Wigner function of two orthogonal states: quantum harmonic oscillator

[kd6ac]

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

 

[fzero]

I am guessing that your post in the HW section was deleted because you didn't show more work on your attempt at a solution. If this is for a HW assignment, you should show that you've tried to evaluate that integral and pinpoint exactly what is giving you trouble. I'm sure that the method used to evaluate the function for a single state would be helpful to review in order to solve the present problem. It looks like you have to use a certain addition formula for (associated) Laguerre polynomials, in particular given as the first equation at http://en.wikipedia.org/wiki/Laguerre_polynomials#Recurrence_relations.

 

[kd6ac]

I am new to physics forums and my frist post was actually this one but then I read somewhere that questions including your own study questions should go under homework, and therefore I posted it again under homework. The answer is not so important to have as I can ask any mathematical package for the answer for specific orthogonal states. It would just be nice to have a closed analytical expression for this like the one I posted for the wigner function of individual eigenstates, which I got from a paper by N Rowley, with title The oscillatory structure of the quantum harmonic oscillator Wigner function. I work as a researcher in physics and the derivation of the result is more or less a mathematicians work. I just need the result and if possible the reference for it so that I can reference the result in my own work. Thanks

 

[fzero]

I wasn't aware there were groups of mathematicians out there that I could outsource my troublesome integrals to...

An expression for the functions you want is given as eq 74 in http://server.physics.miami.edu/~curtright/QMPSoverview.pdf. They give journal references there. I wouldn't be particularly enthusiastic about using the result without rederiving it, since that would just tend to propagate typos and other errors.

 

[kd6ac]

Thanks a lot, yes you are quite right, one should check the result and even derive it themselves. Wigner's function has certain properties which should always fulfil and if that is not enough you can derive the result for a few orthogonal states and then do the same with the formula you just showed me.

Thanks a lot really appreciated.

 

[kd6ac]

It turns out that the equation you just showed me is a more general equation, so if k=n then the equation reduces down to the one I posted, and that is exactly what I wanted, thanks again

 

[Bill_K]

There's an easy way to do all of the integrals at once - the method of generating functions.

Introduce dummy variables t and u, multiply each integral by t^m uⁿ, sum over m and n, and use the generating function for Laguerre functions. The resulting integral is easy - just an exponential - after which, reexpand in powers of t and u.