PHY456H1F: Quantum Mechanics II. Lecture 2 (Taught by Prof J.E. Sipe). Approximate methods.
Posted by peeterjoot on September 15, 2011
Peeter’s lecture notes from class. May not be entirely coherent.
Approximate methods for finding energy eigenvalues and eigenkets.
In many situations one has a Hamiltonian
Here is a “degeneracy index” (example: as in Hydrogen atom).
\item Simplifies dynamics
\item “Applied field”‘ can often be thought of a driving the system from one eigenstate to another.
\item Stat mech.
In thermal equilibrium
Consider any ket
(perhaps not even normalized), and where
but we don’t know these.
So for any ket we can form the upper bound for the ground state energy
There’s a whole set of strategies based on estimating the ground state energy. This is called the Variational principle for ground state. See section 24.2 in the text .
We define the functional
If where is the normalized ground state, then
Here is the reduced mass.
We know the exact solution:
Or guess shape
Using the trial wave function
maybe not too bad…
Assume an infinite nuclear mass with nucleus charge
ground state wavefunction
The problem that we want to solve is
Nobody can solve this problem. It is one of the simplest real problems in QM that cannot be solved exactly.
Suppose that we neglected the electron, electron repulsion. Then
This is the solution to
Now we want to put back in the electron electron repulsion, and make an estimate.
expect that the best estimate is for .
This can be calculated numerically, and we find
The comes from the kinetic energy. The is the electron nuclear attraction, and the final term is from the electron-electron repulsion.
The actual minimum is
Whereas the measured value is .
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.