“Rocking Floor” Perturbation

τ=2.0;
ℓ=0.5;
(Subscript[v, o])=0.25;
v[x_,t_]=(Subscript[v, o]) (x/ℓ) (t/τ) (E^(-((t^2)/(2 (τ^2)))));
Animate[Plot[v[x,t],{x,-.5,.5},PlotRange->{-0.3,0.3}],{t,-10,10}]

(Subscript[ψ, 1])[x_]=(Sqrt[(2/ℓ)]) Cos[π (x/ℓ)];
(Subscript[ψ, 2])[x_]=(Sqrt[(2/ℓ)]) Sin[π  ((2 x)/ℓ)];
(Subscript[ψ, 3])[x_]=(Sqrt[(2/ℓ)]) Cos[π ((3 x)/( ℓ))];

Plot[{(Subscript[ψ, 1])[x],(Subscript[ψ, 2])[x],(Subscript[ψ, 3])[x]},{x,-(ℓ/2),(ℓ/2)}]/. ℓ->1

ℓ=.;
space=(2/ℓ) (∫_(-ℓ/2)^(ℓ/2))Sin[((2 π x)/ℓ)] (x/ℓ) Cos[((π x)/ℓ)]\[DifferentialD]x

τ=.
Assuming[τ>0&&ω>0,time[τ_]=I 2 (∫_0^∞)(t/τ) (E^(-((t^2)/(2 (τ^2))))) Sin[ω t]\[DifferentialD]t]

The integrand is “odd”  “even”   “odd”   =  “even”

f[τ_,ω_,t_]:=(t/(τ^2)) (E^(-((t^2)/(2 (τ^2))))) Sin[ω t]

Plot[f[2,1,t],{t,-10,10}]

Assuming[ωo>0&&τ>0,(∫_(-∞)^∞)(E^(I ωo t)) (t/τ) (E^(-((t^2)/(2 (τ^2)))))\[DifferentialD]t]

cmOfT=(v0/(I ℏ)) space time[τ]

(cmOfT^2)

(cmOfT^2)/.{τ-> (10^(-9)),ω->4.27 (10^16),v0->0.25 9.37,ℏ-> 6.5821 (10^(-16))}//N

Solve[(∂_τ)(cmOfT^2)==0,τ]

Here is the maximum probability at:   τ = ((Sqrt[2])/(Subscript[ω, 0]))

(cmOfT^2)/.{τ->(( (Sqrt[2]))/ω)}//Simplify

((2048 )/(16 81 (E^2) (π^2) 9))//N

e1Calc=(((π^2) (197^2))/(2 511000 0.01))

omega = 3 e1Calc/(6.5821 (10^(-16)))

((Sqrt[2])/omega)

λ=((2 π 3.0 (10^8))/omega)

If (Subscript[v, 0])=(1/2)(Subscript[E, 0])   then

maxProb=((2^9)/(3^6)) ((E^(-2))/(π^3))//N