Assignment 6 solutions

We will need the 3s, 3p0 and 3d0 wavefunctions:

> rho := r -> r/a0;

> psi3s := (r,theta,phi) ->
> 1/243*sqrt(3/(Pi*a0^3))*(27-18*rho(r)+2*rho(r)^2)*exp(-rho(r)/3);

> psi3p0 := (r,theta,phi) ->
> 1/81*sqrt(2/(Pi*a0^3))*rho(r)*(6-rho(r))*exp(-rho(r)/3)*cos(theta);

> psi3d0 := (r,theta,phi) ->
> 1/486*sqrt(6/(Pi*a0^3))*rho(r)^2*exp(-rho(r)/3)*(3*cos(theta)^2-1);

> assume(a0,positive);

Let's now calculate the radial probability densities:

> rpd3s :=
> int(int(psi3s(r,theta,phi)^2*r^2*sin(theta),theta=0..Pi),phi=0..2*Pi);

> rpd3p :=
> int(int(psi3p0(r,theta,phi)^2*r^2*sin(theta),theta=0..Pi),phi=0..2*Pi);

> rpd3d :=
> int(int(psi3d0(r,theta,phi)^2*r^2*sin(theta),theta=0..Pi),phi=0..2*Pi);

Verification: If we did everything right, the probability densities should integrate out to 1:

> int(rpd3s,r=0..infinity);

> int(rpd3p,r=0..infinity);

> int(rpd3d,r=0..infinity);

We can go ahead and plot the radial probability densities:

> plot(subs(a0=1,rpd3s),r=0..30);

> plot(subs(a0=1,rpd3p),r=0..30);

> plot(subs(a0=1,rpd3d),r=0..30);