlier by picking the optimal angle in regards to stage positioning.

However, Greninja is also flawed in many ways. One of Greninja's primary flaws is its inability to break out of disadvantage state. While not as bad as the previous game, Greninja still has difficulties escaping combos due to its fast falling speed and its aerials still having relatively high startup. This can sometimes be alleviated with its back or down aerials, but both are not very effective due to back aerial's almost entirely horizontal hitboxes and down aerial's landing lag. Another option is aerial Water Shuriken, which stalls Greninja in the air and can be used as a landing mix up, as well as Hydro Pump landing mix ups.

Another notable weaknesses is Greninja's multihit moves, especially back air and up air, being notoriously inconsistent, allowing opponents to escape out of these moves. As both aerials are some of Greninja's best combo tools, the inconsistency greatly affects its combo game.

Greninja's biggest weakness, however, is its terrible out of shield game, which is arguably the worst of the entire cast. Because of its high short hop, its aerials slow startup, and lacking a fast grab (although it has good range), Greninja lacks an effective out of shield option faster than frame 14. While its back aerial is fairly quick at frame 5 (making it frame 8 out of shield), it is unable to hit opponents in front of Greninja and is very inconsistent at hitting opponents behind Greninja due to its high short hop. Jumping or rolling out of shield are potential options to reset neutral, but they are very predictable and easily read. Thus, when Greninja is pinned down in shield, it has difficulty escaping the situation without being heavily punished. Combined with its vulnerability to combos, this gives it an atrocious defensive game.

Altogether, Greninja's playstyle requires players to think like an actual ninja: utilizing Greninja's superb mobility and fast attacks to rush down opponents, saving the slower attacks for potential mixups, mindgames and surprise KO options, and remaining unpredictable to prevent being trapped into disadvantageous positions.

Changes from Super Smash Bros. 4Edit

Greninja has been greatly buffed in the transition from SSB4 to Ultimate. An assortment of its traits, such as its strong combo game and great mobility, that made up its playstyle have been further improved, while the general engine changes benefit said playstyle.

Most of the universal changes notably benefit Greninja. As with all other characters in the game, Greninja's already quick mobility is faster like most characters, which benefits its hit-and-run playstyle. The ability to run cancel into any ground move allows Greninja to further exploit its amazing ground mobility, allowing for easier setups into its combo starters, such as up and down tilts and dash attack. The universally reduced landing lag on aerial attacks gives Greninja an easier time landing and comboing, while the universal 3-frame jumpsquat improves Greninja's ground-to-air potential. The implementation of spot dodge canceling and the changes to perfect shielding improve its potential punish game, due to its wide variety of combo starters and fast frame data.

Aside from the universal changes, Greninja has also received notable direct buffs. The biggest ones were to its grab game; its standing grab is faster and its pummel, previously one of the worst in SSB4, deals less damage but is significantly faster, making it better all-around. Greninja's forward throw has higher knockback, allowing it to KO in an emergency, as with up throw. Its up and down throws also have better combo and juggling potential due to the universal changes to mobility — down throw notably now allows for potential KO confirms into forward and back aerials Outside of its improved grab game, Greninja now has a new down tilt that has lower ending lag and sends at more favorable angles, and its dash attack sends at a higher angle, further improving Greninja's combo game. Water Shuriken has more range, improving Greninja's camping ability combined with its faster mobility. Substitute now slows opponents down and offers Greninja intangibility during all of its attack variations, bringing it in line with other counterattacks. Greninja's KO power has also been buffed, with forward smash and forward aerial receiving higher knockback, up smash connecting better into its second hit, and down smash having faster startup.

On the other hand, Greninja is not without a few nerfs. Notably, the ability to tech footstools has made footstool combos much harder to pull off, which hinders Greninja's combo ability (specifically from its down aerial), though this is alleviated by the various buffs to Greninja's moveset. Substitute's attack variants are all weaker while also being more laggy overall, which compensates for the attack's new intangibility. In exchange for its buffed mobility, Greninja is now lighter, which brings it slightly more in-line with other combo-centric and/or hit-and-run characters, while not compensating much for its vulnerability to combos.

In terms of universal changes, some of them are as benefitial as they are detrimental to Greninja. The changes to air dodge mechanics slightly improve its previously below average edgeguarding game, but also make it more vulnerable to combos. The increased shieldstun makes Greninja's own attacks much safer on shield, but also weakens its already poor out of shield game, forcing it to abuse the revamped perfect shielding. Lastly, the weakening of rage makes it slightly harder for Greninja to take advantage of it, despite compensating for its lighter weight.

Collectively, Greninja's strengths and weaknesses are more defined. It now performs much better as a hit-and-run, combo-centric character, with excellent mobility and various KO options and set-ups at a variety of percentages, but with its light weight and attributes making it just as vulnerable to combos while keeping its terrible out of shield game. Despite its nerfs to its previous footstool combos, Greninja has been generally compensated due to the gameplay changes to the engine and the buffs to its moveset giving it a more traditional combo game. That said, various veterans have also been buffed to varying degrees, especially through game updates, while Greninja has comparatively received minor, unimpactful changes from them. Nevertheless, Greninja has improved from SSB4, keeping up with the rest of the cast in Ultimate.

AestheticsEdit

  •   Greninja's model features a slightly more subdued color scheme, more closely resembling its appearance in recent mainline Pokémon games. Its body and tongue appear to have a glossy, wet sheen. The textures on Greninja's tongue have also been adjusted.
  •   Greninja is slightly more expressive, squinting akin to how it does in the Pokémon anime in a few of his animations.
  •   Every move that once used water katanas now uses water kunai, similar to when Ash's Greninja uses Cut in the Pokémon anime.
  •   Greninja's tongue has less physics-based movement than the previous game.
  •   Greninja has altered animations for its sidestep, roll, and airdodge animations; it now disappears in a small whirlwind with leaves scattering, similarly to when Substitute is successful.
  •   Greninja's victory pose where it flips in the air and lands with its arms crossed has been altered: the camera doesn't move as much, causing Greninja's body to face away from the camera.

