Squeezed states, generalized Hermite polynomials and pseudo-diffusion equation

Abstract

We show that the wavefunctions 〈 pq; λ| n〈, of the harmonic oscillator in the squeezed state representation, have the generalized Hermite polynomials as their natural orthogonal polynomials. These wavefunctions lead to generalized Poisson Distribution P n ( pq;λ), which satisfy an interesting pseudo-diffusion equation: ∂P np,q;λ) ∂λ = 1 4 [ ∂ 2 ∂p 2 −( 1 λ 2 ) ∂ 2 ∂q 2 ]P 2(p,q;λ) , in which the squeeze parameter λ plays the role of time. Th entropies S n (λ) have minima at the unsqueezed states (λ=1), which means that squeezing or stretching decreases the correlation between momentum p and position q.