The Wigner function,(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

W(x,p)\equiv\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}

\psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\, dy\; ,

[/itex]

of the quantum harmonic oscillator eigenstates is given by,

[itex]

W(x,p) = \frac{1}{\pi\hbar}\exp(-2\epsilon)(-1)^nL_n(4\epsilon)\; ,

[/itex]

where

[itex]

\epsilon = \frac{1}{\omega\hbar}\left(\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}\right)

[/itex]

and

[itex] L_n(x) [/itex] are the Laguerre polynomials.

I would like to know the Wigner function of two orthogonal states of the quantum harmonic oscillator (in analytical form as above), as in, the Wigner function of

[itex]

W(x,p) = \frac{1}{\pi\hbar}\int_{-\infty}^{\infty}

\psi_m^*(x+y)\psi_n(x-y)e^{2ipy/\hbar}\, dy\; ,

[/itex]

where [itex] m\ne n[/itex].

Any ideas are most welcome.

Thanks in advance, Jimmylok

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# Wigner function of two orthogonal states: quantum harmonic oscillator

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