Hi. Just going over some past exam papers and can't work out how to answer one of the questions as the course notes only use simple examples.
I need to use variational method to estimate the ground state energy of a physical system using a trial wavefunction:
phi = Aexp(-Br/2a) [B and a constants]
I'm then asked to normalise phi using spherical polar coordinates to show
A^2 = (B^3)/(8.pi.a^3)
I know this is done by finding the integral of phi*phi d^3r between 0<r<a, 0<theta<pi and 0<phi<2pi but I'm not sure what to do with the exponental function in this integral.
Anyone have any ideas? Thanks for your help.
Gah. Just found a standard integral at the bottom of the question:
Take A^2 out of integral as constant
Convert integral to r^2exp[-Br/a] sin theta phi dr dtheta dphi
Integral of r^2exp[-Br/a] = (2a^3)/B^3
multiply this by 2 from sin theta integral
multuply by 2pi from phi integral
multiply by the A^2
Equate to 1 for the normalisation and rearrange to get B^3/8pia^3
Simple. Sorry if any of you wasted any of your time trying to help without the standard integral.
In case you still need help with the variational part:
where is the trial state. In this case we have
The general method is to evaluate this, then minimize in terms of B and a. The problem is that you haven't mentioned what the Hamiltonian is.