AttributesEdit

  •   Like all characters, Greninja's jumpsquat animation takes 3 frames to complete (down from 4).
  •   Greninja walks faster (1.43 → 1.502).
  •   Greninja dashes faster (2.08 → 2.288).
    •   Its initial dash is significantly faster (1.6 → 2.178).
  •   Greninja's air speed is faster (1.18 → 1.239).
  •   Greninja is lighter (94 → 88), worsening its survivability.
  •   Greninja's traction is higher (0.045 → 0.087).
  •   Forward roll has less ending lag (FAF 30 → 29).
  •   Back roll grants more intangibility (frames 4-14 → 4-15).
  •   Back roll has more ending lag (FAF 33 → 34).
  •   Spot dodge has less ending lag (FAF 26 → 25).
  •   Spot dodge has more startup with less intangibility (frames 2-16 → 3-16).
  •   Air dodge grants more intangibility (frames 2-26 → 2-27).
  •   Air dodge has significantly more ending lag (FAF 32 → 42).
  •   The changes to locking and the ability to tech aerial footstools hurts Greninja more than the rest of the cast, weakening notorious damage racking and KO setups it possessed in Smash 4.
  •   The increased shieldstun, shield grab startup and shield drop lag further hinder Greninja's poor out of shield game, now being arguably the worst out of the cast, as unlike most other characters, all its options that can bypass shield drop lag are either too slow or, in the case of its back aerial, cannot hit most opponents on the ground.

Ground attacksEdit

  • Neutral attack:
    •   The first two hits have lower knockback (hit 1: 30 base/30 scaling → 20 / 25/25/15/15; hit 2: 30 base/50 scaling → 20/ 25/25/20), and keep opponents on the ground (hit 1: 70°/60°/80° → 361°/361°/180°/361°; hit 2: 70°/60°/90° → 361°). This allows them to connect better and jab lock.
    •   The first three hits have smaller hitboxes overall (hit 1: 3u/2.5u/3.5u → 2.0u/2.0u/2.2u/2.2u; hit 2: 3u/2.5u/4u → 2.5u/2.8u/3.0u; hit 3: 5u/3u/3u → 3u/3.4u/4u). This reduces their vertical range.
    •   The second hit deals more damage (1.6% → 2%).
    •   The third hit deals less damage (3.5% → 3%), and thus less knockback.
    •   The third hit uses stationary hitboxes rather than hitboxes connected to Greninja's arms.
    •   The infinite has faster startup (frame 6 → 4), a shorter gap between hits (4 frames → 3), and has reduced knockback scaling (40 → 10) while gaining a hitstun modifier of 2 on each hit, allowing it to connect much more reliably.
    •   The infinite has a lower hitlag multiplier (1× → 0.5×) and SDI multiplier (1.1× → 0.4×), making it harder to escape.
    •   The infinite is comprised of one extended hitbox rather than two normal ones (size: 5.2u/6.2u → 5.5; Y/Z-offset: 7/7 → 7.5/7.5; Z-stretch: 13 → 14).
    •   The infinite has gained a shieldstun multiplier of 4×. This allows it to lock opponents into their shields between each hit, and thus pressure them more effectively, but also allows them to cancel shieldstun and punish Greninja more easily if they shield 10 hits or more.
    •   The infinite's finisher has three hitboxes instead of two, with different sizes (5.6u/6.6 → 4.2u/4.2u/4.8u) and positions (ID 0: 7 Y-offset/8 Z-offset → 7.5/9; ID 1: 7 Y-offset/16 Z-offset → 7.3/13 → ID 2: 7.5 Y-offset/16.5 Z-offset. This increases its horizontal range, but lowers its vertical range.
    •   The infinite's finisher has reduced hitlag (3× → 2×).
  • Forward tilt:
    •   It deals more damage when angled (7.3% → 8.3%).
    •   It has significantly altered knockback (angled up: 20/30/40 base / 110/90/70 scaling → 50 / 100/79/58; non-angled/angled down: 20/30/40 base / 110/90/70 scaling → 50 / 98/77/56).
      •   This improves its KO power if sweetspotted, especially when angled up, allowing it to KO middleweights at around 170% from center stage.
      •   However, this hinders its KO power if sourspotted, and removes its ability to lock opponents at low percents.
  • Down tilt:
    •   Greninja has a new down tilt: a downward hand sweep instead of a shin kick.
    •   It has less ending lag (FAF 27 → 23).
    •   It has altered knockback (60 base/40 scaling → 30/110). This slightly narrows its combo range at low percentages, but increases it from mid to higher ones, to the point it now allows for KO confirms at high percents.
    •   It launches opponents at a more upward angle (70° → 79°/77°/75°), further improving its combo potential at high percents.
    •   It deals less damage (7% → 4%).
    •   It has smaller hitboxes (4u/3.5u/3u → 3u/2.5u/3u), reducing its vertical range.
  • Dash attack:
    •   The move has an altered animation with Greninja stopping halfway through the spin.
    •   It deals more damage (7% → 8%), with base knockback compensated (100 → 90).
    •   It has a longer hitbox duration (frames 7-10 → 7-11) and less ending lag (FAF 31 → 29).
    •   It sends at a higher angle (60° → 70°). Combined with its lower ending lag and base knockback, this improves its combo ability.
  • Forward smash:
    •   It deals more knockback (30 base/101 scaling → 35/104).
  • Up smash:
    •   The first hit has a new hitbox that only hits grounded opponents with significantly reduced knockback (30 base/120 scaling → 20/10) and SDI multiplier (1× → 0.5×), whereas the previous hitbox can now only hit aerial opponents and has reduced knockback scaling (120 → 90). This makes the first hit connect more consistently into the second hit, no longer failing against opponents on platforms above Greninja.
  • Down smash:
    •   The move has much less startup lag (frame 16 → 11), with its total duration reduced as well (FAF 55 → 50).
    •   The arms' hitboxes have been extended towards Greninja (Z-stretch: 0 → 7), and the move has one additional hitbox at each side during the last active frame, removing its blindspots.

