Oscillatore  armonico in MQ

m=1;  ω=1; ℏ=1

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

H=  ((p^2)/2)+ ((x^2)/2)

(∫_(-∞)^∞)((ψ[1,x])^2)\[DifferentialD]x

HermiteH[2,x]

(∫_(-∞)^∞)((ψ[2,x])^2)\[DifferentialD]x

Plot[ψ[0,x],{x,-3,3}]

Plot[ψ[1,x],{x,-3,3}]

Plot[ψ[5,x],{x,-5,5}]

Plot[ψ[10,x],{x,-10,10}]

Plot[ψ[50,x],{x,-20,20}]

Plot[F[50,x],{x,-20,20},PlotRange->{0,10000000}]

Plot[ψ[49,x],{x,-20,20}]

Plot[((ψ[50,x])^2),{x,-20,20}]

Plot[ψ[100,x],{x,-30,30}]

Confronto con  un oscillatore  classico

x= A  sin ωt;  E=  (((A^2)(ω^2)m)/2);
Probabilita'  di soggiorno∝ | (1/v)|=|(dt/dx)|=(1/(ω(Sqrt[(A^2)-(x^2)])))->(1/(Sqrt[(A^2)-(x^2)]))

Identify     E=  (((A^2)(ω^2)m)/2)= ωℏ(n+(1/2));   ->  A=  (Sqrt[2n+1])

P[n_, x_]:= (1/(π (Sqrt[2n+1-(x^2)])))

Plot[P[50,x],{x,-10,10}]

Plot[((ψ[50,x])^2),{x,-20,20},PlotRange->{0,0.25}]

Plot[{P[0,x],((ψ[0,x])^2)},{x,-1.5,1.5}]

Plot[{P[5,x],((ψ[5,x])^2)},{x,-10,10}]

Plot[{((ψ[50,x])^2),P[50,x]},{x,-20,20},PlotRange->{0,0.25}
]