SHO.nb

$K 2 RowBox[{StyleBox[RowBox[{The, , simple, , harmonic, , oscillator, , is, , one, , of, , the, , most, , classical, , problems, , in, , quantum, , mechanics, , and, , it, , is, , used, , to, , model, , numerous, , systems, , from, , the, , band, , structure, , of, , semiconductors, , to, , the, , index, , profile, , of, , a, , fiber, , optic, , cable . In, , one, , dimension, , the, , harmonic, , oscillator, , is, , described, , by, , the, , RowBox[{potential, :, , , Cell[TextData[{V(x)=, Cell[BoxData[- x ]]}]]}]}], Background -> RGBColor[0., 1., 0.]], }] 2 d RowBox[{StyleBox[RowBox[{For a conservative system, ,, , RowBox[{the, , force, , acting, , on, , such, , a, , potential, , is, , given, , by, , the, , negative, , of, , the, , gradient, , of, , the, , RowBox[{potential, :, , , Cell[TextData[{F(x)=-∇V(x)=, Cell[BoxData[--- V(x)]], =-Kx}]]}]}]}], Background -> RGBColor[0., 1., 0.]], }] dx RowBox[{StyleBox[RowBox[{Which is the familiar restoring force of a spring displaced from it ' s equilibrium position by some amount x . In quantum mechanics the Hamiltonian for a particle of mass, , ,, m, ,, RowBox[{in, , such, , a, , system, , containing, , the, , harmonic, , oscillator, , potential, , is, , given, , RowBox[{by, :, RowBox[{Cell[], Cell[]}]}]}]}], Background -> RGBColor[0., 1., 0.]], }] ^2 ^2 2 ^2 ^ p ^ p m ω x RowBox[{StyleBox[RowBox[{Cell[], Cell[], Cell[], Cell[], Cell[TextData[Cell[BoxData[H = --- + V(x) = --- + ------------]]]]}], Background -> RGBColor[0., 1., 0.]], }] 2 m 2 m 2 ^ RowBox[{StyleBox[RowBox[{The, , Hamiltonian, , operating, , on, , an, , eigenvector, , of, , the, , system, , yields, , energy, , eigenvalues, , RowBox[{or, :, , , Cell[TextData[Cell[BoxData[H | ψ > = E | ψ >]]]]}]}], Background -> RGBColor[0., 1., 0.]], }] m m m ^ RowBox[{StyleBox[RowBox[{This, , can, , be, , put, , into, , the, , coordinte, , representation, , RowBox[{by, :, , Cell[TextData[{<x|, Cell[BoxData[H | ψ > = < x | E | ψ > = E < x | ψ >]]}]]}]}], Background -> RGBColor[0., 1., 0.]], }] m m m m m The left hand side can be expanded to yield : < x | Overscript[H,^] | ψ _ m > = < x | Overscript[p,^]^2/(2 m) + (m ω^2 Overscript[x,^]^2)/2 | ψ > Since 2 ^ * ^2 d RowBox[{StyleBox[RowBox[{Cell[TextData[{<x|, Cell[BoxData[x = x < x |]]}]], , and, , Cell[TextData[Cell[BoxData[< x | p | ψ > = ---- < x | ψ >]]]], (See Schleich, Ch 2, P .35 - P .44), Cell[]}], Background -> RGBColor[0., 1., 0.]], }] m 2 m dx 2 2 h d 1 2 2 RowBox[{StyleBox[RowBox[{The, , eigenvalue, , equation, , for, , the, , particle, , in, , the, , simple, , harmonic, , potential, , is, , written, , in, , the, , position, , representation, , RowBox[{as, :, , , Cell[TextData[{-, Cell[BoxData[--- --- ψ (x) + - m ω x ψ (x) = E ψ (x)]]}]]}]}], Background -> RGBColor[0., 1., 0.]], }] 2 m 2 m 2 m m m dx RowBox[{StyleBox[RowBox[{RowBox[{Where, , the, , replacement, , Cell[TextData[Cell[BoxData[ψ (x) = < x | m >]]]], has, , been, , made . For, , brevity}], ,, , RowBox[{the solutions to the above differential equations are of the form (See Robinett, Ch .10, P .196 - P .199), :, , , RowBox[{Cell[TextData[{Cell[BoxData[ψ ]], (x)=}]], N _ m, h _ m, (ηx), e^(-(ηx)^2/2)}]}]}], Background -> RGBColor[0., 1., 0.]], }] m m where h _ m (ηx) are the Hermite polynomials, η and N _ m are scaling and normalization constants given by : η = ((m ω)/h)^1/4 RowBox[{StyleBox[RowBox[{Cell[TextData[Cell[BoxData[N =]]]], (η^2/π)^1/4, 1/(2^m m !)^(1/2)}], Background -> RGBColor[0., 1., 0.]], }] m The energy eigenvalues for the system are given by : E _ m = (m + 1/2) hω The eigenfunctions , ψ _ m (x), are also known as number states . The notebook below plots a few of the number states and allows one to see individual number states by changing the value of n .$
$"Energy Eigenfunctions and Eigenvalues of the Harmonic Oscillator" ; ω = 2 * 10^14 ; "Angular frequency" ; Ma = 9.109 * 10^(-31) ; "Mass of electron" ; n = 1 ; "Set the single eigenfunction that you would like to plot below" ; h = 1 * 10^(-34) ; "Planck's Constant" ; η = ((Ma * ω)/h)^1/4 ; V[x_] := (Ma * ω^2 * (x)^2)/2 ; "Simple Harmonic Oscillator Potential" ; Nc[m_] := (η^2/π)^1/4 1/(2^m * m !)^(1/2) ; "Normalization constant" ; ψ[m_, x_] := Nc[m] * HermiteH[m, η * x] * e^(-(η * x)^2/2) ; "Simple harmonic oscillator number state" ; A = Table[Plot[ψ[m, x] + 350 * m, {x, -2 * 10^(-4), 2 * 10^(-4)}, Frame -> True, FrameLabel -> {"Position-x", "ψ(x)", "Harmonic Oscillator Eigenfunctions", o}, DisplayFunction -> Identity], {m, 0, 10}] ; B = Plot[(10^13) * V[1.1 * x], {x, -2 * 10^(-4), 2 * 10^(-4)}, Frame -> True, FrameLabel -> {"Position-x", "ψ(x)", "Harmonic Oscillator Eigenfunctions", o}, DisplayFunction -> Identity] Show[A, B, DisplayFunction -> $DisplayFunction, PlotRange -> {0, 4000}] ; Plot[ψ[n, x], {x, -2 * 10^(-4), 2 * 10^(-4)}, Frame -> True, FrameLabel -> {"Position-x", "ψ(x)", "Single Eigenfunction n=", n}, PlotRange -> {-200, 200}]$

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$W[m_, x_, p_] := (-1)^m/π e^(-(p^2 + x^2)) * LaguerreL[m, 2 * (p^2 + x^2)] ; Table[{Print["Wigner function for the number state m=", m], Plot3D[W[m, x, p], {x, -4, 4}, {p, -4, 4}, PlotPoints -> 100, PlotRange -> {-.35, .35}, Mesh -> False, AxesLabel -> {"x", "p", "W(x,p)"}]}, {m, 0, 5}] ; "Four Wigner Functions"$