ψにΦ

Postscript

-%5cfrac%7b%5chbar%5e2%7d%7b2m%7d%5cfrac%7bd%5e2%5cPsi%7d%7bdx%5e2%7d=i%5chbar%5cfrac%7bd%5cPsi%7d%7bdt%7dの解の一つ%5cPsi(t,x)=(2%5cpi)%5e%7b-%5cfrac%7b1%7d%7b4%7d%7d%5csqrt%7b%5cfrac%7b%5csigma%7d%7b%5csigma%5e2+i%5chbar t/2m%7d%7d%5cexp%5cleft%5b-%5cfrac%7b(x-x_0-v_0t)%5e2%7d%7b4(%5csigma%5e2+i%5chbar t/2m)%7d-i%5comega_0t+ik_0x%5cright%5d, v_0=%5cfrac%7b%5chbar k_0%7d%7bm%7d, %5comega_0=%5cfrac%7b%5chbar k_0%5e2%7d%7b2m%7d(正しく計算できていれば)を図にしてみた。
赤が実部、青が虚部、黒が絶対値です。
ImageMagick で PostScript から gif に変換した時のコマンド

convert -delay 2 -transparent white -dispose Background -page 100x100+0+0 foo.ps foo.gif

ソース

%!PS
%%BoundingBox: 0 0 100 100
gsave
% 図のサイズ
/halfwidth 50 def
% 数学の定数
/pi 3.141592654 def
/e  2.718281828 def
% ラジアン単位で
/rsin {180 mul pi div sin} def
/rcos {180 mul pi div cos} def
/ratan {atan dup 180 gt {360 sub} if pi mul 180 div} def
% 物理定数
/hbar 1 def
% 物理パラメータ
/m 0.5 def
/x0 0 def
/k0 0.5 def
/v0 hbar k0 mul m div def
/ω0 hbar k0 dup mul mul 2 m mul div def
/σ 2.5 def
% 初期時刻
/t -50 def
100 {
    halfwidth dup translate
    /tt hbar t mul  2 m mul σ dup mul mul div def
    /σt tt dup mul 1 add sqrt σ mul def
    /arg tt 1 ratan def
    /oldx halfwidth neg 1 sub def /oldψabs 0 def /oldψR 0 def /oldψI 0 def
    halfwidth neg 0.5 halfwidth {
        /x exch def
        /dx2 x x0 sub v0 t mul sub dup mul 4 σt dup mul mul div def
        /ψabs 2 pi σt dup mul mul mul -0.25 exp e dx2 neg exp mul 100 mul def
        -0.5 arg mul tt dx2 mul sub ω0 t mul sub k0 x mul add dup
        rcos ψabs mul /ψR exch def
        rsin ψabs mul /ψI exch def
        % 実部
        1 0 0 setrgbcolor
        oldx oldψR moveto x ψR lineto stroke
        % 虚部
        0 0 1 setrgbcolor
        oldx oldψI moveto x ψI lineto stroke
        % 絶対値
        0 0 0 setrgbcolor
        oldx oldψabs moveto x ψabs lineto stroke
        /oldx x def /oldψabs ψabs def /oldψR ψR def /oldψI ψI def
    } for
    showpage
    /t t 1 add def
} repeat
grestore