Editing Quantum harmonic oscillator (section)

==One-dimensional harmonic oscillator== 

===Hamiltonian and energy eigenstates===
[[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.]]

The [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the particle is:
<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]].{{sfnp|Zwiebach|2022|pp=233-234}}

The time-independent [[Schrödinger equation]] (TISE) is,
<math display="block"> \hat H \left| \psi \right\rangle = E \left| \psi \right\rangle  ~,</math>
where <math>E</math> denotes a real number (which needs to be determined) that will specify a time-independent [[energy level]], or [[eigenvalue]], and the solution  <math>| \psi \rangle</math> denotes that level's energy [[eigenstate]].{{sfnp|Zwiebach|2022|p=234}}

Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the [[wave function]] <math>\langle x | \psi \rangle = \psi (x) </math>, 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]],{{sfnp|Zwiebach|2022|p=241}}<ref>{{cite book|first=Gregory J. |last=Gbur |author-link=Greg Gbur |title=Mathematical Methods for Optical Physics and Engineering |publisher=Cambridge University Press |year=2011 |isbn=978-0-521-51610-5 |pages=631–633}}</ref>
<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]],
<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{{sfnp|Zwiebach|2022|p=240}}
<math display="block"> E_n = \hbar \omega\bigl(n + \tfrac{1}{2}\bigr).</math>The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be <math display="inline">\langle \hat{x} \rangle = 0 </math> and <math display="inline">\langle \hat{p} \rangle = 0  </math> owing to the symmetry of the problem, whereas:

<math>\langle \hat{x}^2 \rangle = (2n+1)\frac{\hbar}{2m\omega} = \sigma_x^2 </math>

<math>\langle \hat{p}^2 \rangle =  (2n+1)\frac{m\hbar\omega}{2} = \sigma_p^2  </math>

The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value of <math display="inline">\sigma_x \sigma_p = \frac{\hbar}{2}   </math> which is its minimum value due to uncertainty relation and also corresponds to a gaussian wavefunction.{{sfnp|Zwiebach|2022|pp=249-250}}

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]].

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.

===Ladder operator method===
[[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.{{sfnp|Zwiebach|2022|pp=246-249}} 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>}},
<math display="block">\begin{align}
          a &=\sqrt{m\omega \over 2\hbar} \left(\hat x + {i \over m \omega} \hat p \right) \\
  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.{{clarify|reason=Probably correct, e.g. phase space is incompressible and therefore this is a rotation, Obscure reference, expand argument about time evolution operator, representation theory and unitarity on [[Ladder operators]] page and link here|date=June 2024}}

These operators lead to the following representation of <math>\hat{x}</math> and  <math>\hat{p}</math>,
<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''}}}}, when operated on by these ladder operators, give
<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>

From the relations above, we can also define a number operator {{mvar|N}}, which has the following property:
<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]],
<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
<math display="block">\hat H = \hbar\omega\left(N + \frac{1}{2}\right),</math>

so the eigenstates of {{mvar|N}} are also the eigenstates of energy.
To see that, we can apply <math>\hat{H}</math> to a number state <math>|n\rangle</math>:

<math display="block">
\hat{H} |n\rangle = \hbar \omega \left(\hat{N} + \frac{1}{2}\right) |n\rangle.
</math>

Using the property of the number operator <math>\hat{N}</math>:

<math display="block">
\hat{N} |n\rangle = n |n\rangle,
</math>

we get:

<math display="block">
\hat{H} |n\rangle = \hbar \omega \left(n + \frac{1}{2}\right) |n\rangle.
</math>

Thus, since <math>|n\rangle</math> solves the TISE for the Hamiltonian operator <math>\hat{H}</math>, is also one of its eigenstates with the corresponding eigenvalue:

<math display="block">
E_n = \hbar \omega \left(n + \frac{1}{2}\right) .
</math>

QED.

The commutation property yields
<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,
<math display="block">Na|n\rangle = (n - 1)a | n \rangle.</math>

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 an '''annihilation operator''' ("lowering operator"), and {{math|''a''<sup>†</sup>}} a '''creation operator''' ("raising operator"). The two operators together are called [[ladder operator]]s. 

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
<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>

the smallest eigenvalue of the number operator is 0, and
<math display="block">a \left| 0 \right\rangle = 0. </math>

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

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 
<math display="block">\left\{\left| 0 \right\rangle, \left| 1 \right\rangle, \left| 2 \right\rangle, \ldots , \left| n \right\rangle, \ldots\right\},</math>

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

Arbitrary eigenstates can be expressed in terms of |0⟩,{{sfnp|Zwiebach|2022|p=248}}
<math display="block">|n\rangle = \frac{(a^\dagger)^n}{\sqrt{n!}} |0\rangle. </math>
{{math proof|<math display="block">\begin{align}
   \langle n | aa^\dagger | n \rangle &= \langle n|\left([a, a^\dagger] + a^\dagger a\right) \left| n \right\rangle = \langle n| \left(N + 1\right) |n\rangle = n + 1 \\[1ex]
     \Rightarrow a^\dagger | n\rangle &= \sqrt{n + 1} | n + 1\rangle \\[1ex]
                    \Rightarrow|n\rangle &= \frac{1}{\sqrt{n}} a^\dagger \left| n - 1 \right\rangle = \frac{1}{\sqrt{n(n - 1)}} \left(a^\dagger\right)^2 \left| n - 2 \right\rangle = \cdots = \frac{1}{\sqrt{n!}} \left(a^\dagger\right)^n \left|0\right\rangle.
\end{align}</math>}}