Aerial attacksEdit

  •   All aerials have less landing lag (12 frames → 7 (neutral), 15 → 11 (forward), 13 → 10 (back), 15 → 14 (up), 32 → 30 (down)).
  • Neutral aerial:
    •   The move has considerably less ending lag (FAF 65 → 53), making it much less likely to cause a self-destruct offstage.
    •   The late hit has a longer duration (frames 14-16 → 14-19).
  • Forward aerial:
    •   It has much more knockback scaling (84 → 95), allowing it to KO roughly 20% earlier.
    •   The move has a new, smaller extended hitbox directly in front of Greninja (size: 3u; Y/Z-offset: 7.5/8; Y-stretch: 8.5), removing its blindspot.
    •   The move's two other hitboxes are smaller (5.4u/4.4u → 4.7u/3.7u), and one of them is offset further from Greninja (Y-offset: -7 → -8), creating a new blindspot between them.
  • Back aerial:
    •   The move has less ending lag (FAF 46 → 41).
    •   It has bigger hitboxes (hits 1 and 2: 5u/4u/2u → 5.5u/4.3u/2.5u; hit 3: 5.8u/4.5u/2u → 5.8u/4.5u/2.5u).
    •   The last two hits have less startup (hit 2: frame 8 → 7; hit 3: frame 13 → 11).
    •   The first hit sends at different angles, one of them being a different autolink angle (365°/30°/30° → 367°/35°/35°). This allows it to connect better into the second hit.
    •   The third hit deals more damage (4% → 6%), with its knockback scaling not fully compensated (120 → 95), allowing it to KO slightly earlier.
  • Up aerial:
    •   The fifth hit uses the autolink angle like the previous hits (85° → 367°), allowing it to connect more reliably into the final hit, and increasing the move's window for drag-down setups.
    •   The first five hits have more base knockback (45 → 55), and the fifth hit no longer uses set knockback. This is overall beneficial to the move, as it inflicts more hitstun to opponents that further facilitates drag-down setups, without drastically disrupting its linking ability due to using the autolink angle.
  • Down aerial:
    •   The move has less ending lag if it misses (FAF 57 → 52), making it slightly safer to use offstage.
    •   Greninja no longer loses its double jump when hitting with the move.
    •   It has more ending lag when it hits an opponent (FAF 13 → 21), hindering its combo potential and making it less safe on shield.
    •   Its hitboxes are smaller (clean: 6u → 4.3u; late: 7u → 5.2u) and higher up (Y-offset: -2 → 0.1), reducing its range.

Throws and other attacksEdit

  • Grab:
    •   All grabs have more ending lag (standing: FAF 30 → 39; dash: 40 → 47; pivot: 36 → 42).
    •   Standing and pivot grab have one frame faster startup (standing: frame 11 → 10; pivot: frame 15 → 14).
    •   Dash grab has slower startup (frame 9 → 13).
    •   Dash grab's grabbox doesn't extend out as far (Z-stretch: 20 → 17.5), reducing its range.
    •   Pivot grab's grabbox extends further (Z-stretch: -22 → -25.3), increasing its range. It is now the largest non-tether pivot grab in the game.
  • Pummel:
    •   Pummel deals more hitlag (4 frames → 11), but has less startup (frame 6 → 2) and significantly ending lag (FAF 27 → 8). It is now one of the fastest pummels in the game, rather than one of the slowest.
    •   It deals less damage (2% → 1%).
  •   The speed of Greninja's forward, back, and up throws is no longer weight-dependent. This improves up throw's combo potential against heavyweights, but worsens it against lightweights.
  • Forward throw:
    •   The move has gained a hitbox before the throw, increasing its damage (5% → 3.5% (hit 1), 4.5% (throw); 8% total) and allowing it to hit bystanders.
    •   It launches at a lower angle (50° → 40°) with much more knockback overall (70 base/45 scaling → 65/100), to the point it has above-average knockback for a forward throw rather than being one of the weakest in the game, greatly improving its utility for setting up edgeguards and KOing at very high percents.
  • Back throw:
    •   Back throw deals more damage (8% → 9%).
    •   It has an altered animation: Greninja turns towards the screen instead of remaining facing forward, and swings its arms horizontally towards its back to toss the opponent.
  • Up throw:
    •   Up throw has one frame less ending lag (FAF 44 → 43).
  • Down throw:
    •   Down throw has less ending lag (FAF 43 → 38), improving its combo potential.
  • Edge attack:
    •   Edge attack deals more damage (7% → 9%).
  • Down taunt:
    •   Down taunt has much more ending lag (FAF 80 → 110), further reducing its very limited utility.
    •   Greninja now suffers hitlag when down taunt connects, but the opponent only suffers hitlag on the ground, as the air only hitboxes are flinchless, and thus cannot inflict hitlag. This effectively reduces the move's hit rate.

