Quantum-Classical correspondence

In this Mathematica notebook everything is expressed in terms of the harmonic oscillator length parameter b=(Sqrt[(ℏ/(m ω))]).

Some useful definitions

A(n) is the amplitude of classical motion for given n.

A[n_]=b (Sqrt[2 n+1])

(Subscript[ϕ, n])is the normalised wave function of the harmonic oscillator

(Subscript[ϕ, n_])[x_]:=(2^(-(n/2))) (1/(Sqrt[n!])) (1/(Sqrt[b])) (π^(-1/4)) (E^(-((x^2)/(2 (b^2))))) HermiteH[n,(x/b)]

For example, look at (Subscript[ϕ, 0]):

(Subscript[ϕ, 0])[x]

I might as well take b=1, since everything is expressed in it.

b=1

Let me now define a function that plots the classical and quantum probabilities for given n,

pl[n_]:=Plot[{(((Subscript[ϕ, n])[x])^2),(1/(π (Sqrt[((A[n])^2)-(x^2)])))},{x,-A[n],A[n]},PlotStyle->{Hue[0.],Hue[0.6]}]

Investigate this for various values of n.

First for n=0. Do you understand this behaviour?

pl[0]

Then for n=10.

pl[10]

Manipulate[pl[i],{i,10,100,1}]

Assignments

Can you take a running average of the quantum result?

Hint: Define Av[f_,x_] = (∫_(x-ϵ)^(x+ϵ))fdx, and choose ϵ the right way

What conclusion(s) can you draw from these results?