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kd6ac
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The Wigner function,
<br /> W(x,p)\equiv\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}<br /> \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\, dy\; ,<br />
of the quantum harmonic oscillator eigenstates is given by,
<br /> W(x,p) = \frac{1}{\pi\hbar}\exp(-2\epsilon)(-1)^nL_n(4\epsilon)\; ,<br />
where
<br /> \epsilon = \frac{1}{\omega\hbar}\left(\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}\right)<br />
and
L_n(x) 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
<br /> W(x,p) = \frac{1}{\pi\hbar}\int_{-\infty}^{\infty}<br /> \psi_m^*(x+y)\psi_n(x-y)e^{2ipy/\hbar}\, dy\; ,<br />
where m\ne n.
Any ideas are most welcome.
Thanks in advance, Jimmylok
<br /> W(x,p)\equiv\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}<br /> \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\, dy\; ,<br />
of the quantum harmonic oscillator eigenstates is given by,
<br /> W(x,p) = \frac{1}{\pi\hbar}\exp(-2\epsilon)(-1)^nL_n(4\epsilon)\; ,<br />
where
<br /> \epsilon = \frac{1}{\omega\hbar}\left(\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}\right)<br />
and
L_n(x) 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
<br /> W(x,p) = \frac{1}{\pi\hbar}\int_{-\infty}^{\infty}<br /> \psi_m^*(x+y)\psi_n(x-y)e^{2ipy/\hbar}\, dy\; ,<br />
where m\ne n.
Any ideas are most welcome.
Thanks in advance, Jimmylok
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