Special movesEdit

  • Water Shuriken:
    •   Water Shuriken has slightly increased range.
    •   Opponents don't fall out of the fully charged shuriken as much as before.
    •   Uncharged Water Shuriken has less knockback scaling (85 → 45).
    •   Water Shuriken looks sharper.
  • Shadow Sneak:
    •   It can now be used multiple times in midair.
    •   Shadow Sneak's shadow travels faster.
    •   Both attack variants have less startup.
    •   Forward Shadow Sneak has a new animation: it is a forward kick instead of a handstand kick. It launches at a much lower angle (48° → 36°), improving its KO potential.
    •   Greninja now cloaks itself in a whirlwind before disappearing.
    •   Shadow Sneak stalls less in the air (initial vertical boost: 2.1 → 1.8), hindering its recovery usage.
    •   The shadow is notably darker, and the shadow's eyes flash red just before release. This makes it easier to predict the attack.
  • Hydro Pump:
    •   Hydro Pump now has an arrow pointing in the direction of travel, similar to Pikachu's Quick Attack and Pichu's Agility.
    •   Hydro Pump can now be ledge-cancelled.
    •   Greninja no longer loses its double jump if it is hit out of Hydro Pump.
    •   The water shots have considerably reduced knockback (65 (base)/100(scaling) → 60/80), and smaller hitboxes (6u → 4.7u), greatly reducing the move's offensive utility.
  • Substitute:
    •   The counter window is longer (frames 8-29 → 8-34).
    •   It has more ending lag if it misses (FAF 65 → 70).
    •   Substitute slows down the opponent when the attack activates, allowing the attack to land more consistently.
    •   All of the attack's variants have one frame faster startup (frame 41 → 40).
    •   All of the attack's variants now grant Greninja intangibility during the attack (frames 1-37 → 1-56).
    •   All of the attack's variants have slightly more ending lag (FAF 85 → 87).
    •   The side variant has much less knockback scaling (100 → 85).
    •   The up and diagonal up variants deal less damage (up: 14% → 13%; diagonal up: 13% → 12%) and have less knockback scaling (both: 100 → 95), noticeably hindering their KO power.
    •   The down variant deals less damage (14% → 13%) and has less knockback (grounded: 100 scaling → 95; aerial: 50 base/100 scaling → 25/65). This hinders its KO power; particularly, the aerial version is no longer among the strongest meteor smashes in the game.
    •   The diagonal down variant deals less damage (13% → 12%) without compensation on knockback, hindering its KO power.
    •   Substitute has more vibrant particle effects.
  • Secret Ninja Attack:
    •   Greninja now turns into Ash-Greninja during its Final Smash.
    •   Greninja and its opponents are no longer silhouetted during the attack.
    •   The final hit has considerably less knockback scaling (160 → 125), reducing the move's KO potential to the point where it is now weaker than the similar final smashes of Robin and Mii Brawler, both of which also deal less knockback now.

Update historyEdit

Greninja has received a mix of minor buffs and nerfs and glitch fixes via game updates. Some important nerfs include removing the ability to cancel endlag with Hydro Pump and the universal projectile shield damage nerf affected Greninja's safety and neutral game making actions in general less safe. To partially compensate, other aspects like down smash and pummel were made more reliable and increased shield size improve the potency of rushdown.

These changes did not affect Greninja in significant ways, and thus Greninja's competitive standing has remained virtually the same since the game's launch.

  1.2.0

  •   A visual glitch with Greninja's screen KO has been fixed.

  2.0.0

  •   Down smash has a larger internal hitbox (Z offset: 0u → 0u-7u).
  •   Hydro Pump no longer instantly cancels Greninja's endlag when landing on a moving or slanted platform.

  3.0.0

  •   Water Shuriken deals less shield damage (0 → (-1.5 to -5.5)/-4.5 (uncharged/fully charged, hit 6)).

  3.1.0

  •   Fixed an unknown glitch that resulted in Greninja sliding while shielding.

  4.0.0

  •   Greninja nows immediately descends when using down aerial after being launched, rather than air stalling.

  7.0.0

  •   Overall shield size has been increased by 1.206×.

  8.0.0

  •   Pummel has a larger hitbox (5u → 6u), allowing it to connect more consistently.

MovesetEdit

For a gallery of Greninja's hitboxes, see here.

Note: All numbers are listed as base damage, without the 1v1 multiplier.

