## 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).

**Why?**

\begin{itemize}

\item Simplifies dynamics

take

Then

\item “Applied field”‘ can often be thought of a driving the system from one eigenstate to another.

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL2fig1}

\caption{qmTwoL2fig1}

\end{figure}

\item Stat mech.

In thermal equilibrium

where

and

\end{itemize}

# Variational principle

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 [1].

We define the functional

If where is the normalized ground state, then

## Examples.

### Hydrogen atom

where

Here is the reduced mass.

We know the exact solution:

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL2fig2}

\caption{qmTwoL2fig2}

\end{figure}

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL2fig3}

\caption{qmTwoL2fig3}

\end{figure}

estimate

Or guess shape

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL2fig4}

\caption{qmTwoL2fig4}

\end{figure}

Using the trial wave function

find

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL2fig5}

\caption{qmTwoL2fig5}

\end{figure}

Minimum at

So

maybe not too bad…

### Helium atom

Assume an infinite nuclear mass with nucleus charge

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL2fig6}

\caption{qmTwoL2fig6}

\end{figure}

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

where

with

This is the solution to

Now we want to put back in the electron electron repulsion, and make an estimate.

Trial wavefunction

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 .

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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