====Analytical questions====

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
<math display="block">\left(x+\frac{\hbar}{m\omega}\frac{d}{dx}\right)\psi_0 = 0,</math>
whose solution is easily found to be the [[Gaussian_function|Gaussian]]<ref group="nb">The normalization constant is <math>C = \left(\frac{m\omega}{\pi \hbar}\right)^{{1}/{4}}</math>, and satisfies the normalization condition <math>\int_{-\infty}^{\infty}\psi_0(x)^{*}\psi_0(x)dx = 1</math>.</ref>
<math display="block">\psi_0(x)=Ce^{-\frac{m\omega x^2}{2\hbar}}.</math>
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>{{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 |bibcode=2013qtm..book.....H | at = Theorem 11.4}}</ref>

Explicitly connecting with the previous section, the ground state  |0⟩  in the position representation is determined by <math> a| 0\rangle =0</math>,
<math display="block">  \left\langle x \mid a \mid 0 \right\rangle = 0 \qquad
  \Rightarrow \left(x + \frac{\hbar}{m\omega}\frac{d}{dx}\right)\left\langle x\mid 0\right\rangle = 0 \qquad
  \Rightarrow           </math>
<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) 
    = \psi_0  ~,</math>
hence
<math display="block"> \langle x \mid a^\dagger \mid 0 \rangle = \psi_1 (x) ~,</math>
so that <math>\psi_1(x,t)=\langle x \mid e^{-3i\omega t/2} a^\dagger \mid 0 \rangle </math>, and so on.

===Natural length and energy scales===
{{see also|Path integral formulation#Simple harmonic oscillator}}

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]].

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
<math display="block"> H = -\frac{1}{2} {d^2 \over dx^2} +\frac{1}{2}  x^2 ,</math>
while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half,
<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>
<math display="block">E_n = n + \tfrac{1}{2} ~,</math>
where {{math|''H''<sub>''n''</sub>(''x'')}} are the [[Hermite polynomials]].

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.

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>
<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>
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
<math display="block">\psi(x,t)=\int dy~ K(x,y;t) \psi(y,0) \,.</math>
===Coherent states===

{{main|Coherent state}}

[[File:Coherent state gif.gif|thumb|right|450px|Coherent state dynamics for <math>\alpha = \sqrt{10}</math>, in units of the harmonic oscillator length <math>x_0=\sqrt{\hbar/m\omega}</math>, showing the probability density <math>|\psi(x,t)|^2</math> and the quantum phase (color).]]

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.{{sfnp|Zwiebach|2022|pp=481-492}}

The coherent states are indexed by <math>\alpha \in \mathbb{C}</math> and expressed in the {{math|{{braket|ket|''n''}}}} basis as

<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>

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} = \alpha_0 e^{-i\omega t}</math>.<math display="block">|\alpha(t)\rangle = \sum_{n=0}^\infty e^{-i\left(n+\frac{1}{2}\right) \omega t}|n\rangle \langle n | \alpha \rangle =  e^{\frac{-i\omega t}{2}}e^{-\frac{1}{2} |\alpha|^2} \sum_{n=0}^\infty\frac{(\alpha e^{-i\omega t})^n}{\sqrt{n!}} |n\rangle = e^{-\frac{i\omega t}{2}}|\alpha e^{-i\omega t}\rangle </math>

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>. Calculating the expectation values:

<math>\langle \hat{x} \rangle_{\alpha(t)} = \sqrt{\frac{2\hbar}{m\omega}}|\alpha_0|\cos{(\omega t - \phi)} </math>

<math>\langle \hat{p} \rangle_{\alpha(t)} = -\sqrt{2m\hbar \omega}|\alpha_0|\sin{(\omega t - \phi)} </math>

where <math>\phi </math> is the phase contributed by complex {{mvar|α}}. These equations confirm the oscillating behavior of the particle. 

The uncertainties calculated using the numeric method are:

<math>\sigma_x(t)=\sqrt{\frac{\hbar}{2m\omega}}  </math>

<math>\sigma_p(t) = \sqrt{\frac{m\hbar\omega}{2}} </math>

which gives <math display="inline">\sigma_x(t)\sigma_p(t) = \frac{\hbar}{2}  </math>. Since the only wavefunction that can have lowest position-momentum uncertainty, <math display="inline">\frac{\hbar}{2}  </math>, is a gaussian wavefunction, and since the coherent state wavefunction has minimum position-momentum uncertainty, we note that the general gaussian wavefunction in quantum mechanics has the form:<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{\langle\hat{x}\rangle_\alpha}{2}) - \frac{m\omega}{2\hbar}(x' - \langle\hat{x}\rangle_\alpha)^2} .</math>Substituting the expectation values as a function of time, gives the required time varying wavefunction.<br />

The probability of each energy eigenstates can be calculated to find the energy distribution of the wavefunction:

<math>P(E_n)=|\langle n | \alpha \rangle|^2 =  \frac{e^{-|\alpha|^2}|\alpha|^{2n}}{n!}</math>

which corresponds to [[Poisson distribution]].

===Highly excited states===
{{multiple image
| width = 320
| direction = vertical
| image1 = Excited_state_for_quantum_harmonic_oscillator.svg
| image2 = QHOn30pdf.svg
| 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.
}}
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.