  Name Damage Description
Neutral attack Single Hit (たんだ) / Palm Strike (しょうだ) / Water Palm Heel (しょうてい) / Water Cutter (みずカッター) / Cutter Finish (カッターフィニッシュ) 2% Two alternating palm thrusts followed by a double palm thrust that emits a small blast of water. If button mashed, it is instead followed by a series of knifehand strikes that emit blade-shaped water blasts that concludes with an outward knifehand strike that emits a wide blast of water. It can also be jab canceled, such as into forward tilt, down tilt and forward smash.
2%
3%
0.5% (loop), 2% (last)
Forward tilt Rolling Kick (まわしげり, Roundhouse Kick) 7.3% A hook kick which stops half way. It can be angled and can lock opponents.
Up tilt Anti-Air Tongue (ぜつたいくう) 4.5% Swings its tongue upwards. A good aerial combo starter and juggling tool due to its low knockback and somewhat disjointed hitbox. It can combo into itself at low percents and is a reliable way to connect into up aerial.
Down tilt Shin Slash (すねぎり) 4% Does a downward hand sweep. It sends opponents at an upward angle, making it a versatile combo starter. Notably, it can confirm a KO into an up smash rather easily.
Dash attack Surface Kick (すいめんげり) 8% Does a sweep kick. Is arguably one of the best dash attacks in the game, as it launches opponents at an excellent angle for combos, making it one of Greninja's best combo starters. Reliably combos into back aerial at virtually any percent, and can set up up aerial strings or drag-down combos with up aerial.
Forward smash Water Kunai (みずくない) 14% An inward slash with a water kunai. Deals good knockback and has good range. However, it has notable ending lag.
Up smash Double Water Kunai (ダブルみずくない) 5% (hit 1), 14% (hit 2 clean center), 11% (hit 2 clean sides), 10% (hit 2 late) Two reverse gripped inward slashes with water kunai, similar to Sheik's up smash. Greninja's strongest finisher, especially when hit clean. Can be combo'd into from a down-tilt at specific percentages for a KO.
Down smash Water Kunai Shake Off (ダブルふりはらいみずくない) 13% (kunai), 11% (arms) Hits both sides with water kunai. Due to it sending opponents at an semi-spike angle and coming out on frame 11, it is a quick and effective way to set up an edge-guard situation.
Neutral aerial Water Burst (みずはれつ) 11% (clean), 6% (late) Strikes a ninjutsu pose while emitting an exploding water bubble. Despite noticeable start-up lag for a neutral aerial(frame 12), it boasts great combo potential at low to mid percentages with its strong and weak hits and can KO confirm into up-smash at high percents with the weak hit. It can also KO at very high percents with its strong hit.
Forward aerial Aerial Water Kunai (くうちゅうみずくない) 14% Slashes with a water kunai. It has some start-up and suffers from high end lag, but it is a great tool in the neutral for spacing due to its disjointed hitbox and can be used for KOing. It is also safe on shield if spaced correctly.
Back aerial Triple Kick (さんてんげり) 3% (hit 1), 2.5% (hit 2), 6% (hit 3) Kicks backwards three times. It is Greninja's fastest aerial attack, although it is also one of the weakest aerials of its kind. However, this makes it a useful combo tool in return, as it can be followed up from a down throw, down tilt, as well as a dash attack. It can be a situational out-of-shield option if the opponent crosses-up on its shield.
Up aerial Spiral Kick (らせんげり) 1.3% (hits 1-5), 3% (hit 6) Does a upward corkscrew kick, similar to both Sheik's and Joker's up aerials. One of Greninja's best combo and KO tools, as it can juggle and KO effectively due to its great jumping prowess. It also allows for drag-down combos because it's multi-hit properties, although this requires precise timing to land it at the right time, as its final launching hit comes out too fast to set up drag-down combos otherwise.
Down aerial Aerial Kick (きゅうしゅうげり) 8% A diving double foot stomp. It acts as a stall-then-fall and bounces off opponents. The clean hit meteor smashes opponents while the late hit sends the opponent upwards, allowing for some situational combos.
Grab Grab (つかみ) — Grabs with a whirlpool. While its standing grab is slow, it is among the longest-reaching grabs in the game.
Pummel Grab Water Bind (つかみみずしばり) 1% Compresses target with water. Decent speed.
Forward throw Push Away (おしのけ) 3.5% (hit 1), 4.5% (throw) Shoves the opponent forward. Can KO at high percentages near the edge.
Back throw Send-off　(さかおくり) 9% Greninja leans forward and flings the opponent backwards. Like forward throw, its decent knockback gives it good edgeguarding potential at high percentages.
Up throw Toss Up (なげあげ) 5% Tosses the opponent upwards. One of Greninja's most useful throws, as it can combo into a up tilt or up aerial at low and medium percents and can even KO at later percents.
Down throw Slam (たたきつけ) 5% Slams the opponent onto the ground. It can combo into a down tilt, forward tilt, and dash attack at low percentages. It can combo into back aerial at medium percents and later into forward aerial as well at higher percentages with good timing.
Forward roll
Back roll
Spot dodge
Air dodge 		— —
Techs — —
Floor attack (front)
Floor getups (front) 		  8% Sweep kicks around itself while getting up.
Floor attack (back)
Floor getups (back) 		  8% Sweep kicks around itself while getting up.
Floor attack (trip)
Floor getups (trip) 		  8% Sweep kicks around itself while getting up.
Edge attack
Edge getups 		  9% Performs a roundhouse kick while climbing up.
Neutral special Water Shuriken 3%-10.8% (uncharged), 1.0% (fully charged looping hits), 9% (fully charged final hit) Uses Water Shuriken, which can be charged. Depending on how long the move is charged, the shuriken will be larger and do more damage, however the speed and distance will decrease as a result. At full charge, it hits multiple times and can kill with a Shadow Sneak follow-up at low and medium percents. It is also a decent KOing option at higher percents. A fully charged water shuriken that is reflected at a higher speed will have trouble landing the final hit on Greninja because of the looping hits not moving Greninja far enough for it.
Side special Shadow Sneak 10% (normal), 12% (reverse) Greninja moves its shadow along the ground; the way the shadow moves is relative to Greninja's position, and it can be moved further or closer to change how far Greninja teleports. After a short time or when the special button is released, Greninja disappears briefly, then reappears at its shadow while performing either a flip kick forwards or a stronger kick backwards (depending on control stick direction) to attack. While performing Shadow Sneak, Greninja cannot run, attack, grab, dodge or shield. However, it can walk slowly, jump and taunt. Has strong knockback and base knockback, which allows it to kill notably early when hitting the move offstage.
Up special Hydro Pump 2% (per shot) Uses Hydro Pump to propel itself in the inputted direction; it can be used twice. Each shot has a windbox effect that pushes opponents, making it a decent option for gimping predictable recoveries.
Down special Substitute 13% (up or down), 12% (diagonally), 11% (left or right) Does a pose, and if anyone hits it while posing, Greninja will temporarily disappear, get replaced by a wooden log or a Substitute doll, and then appear behind the opponent and strike them. It deviates noticeably from other counters, as the attack itself can be angled in an inputted direction and launching the opponent in that direction (with the down input meteor smashing opponents, which makes it a good punish against reckless edgeguarders).
Final Smash Secret Ninja Attack 6% (the mat), 50% (Total in the attack after mat) Turns into Ash-Greninja and attacks the opponent with a mat. If an opponent is caught in the mat, Greninja will send them up into the air and strike the opponent repeatedly in midair with its kunai, before slamming them down with a final hit.
Down taunt 0.5% While standing on one foot, Greninja holds out its hands, faces the screen, and summons small sprays of water. The sprays produce some knockback, though they're only able to KO Mario at 599% percent. A video showing the exact KO percentages at which each character can be KO'd can be found here. Unlike the rest of Greninja's taunts, this one cannot be cancelled (unless Shadow Sneak is being charged before using it).

