(1252) Wigner function for superposition of Gaussians

Consider a superposition of two Gaussian wavepackets that are distance \( d \) apart. Assume that they are positioned at \( x= \pm d/2 \). Each Gaussian has a spatial width \( \sigma \).

(1) Find the Wigner function for this supposition. Write it as a sum of two phase-space Gaussians and an interference term.

(2) Write an expression for the momentum distribution.

(3) Write what is the Wigner function after time \( t \).

(4) Write what is the \( x \) distribution after time \( t \).

Based on Lecture notes [52.7]