♦ Lösung zu Quantenmechanik II, Hausübung 1.1:
Das Energiefunktional nimmt für die Form an.

> assume(b>=0);

> E := proc(u)
local zaehler,nenner,energie,b_min,energie_min,u_min,ueberlapp,ergebnis;
zaehler := simplify(int(u*(diff(u,rho$2) +2/rho*u),rho=0..infinity));
nenner := simplify(int(u^2,rho=0..infinity));
energie := simplify(zaehler/nenner);
b_min := solve(diff(energie,b),b);
energie_min := simplify(subs(b=b_min,energie));
u_min := simplify(subs(b=b_min,u/abs(sqrt(nenner))));
ueberlapp := simplify(int(2*rho*exp(-rho)*u_min,rho=0..infinity));
ergebnis := matrix(9,2,[
Int(u*(Diff(u,rho$2) +2/rho*u),rho=0..infinity)*`=`,zaehler,
Int(u^2,rho=0..infinity)*`=`,nenner,
E*[Phi]*`=`,-energie*E[H],
b[min]*`=`,b_min,
E[min]*`=`,-energie_min*E[H],
` =`,evalf(-energie_min)*E[H],
` u`[var]*`=`,u_min,
`<u`[0]*`|`*`u>=`,ueberlapp,
` =`,evalf(ueberlapp)]);
RETURN(evalm(ergebnis));
end:

> energy1 := E(rho*exp(-b*rho));

> energy2 := E(rho^2*exp(-b*rho));

> energy3 := E(rho/(b^2+rho^2));

> pp := plot([energy1[7,2],energy2[7,2],energy3[7,2]],
rho=0..6.5,color=[red,green,blue],linestyle=[1,2,3],labels=[rho,u]):
pp1 := PLOT(TEXT([5.3,0.7],`2 rho exp(-rho)`,ALIGNLEFT),
CURVES([[5.5,0.7],[6,0.7]],COLOR(RGB,1.0,0.0,0.0),LINESTYLE(1)),
TEXT([5.3,0.65],`9/4 sqrt(2) rho^2 exp(-3/2 rho)`,ALIGNLEFT),
CURVES([[5.5,0.65],[6,0.65]],COLOR(RGB,0.0,1.0,0.0),LINESTYLE(2)),
TEXT([5.3,0.6],`rho/(pi^2/16 + rho^2)`,ALIGNLEFT),
CURVES([[5.5,0.6],[6,0.6]],COLOR(RGB,0.0,0.0,1.0),LINESTYLE(3))):

> plots[display]([pp,pp1]);

>