StatsEdit

Stats Weight Dash speed Walk speed Traction Air friction Air speed Air acceleration Gravity Falling speed Jumpsquat Jump Height Double jump Height
Value 88 2.178 – Initial dash
2.288 – Run		 1.502 0.087 0.015 1.239 0.01 – Base
0.07 – Additional		 0.18 1.85 – Base
2.96 – Fast-fall		 3 46 - Base
22.11 - Short hop		 46

Announcer callEdit

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  • English

  • Japanese/Chinese

  • Korean

  • French

  • Russian

On-screen appearanceEdit

  • Emerges from a Poké Ball, then performs a ninjutsu hand sign that emits a small burst of water from its hands.

  • Greninja's on-screen appearance

TauntsEdit

  • Up Taunt: Stands upright, clasping its hands together before assuming a ninjutsu stance. The stance resembles one of its attack animations from the Pokémon series.
  • Side Taunt: Shakes head from side to side, causing its tongue to whip out in the same directions. Particles of saliva fly off with each whip.
  • Down Taunt: Poses with arms out and palms upward, and summons small sprays of water from them, which deal 1% or 2% damage and produce some knockback, though they're able to KO only at above 420%. A video showing the exact KO percentages at which each character can be KO'd can be found here.

  • Greninja's up taunt.

  • Greninja's side taunt.

  • Greninja's down taunt.

Idle PosesEdit

  • Crosses arms over its body, then separates them with a flourish.
  • Hunches over and assumes a ninjutsu stance.

  • Greninja's first idle pose.

  • Greninja's second idle pose.

Crowd cheerEdit

Cheer (English) Cheer (Japanese/Chinese) Cheer (Italian) Cheer (Dutch) Cheer (French)
Cheer
 

 
Description Gre - ninja! Ge - kkou -ga! Gre - nin - ja! Gre - ninja! Am - phi - no - bi!
Cheer (German) Cheer (Spanish) Cheer (Russian) Cheer (Korean)
Cheer
 

 
Description Quaaaaa - jutsu! Greninja! Greninja! Ya ya ya! Gre - ninja! Gae - gul - nin - ja!

Victory posesEdit

  • Left: Does a few hand seals with splashing water, and then a ninja pose. It resembles one of its attack animations in Pokémon X and Y.
  • Up: Performs Double Team to briefly create three afterimages of itself.
  • Right: Does a flip, lands in a spinning pose, and crosses its arms.
A small excerpt of the title theme of Pokémon Red, Blue, Yellow, and Green Versions, a track which would go on to become the Pokémon main theme and the title theme for the entire series.

In competitive playEdit

Most historically significant playersEdit

See also: Category:Greninja players (SSBU)

Tier placement and historyEdit

In the early metagame, players quickly noticed that Greninja had been buffed from Smash 4, with improved versatility and speed, and despite losing its footstool combos, it gained a stronger combo game thanks to improved frame data on moves such as its dash attack, up throw, down throw, and neutral air. Greninja's high learning curve did not deter players such as Stroder, Jw, and Lea from seeing success with the character, and Greninja became a popular pick in competitive play, culminating in the character peaking at 10th in terms of representation between July and December 2019. Due to Greninja's buffs and its strong representation, it was considered an upper-high tier character in the early metagame.

This perception shifted following the end of the online metagame, especially as Greninja players began pointing out frustrating traits in the character's moveset, most notably inconsistencies in Greninja's multihit moves. Furthermore, Greninja's best players from before the pandemic either became less active, changed mains, or had worse performance: for example, Lea would go on hiatus in 2022, with mixed results in the events he has attended since, while Stroder would begin playing other characters over Greninja. Although Greninja still saw strong results from players such as Tarik, Greninja's overall representation would drop, bottoming out at 45th between January and June 2022. Greninja's lower representation and more apparent flaws led to a colder reception than in the early metagame, and as such, Greninja was ranked 33rd on the first tier list, towards the lower end of high tier.

Following the release of the first list, Greninja's results would noticeably improve, thanks to Tarik's results remaining strong and the rise of many other Greninja players, including Tsuna and Anarchy, in their respective scenes. Continued efforts from dedicated mains prevented Greninja from dropping further, and he only slid down a spot to 34th on the second list, ranking towards the middle of B+ tier, with this drop mainly being the result of the meteoric rise of Bayonetta and Corrin from the first tier list. Overall, although less common than the early metagame, Greninja remains a rather viable pick in the post-online metagame.

Classic Mode: Your Turn, Greninja!Edit

 
Greninja's congratulations screen.

Greninja battles characters that represent different types from the Pokémon games: for example, Charizard and Bowser represent the Fire type, while Mewtwo, Ness and Lucas represent the Psychic type. Interestingly, the types represented correspond to those initially available in the Pokémon Trading Card Game.

Round Opponent Stage Music Notes
1   Charizard and   Bowser Pokémon Stadium Battle! (Elite Four) / Battle! (Solgaleo/Lunala) Represents Fire-type. Charizard's Pokémon Trainer is absent.
2   Pikachu,   Pichu, and   Zero Suit Samus Pokémon Stadium 2 Battle! (Steven) Represents Electric-type.
3   Lucario,   Ryu, and   Ken Pokémon Stadium Battle! (Reshiram / Zekrom) Represents Fighting-type.
4   Ivysaur Pokémon Stadium Battle! (Gladion) Represents Grass-type. Ivysaur's Pokémon Trainer is absent.
5   Mewtwo,   Ness, and   Lucas Pokémon Stadium 2 Battle! (Dialga/Palkia) / Spear Pillar Represents Psychic-type.
6   Squirtle and   Greninja Kalos Pokémon League Battle! (Champion) - Pokémon X / Pokémon Y Represents Water-type. Squirtle's Pokémon Trainer is absent. The CPU will be the   default Greninja if the player chooses the black costume.
Bonus Stage
Final Master Hand Final Destination Master Hand (Less than 7.0 intensity)
Master Hand / Crazy Hand (Intensity 7.0 or higher)		 On intensity 7.0 and higher, Crazy Hand fights alongside Master Hand.

