There is an obvious property of the Wigner function when one takes the limit p→∞. By the Riemann's lemma, the Wigner function is expected to go to zero in this limit but it does driven by the largest eigenvalue. Your question can be stated in the following way. The Wigner function for a generic operator A can be defined as
WA(x,p)=12πℏ∫+∞−∞dye−iℏpy⟨x−y|A|x+y⟩.
Now, assume that you have diagonalized A so that A|an⟩=an|an⟩, you can write immediately
WA(x,p)=12πℏ∑nan∫+∞−∞dye−iℏpyϕ∗n(x−y)ϕn(x+y)=12πℏ∑nanWn(x,p)
being ϕn(x)=⟨an|x⟩. Of course, if the spectrum of A will not run to infinity, taking the limit of increasing p should grant the work done. This can also be argued from the unbounded case of the Hamiltonian of the harmonic oscillator. In this case you will get
WA(x,p)=∑nEnWH.O.n(x,p)
being now
WH.O.n(x,p)=(−1)nπℏe−p2ℏ2κ2−κ2x2Ln(2p2ℏ2κ2+2κ2x2)
and Ln(x) the Laguerre polynomials and κ2=mω/ℏ. In the limit p→∞, being x fixed, the Laguerre polynomial will give the driving contribution pn. In this particular case, it is the "energy" of the oscillator to drive to zero with the power of n.
This post has been migrated from (A51.SE)