Chemistry 3730 assignment 10 solutions

The potential is

> V := x -> A*abs(x);

The trial wavefunction is

> phi := x -> exp(-b*x^2);

where b is the variational parameter. Maple will need to know that b is positive:

> assume(b>0);

The variational energy is . Since , the numerator of this expression is . We'll compute these terms one at a time:

> Kip := int(phi(x)*(-hbar^2/(2*m))*diff(phi(x),x$2),
  x=-infinity..infinity);

> Vip := int(phi(x)*V(x)*phi(x),x=-infinity..infinity);

> ip := int(phi(x)^2,x=-infinity..infinity);

> Evar := (Kip+Vip)/ip;

To find the best value of b, we minimize the variational energy:

> s1:=solve(diff(Evar,b)=0,b);

The complex solutions are artifacts. Therefore b is the first solution returned above.

> assign(b=s1[1]);
> simplify(b);

Now use the hints:

> assume(A>0); assume(hbar>0); assume(m>0);

The variational energy is then

> simplify(Evar);

The normalization factor is

> simplify(1/sqrt(ip));