Note: All rounds except the sixth round take place on Pokémon Stadium and Pokémon Stadium 2. If applicable, each stage will also shift to their appropriately-typed form at the earliest possible opportunity. (The stages remain in their default form in rounds 3 and 5, as none of the stages have Psychic or Fighting-themed forms.)

Credits roll after completing Classic Mode. Completing it as Greninja has Battle! (Trainer Battle) - Pokémon X / Pokémon Y accompany the credits.

Role in World of LightEdit

 
Finding Greninja in World of Light

Greninja was among the fighters that were summoned to fight the army of Master Hands.

During the opening cutscene, Greninja was present on the cliffside when Galeem unleashed his beams of light. Greninja leaped into the air to avoid one of the beams, which hit Lucario instead. Greninja was hit shortly after and vaporized, getting imprisoned by Galeem afterward along with the other fighters, sans Kirby. A puppet fighter cloned from Greninja is later seen alongside ones cloned from Fox, Samus, Link and other fighters.

Greninja was one of the many fighters that fell under Dharkon's control upon Galeem's first defeat, and it can be found in the Mysterious Dimension at The Dark Realm. It can be seen impeding the path, making it an obligatory unlock.

Greninja is later seen among several other fighters, making their last stand against Galeem and Dharkon. It also shows up in the bad ending where Galeem emerges victorious against Dharkon, witnessing Galeem engulf the world in light.

Fighter BattleEdit

No. Image Name Type Power Stage Music
50
 
 Greninja
 
Shield
 10,600 Kalos Pokémon League (Ω form) Battle! (Trainer Battle) - Pokémon X / Pokémon Y

SpiritEdit

Greninja's fighter spirit can be obtained by completing Classic Mode. It is also available periodically for purchase in the shop for 300 Gold, but only after Greninja has been unlocked. Unlocking Greninja in World of Light allows the player to preview the spirit below in the Spirit List under the name "???". As a fighter spirit, it cannot be used in Spirit Battles and is purely aesthetic. Its fighter spirit has an alternate version that replaces it with its artwork in Ultimate.

  • 418. Greninja

In Spirit BattlesEdit

As the main opponentEdit

Spirit Battle parameters Inspiration
No. Image Name Series Enemy Fighter(s) Type Power Stage Rules Conditions Music Character
154
 
 Winky Donkey Kong Series •Greninja  
 
 1,700 Mushroom Kingdom U N/A •The enemy deals damage when falling
•The enemy has increased jump power 		Jungle Level (Brawl)
199
 
 Zora The Legend of Zelda Series •Greninja  
 
 1,800 Great Bay N/A •The enemy's neutral special has increased power Ocarina of Time Medley
385
 
 Slippy Toad Star Fox Series •Greninja  
•Fox  
 
 9,600 Frigate Orpheon (hazards off) N/A •Defeat the main fighter to win
•Timed battle (1:30)
•The enemy tends to avoid conflict 		Corneria - Star Fox
482
 
 Raikou, Entei, & Suicune Pokémon Series •Greninja  
•Incineroar  
•Pikachu  
 
 9,900 Suzaku Castle •Hazard: Lava Floor •The floor is lava Pokémon Red / Pokémon Blue Medley Suicune
516
 
 Darkrai Pokémon Series •Greninja  
 
 9,900 Luigi's Mansion (Ω form) •Item: Black Hole
•Hazard: Slumber Floor 		•The floor is sleep-inducing
•Only certain Pokémon will emerge from Poké Balls (Darkrai) 		Battle! (Team Galactic)
770
 
 Metal Gear RAY Metal Gear Solid Series •Metal Greninja   (140 HP)
 
 4,200 Shadow Moses Island •Item: Exploding Types •Stamina battle
•Explosion attacks aren't as effective against the enemy
•The enemy is metal 		Yell "Dead Cell"
893
 
 Shadow Man Mega Man Series •Greninja  ×3 (60 HP)
 
 3,500 Norfair (Battlefield form) N/A •The enemy's neutral special has increased power
•Stamina battle
•The enemy favors neutral specials 		Shadow Man Stage
1,014
 
 Luka Bayonetta Series •Greninja  
 
 3,800 New Donk City Hall •Temporary Invincibility •The enemy becomes temporarily invincible when badly damaged Riders Of The Light
1,048
 
 Octoling Octopus Splatoon Series •Greninja Team  ×4
 
 3,900 Moray Towers N/A •Timed battle (2:00) Octoweaponry
1,143
 
 Frog & Snake Kaeru no Tame ni Kane wa Naru •Greninja  
•King K. Rool  
 
 3,600 Dream Land GB (Castle Lololo interior) •Assist Trophy Enemies (Sablé Prince) •Hostile assist trophies will appear Kirby Retro Medley (Castle Lololo) Frog
1,291
 
 Ninjara ARMS •Greninja  
 
 3,600 Suzaku Castle •Item: Boomerang •The enemy has increased move speed Ninja College
•Mii Brawler   (Moveset 2313, Ninjara Wig, Ninjara Outfit)[SB 1]
  1. ^ This alternative occurs when the corresponding DLC has been purchased and downloaded.

