This notebook contains additional material to the Mathematica GuideBooks  (Springer-Verlag, New York, 2004) http://www.MathematicaGuideBooks.org . It uses the stylesheet of the Mathematica GuideBooks, which is embedded.

Symbolics Volume, Chapter 3

Discrete equivalents of the continuous Hermite polynomials times weight function ((((π^(1/2))(2^n)n!))^(-1/2))exp(-(ξ^2)/2)(Subscript[H, n])(ξ) are given by [✶], [✶], [✶], [✶]

(Subscript[f, m]^j)(q):=(((-1))^(j+m)) (2^(-j)) (Sqrt[(({{2 j}, {j+m}})) (({{2 j}, {j+q}}))]) (Subscript[K, j+m])(j+q;(1/2),2 j)

where (Subscript[K, n])(x;p,ν) are the symmetric Kravchuk polynomials (Subscript[K, n])(x;p,ν)(Subscript[=,  2])(Subscript[F, 1])(-n,-x;-ν;1/p), j is a positive integer, q and m are positive integers fulfilling -j≤q,m≤j.

Visualize the resulting discrete functions. As j approaches infinity, the Hermite polynomials times weight function are recovered in the following way:

(lim_(j->∞))(j^(1/4))(Subscript[f, n-j]^j)((Sqrt[j])q)=(1/(Sqrt[(Sqrt[π]) (2^n) n!]))exp(-((ξ^2)/2))(Subscript[H, n])(ξ)

For a fixed j, calculate how the left- and the right- hand sides of the last equation differ as a function of n.

KravtchukK[n_, x_, p_, ν_] = Hypergeometric2F1[-n, -x, -ν, 1/p];
(* let Hypergeometric2F1 autoevaluate in the degenerate case *)
KravtchukK[n_, ξ_, p_, ξ_] = Hypergeometric2F1[-n, -ξ, -ξ, 1/p];

f[m_, j_, q_] := (-1)^(j + m) 2^-j Sqrt[Binomial[2j, j + m] Binomial[2j, j + q]]*
                 KravtchukK[j + m, j + q, 1/2, 2j]
                 
discreteHOWF[j_][n_, q_] := {j^(-1/2) q, j^(1/4) f[n - j, j, q]}
continuousHOWF[n_, ξ_] := {ξ, Exp[-ξ^2/2] HermiteH[n, ξ]/Sqrt[Sqrt[Pi] 2^n n!]}

The next graphic shows all discrete wave functions for j=128 at once.

With[{j = 128}, 
     Show[Graphics[{PointSize[0.002],
     Table[{Hue[0.8 n/j], Point[N[discreteHOWF[j][n, q], 200]]}, 
           {n, 0, j}, {q, -j, j}]},  PlotRange -> All]]]

The following graphic shows the lowest discrete wave functions for j=128.

With[{j = 128, nMax = 12},
Show[Graphics[Table[{Hue[0.8 n/nMax], PointSize[0.006],
                Point /@ #, Line[#]}&[
                 Table[discreteHOWF[j][n, q], {q, -j, j}]],
               {n, 0, nMax}]], PlotRange -> All, AspectRatio -> 1/4]]

The next three graphic show the discrete wave functions as points and the continuous ones as curves. For n≪j, the discrete wave functions are an excellent approximation to the continuous one.

With[{j = 128, n = 6},
Show[GraphicsArray[Function[n, 
Graphics[{{Hue[0], PointSize[0.008], Point /@ #},
               {GrayLevel[0], Thickness[0.0001],
                Line[{#, continuousHOWF[n, #][[2]]}& @@@ #]}}&[
                Table[discreteHOWF[j][n, q], {q, -j, j}]],
                PlotRange -> All]] /@ {6, 20, 40}]]]

The sum of the squares of the differences (between the discrete and the continuous wave functions) at the discrete points shows the features of the last graphic more quantitatively. Interestingly, the sum of squares oscillates around a limiting value.

δ[j_, n_, prec_] := (* sum square of differences *)
Sum[(#2 - continuousHOWF[n, #1][[2]])^2& @@
     discreteHOWF[N[j, prec]][N[n, prec], N[q, prec]], {q, -j, j}]

δData = Table[{n, δ[128, n, 120]}, {n, 128}];
ListPlot[Log[10, δData], PlotRange -> All]

For a (Fock-space-)truncated harmonic oscillator, see [✶].