Infinite Potential Wall

The energy eigenfunctions and eigenvalues of the infinite potetial wall:

ψ[x_,n_] = (Sqrt[(2/L)])Sin[((n π)/L)x];En[n_] = (((ℏ^2)(π^2))/(2 m (L^2)))(n^2);

$Assumptions = n∈Integers && l∈Integers && n>0 && m > 0 && L>0 && ℏ>0 && l>0

Problem 2:

Consider the particle in the initial state ϕ(x,0) = A (sin^3)(((π )/L)x). Compute the state ϕ(x,t) and the corresponding average values of position , momentum etc.

Solution:

Consider the expression of the wave function interms of the energy eigenfunctions:
ϕ(x,0) ≡ϕ(x) =  (∑_(n=1)^∞)(Subscript[c, n])(Subscript[ψ, n])(x), (Subscript[c, n])= (∫_0^L)(Subscript[ψ, n])*(x)ϕ(x)dx
ϕ(x,t) = (∑_(n=1)^∞)(Subscript[c, n])(Subscript[ψ, n])(x) (e^(-i (Subscript[E, n])t/ℏ))
< x > (t) = (∑_(n,m=1)^∞)(Subscript[c, m])*(Subscript[c, n])(e^(-i ((Subscript[E, n])-(Subscript[E, m]))t/ℏ))(∫_0^L)(Subscript[ψ, m])*(x)x (Subscript[ψ, n])(x)dx
< (x^2) > (t) = (∑_(n,m=1)^∞)(Subscript[c, m])*(Subscript[c, n])(e^(-i ((Subscript[E, n])-(Subscript[E, m]))t/ℏ))(∫_0^L)(Subscript[ψ, m])*(x)(x^2) (Subscript[ψ, n])(x)dx

We compute the normalization factor A and define the wave function at t=0

A = (1/(Sqrt[(∫_0^L)(((Sin[(π/L)x]))^6)\[DifferentialD]x]))

ϕ[x_,0] = A  ((Sin[(π/L)x])^3);(∫_0^L)((ϕ[x,0])^2)\[DifferentialD]x

cn[n_] = (∫_0^L)ψ[x,n] ϕ[x,0]\[DifferentialD]x

cn[n_] = (∫_0^L)ψ[x,n] ϕ[x,0]\[DifferentialD]x /. Sin[n π] -> 0

All coefficients are zero except when the denominator is 0:

Solve[9-10 (n^2)+(n^4)==0,n]

cn[1] = (∫_0^L)ψ[x,1] ϕ[x,0]\[DifferentialD]x

cn[3] = (∫_0^L)ψ[x,3] ϕ[x,0]\[DifferentialD]x

ϕ[x_,t_] = (∑_(n=1)^3)cn[n] ψ[x,n] (E^(-I En[n]t/ℏ))

Abs[{ϕ[x,0],ϕ[x,0.3],ϕ[x,0.5]}]/.{x-> 0.4,L-> 1,m-> 1, ℏ-> 1}

Absϕ[x_,t_] = Abs[ϕ[x,t]]/.{L-> 1,m-> 1, ℏ-> 1};
Plot[{Absϕ[x,0],Absϕ[x,0.3],Absϕ[x,0.5],Absϕ[x,0.7],Absϕ[x,1.]},{x,0,1},PlotStyle->{Red,Green, Blue, Yellow, Magenta}]

Here you can see an animation of the time evolution of |ϕ(x,t)((|)^2)

Manipulate[Plot[Absϕ[x,t],{x,0,1},PlotRange->{{0,1},{0,2}}],{t,0,4,0.0001}]

as well as the real and imaginary parts of the wave function Re(ϕ(x,t)), Im(ϕ(x,t)):

Reϕ[x_,t_] = Re[ϕ[x,t]]/.{L-> 1,m-> 1, ℏ-> 1};
Imϕ[x_,t_] = Im[ϕ[x,t]]/.{L-> 1,m-> 1, ℏ-> 1};
Manipulate[Plot[{Reϕ[x,t],Imϕ[x,t]},{x,0,1},PlotRange->{{0,1},{-2,2}},PlotStyle->{Red,Blue}],{t,0,4,0.0001}]

Average energy <E>, <(E^2)>:

avE = (∑_(n=1)^3)((cn[n])^2)En[n]

avE/En[1]

avE2 = (∑_(n=1)^3)((cn[n])^2)((En[n])^2)

avE2/((En[1])^2)

ΔE = (Sqrt[avE2-(avE^2)])

ΔE/En[1]//Simplify

xnm[n_,l_] = (∫_0^L)ψ[x,n]x ψ[x,l]\[DifferentialD]x

The n=m case must be treated differently:

xnm[n_,n_] = (∫_0^L)ψ[x,n]x ψ[x,n]\[DifferentialD]x

x2nm[n_,l_] = (∫_0^L)ψ[x,n](x^2) ψ[x,l]\[DifferentialD]x

x2nm[n_,n_] = (∫_0^L)ψ[x,n](x^2) ψ[x,n]\[DifferentialD]x

<x>(t) = <x>(0) = L/2

xavϕ[t_] = (∑_(n=1)^3)(∑_(l=1)^3) cn[n] cn[l] xnm[n,l] (E^(-I(En[l]-En[n])t/ℏ))

x2avϕ[t_] = (∑_(n=1)^3)(∑_(l=1)^3) cn[n] cn[l] x2nm[n,l] (E^(-I(En[l]-En[n])t/ℏ))//ExpToTrig

Plot[{x2avϕ[t]}/.{L-> 1,m-> 1, ℏ-> 1},{t,0,1}]

Momenta now:

pnm[n_,l_] = (∫_0^L)ψ[x,n](-I ℏ) (∂_x)ψ[x,l]\[DifferentialD]x /. Cos[l π] -> (((-1))^l)

pnm[n_,n_] = (∫_0^L)ψ[x,n](-I ℏ) (∂_x)ψ[x,n]\[DifferentialD]x

p2nm[n_,l_] = (∫_0^L)ψ[x,n](- (ℏ^2)) (∂_(x,x))ψ[x,l]\[DifferentialD]x

p2nm[n_,n_] = (∫_0^L)ψ[x,n](- (ℏ^2)) (∂_(x,x))ψ[x,n]\[DifferentialD]x

pavϕ[t_] = (∑_(n=1)^3)(∑_(l=1)^3) cn[n] cn[l] pnm[n,l] (E^(-I(En[l]-En[n])t/ℏ))

p2avϕ[t_] = (∑_(n=1)^3)(∑_(l=1)^3) cn[n] cn[l] p2nm[n,l] (E^(-I(En[l]-En[n])t/ℏ))

Compare with the exact formula for <(p^2)>(0):

(∫_0^L)ϕ[x,0] (-(ℏ^2))(∂_(x,x))ϕ[x,0]\[DifferentialD]x