Alternate costumesEdit

 
               

GalleryEdit

Fighter Showcase VideoEdit

TriviaEdit

  • In the Pokémon series, Ash-Greninja is only obtainable as a male. The fact that Greninja transforms into Ash-Greninja for its Final Smash, Secret Ninja Attack, implies that Greninja is a male in Ultimate.
  • Greninja's new character portrait resembles its air dodge animation.
    • It also resembles Fox's character portrait from Super Smash Bros. 4 but with the arm and leg positions mirrored.
  • Greninja's down taunt has a unique property where it will seemingly be absorbed by counters without triggering the counterattack, even against grounded opponents despite the grounded hitboxes being standard hitboxes. This is because despite the grounded hitboxes being standard hitboxes that are counterable like any other hitbox, the aerial hitboxes actually take priority over the grounded hitboxes, and the aerial hitboxes are windboxes, which cannot trigger counters but will still be negated by them. Since counters respond to both grounded and aerial hitboxes, the game treats this interaction as a windbox connecting with a counter and completely ignores the regular hitboxes that hit grounded opponents.
    • As Greninja's down taunt is the only attack in the game with direct windboxes that trigger hitlag for the user, this gives the unique appearance of the attack striking the counter but being negated.
  • Greninja's fighter number, 50, is the same as the number of its costume in Super Mario Maker. It shares this distinction with Inkling.
  • Greninja, Ivysaur, Olimar, Little Mac, Ryu and Ken are the only base game characters to never appear as minions in any Spirit battles.
  • Alongside Luigi, Greninja is one of two characters in Ultimate with a taunt that cannot be canceled, due to the fact that their non-cancelable taunts have hitboxes.
    • Strangely, this does not apply for Snake and Kazuya's taunts that have damaging hitboxes
    • Greninja and Luigi are also the only two characters whose Classic Mode titles feature their names.
  • Greninja can also be unlocked immediately after clearing Classic Mode as Sheik, referencing their ninja-like traits and movements.
  • Greninja appears slightly tilted in its damage meter compared to its character artwork. This distinction is shared with fellow Pokémon series character Incineroar.
    • Coincidentally, both are final evolutions of starter Pokémon and both have Dark as their secondary type.
    • Both are also found and unlocked in the Dark Realm in World of Light.
  • Incineroar and Jigglypuff are the only Pokémon that are not encountered in Greninja's Classic Mode route.
  • Oddly, Greninja does not vanish when performing a directional air dodge despite the sound effects playing. It shares this oddity with Rosalina & Luma and Palutena.
  • In Ultimate, Greninja has a weight of 88, which almost matches its weight in Pokémon (in pounds), being 88.2 lbs.


Fighters in Super Smash Bros. Ultimate
Veterans Bayonetta · Bowser · Bowser Jr. · Captain Falcon · Cloud · Corrin · Dark Pit · Diddy Kong · Donkey Kong · Dr. Mario · Duck Hunt · Falco · Fox · Ganondorf · Greninja · Ice Climbers · Ike · Jigglypuff · King Dedede · Kirby · Link · Little Mac · Lucario · Lucas · Lucina · Luigi · Mario · Marth · Mega Man · Meta Knight · Mewtwo · Mii Fighter (Mii Brawler · Mii Gunner · Mii Swordfighter) · Mr. Game & Watch · Ness · Olimar · Pac-Man · Palutena · Peach · Pichu · Pikachu · Pit · Pokémon Trainer (Charizard · Ivysaur · Squirtle) · R.O.B. · Robin · Rosalina & Luma · Roy · Ryu · Samus · Sheik · Shulk · Snake · Sonic · Toon Link · Villager · Wario · Wii Fit Trainer · Wolf · Yoshi · Young Link · Zelda · Zero Suit Samus
Newcomers Banjo & Kazooie · Byleth · Chrom · Daisy · Dark Samus · Hero · Incineroar · Inkling · Isabelle · Joker · Kazuya · Ken · King K. Rool · Min Min · Piranha Plant · Pyra / Mythra  · Richter · Ridley · Sephiroth · Simon · Sora · Steve · Terry
  Pokémon universe
Fighters Pikachu (SSB · SSBM · SSBB · SSB4 · SSBU) · Jigglypuff (SSB · SSBM · SSBB · SSB4 · SSBU) · Pichu (SSBM · SSBU) · Mewtwo (SSBM · SSB4 · SSBU) · Pokémon Trainer (SSBB · SSBU) · Squirtle (SSBB · SSBU) · Ivysaur (SSBB · SSBU) · Charizard (SSBB · SSB4 · SSBU) · Lucario (SSBB · SSB4 · SSBU) · Greninja (SSB4 · SSBU) · Incineroar (SSBU)
Boss Rayquaza
Stages Saffron City · Pokémon Stadium · Poké Floats · Pokémon Stadium 2 · Spear Pillar · Unova Pokémon League · Prism Tower · Kalos Pokémon League
Items Poké Ball · Master Ball
Poké Ball Pokémon Abomasnow · Abra · Arceus · Articuno · Beedrill · Bellossom · Bewear · Blastoise · Bonsly · Celebi · Chansey · Charizard · Chespin · Chikorita · Clefairy · Cyndaquil · Darkrai · Dedenne · Deoxys · Ditto · Eevee · Electrode · Entei · Exeggutor · Fennekin · Fletchling · Gardevoir · Genesect · Giratina · Gogoat · Goldeen · Groudon · Gulpin · Hitmonlee · Ho-Oh · Inkay · Jirachi · Keldeo · Koffing · Kyogre · Kyurem · Latias & Latios · Lugia · Lunala · Manaphy · Marill · Marshadow · Meloetta · Meowth · Metagross · Mew · Mimikyu · Moltres · Munchlax · Onix · Oshawott · Palkia · Piplup · Porygon2 · Pyukumuku · Raichu · Raikou · Scizor · Snivy · Snorlax · Solgaleo · Spewpa · Starmie · Staryu · Suicune · Swirlix · Tapu Koko · Togedemaru · Togepi · Torchic · Unown · Venusaur · Victini · Vulpix · Weavile · Weezing · Wobbuffet · Xerneas · Zapdos · Zoroark
Enemies Chandelure · Cryogonal · Gastly · Koffing · Petilil
Other Charmander · Cresselia · Dialga · Porygon · Registeel · Reshiram · Team Rocket · Zekrom · List of Pokémon
Trophies, Stickers and Spirits Trophies (SSBM · SSBB · SSB4) · Stickers · Spirits
Music Brawl · SSB4 · Ultimate

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