Editing Quantum harmonic oscillator (section)

===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.