## PHY456H1F: Quantum Mechanics II. Recitation 3 (Taught by Mr. Federico Duque Gomez). WKB method and Stark shift.

Posted by peeterjoot on October 28, 2011

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# WKB method.

Consider the potential

latex x \in [0,a]$} \\ \infty & \quad \mbox{otherwise} \\ \end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.1)$

as illustrated in figure (\ref{fig:qmTwoR3:qmTwoR3fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoR3fig1}

\caption{Arbitrary potential in an infinite well.}

\end{figure}

Inside the well, we have

where

With

We have

Where

Setting boundary conditions we have

Noting that we have , we have

So

At the other boundary

So we require

or

This is called the Bohr-Sommerfeld condition.

**Check** with .

We have

or

# Stark Shift

Time independent perturbation theory

where is the electric field.

To first order we have

and

With the default basis , and we have a 4 fold degeneracy

but can diagonalize as follows

FIXME: show.

where

We have a split of energy levels as illustrated in figure (\ref{fig:qmTwoR3:qmTwoR3fig2})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoR3fig2}

\caption{Energy level splitting}

\end{figure}

Observe the embedded Pauli matrix (FIXME: missed the point of this?)

Proper basis for perturbation (FIXME:check) is then

and our result is

# Adiabatic perturbation theory

Utilizing instantaneous eigenstates

where

We found

where

and

Suppose we start in a subspace

Now expand the bra derivative kets

To first order we can drop the quadratic terms in leaving

so

## A different way to this end result.

A result of this form is also derived in [1] section 20.1, but with a different approach. There he takes derivatives of

Bra’ing into this we have, for

or

so without the implied perturbation of we can from 4.36 write the *exact* generalization of 4.32 as

# References

[1] D. Bohm. *Quantum Theory*. Courier Dover Publications, 1989.

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