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Heisenberg uncertainty for the harmonic oscillator?

Posted 2 months ago
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I have to do a program about the Heisenberg uncertainty for the harmonic oscillator. I wrote all the integrals that I need to use for the medium values and the specific function of the oscillator, but it doesn't work...Could someone help me please?

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

mediex[i_, x_] := \!\(\*SubsuperscriptBox[\(∫\), \(0\), \(∞\)]\(\((Abs[\ψ[i, x]])\)^2*x \[DifferentialD]x\)\)
mediex[5, x]

15/(8 Sqrt[π])

mediex2[i_, x_] := \!\(\*SubsuperscriptBox[\(∫\), \(0\), \(∞\)]\(\((Abs[\ψ[i, x]])\)^2*x^2 \[DifferentialD]x\)\)

mediex2[5, x]

11/4

mediep[i_, x_] := \!\(\*SubsuperscriptBox[\(∫\), \(0\), \(∞\)]\(ψ[i, x]*\((\(-iℏ\)\ )\) D[ψ[i, x], 
    x] \[DifferentialD]x\)\)

mediep[5, x]

0

mediep2[i_, x_] := \!\(\*SubsuperscriptBox[\(∫\), \(0\), \(∞\)]\(ψ[i, x]*\((ℏ^2\ )\) D[D[ψ[i, x], x], 
    x] \[DifferentialD]x\)\)

mediep2[5, x]

-((11 ℏ^2)/4)

2 Replies

I am by no means sure, but I think you should write

mediep2[i_, x_] := -\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Infinity]\)]\(\[Psi][
     i, x]*\((\[HBar]^2\ )\) D[D[\[Psi][i, x], x], 
     x] \[DifferentialD]x\)\)

and then

deltax[i_, x_] := Sqrt[mediex2[i, x] - mediex[i, x]^2]
deltap[i_, x_] := Sqrt[mediep2[5, x] - (mediep[5, x])^2]

and

relatie[5, x] // PowerExpand // FullSimplify
% // N
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