## PHY456H1F: Quantum Mechanics II. Lecture 9 (Taught by Prof J.E. Sipe). Adiabatic perturbation theory (cont.)

Posted by peeterjoot on October 9, 2011

# Disclaimer.

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

# Adiabatic perturbation theory (cont.)

We were working through Adiabatic time dependent perturbation (as also covered in section 17.5.2 of the text [1].)

Utilizing an expansion

where

and found

where

Look for a solution of the form

where

Taking derivatives of and after a bit of manipulation we find that things conveniently cancel

We find

so

With a last bit of notation

the problem is reduced to one involving only the sums over the terms, and where all the dependence on has been nicely isolated in a phase term

## Looking for an approximate solution.

**Try** an approximate solution

For this is okay, since we have which is consistent with

However, for we get

But

FIXME: I think we argued in class that the contributions are negligible. Why was that?

Now, are energy levels will have variation with time, as illustrated in figure (\ref{fig:qmTwoL9fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL9fig1}

\caption{Energy level variation with time}

\end{figure}

Perhaps unrealistically, suppose that our energy levels have some “typical” energy difference , so that

or

Suppose that is much less than a typical time over which instantaneous quantities (wavefunctions and brakets) change. After a large time

so we have our phase term whipping around really fast, as illustrated in figure (\ref{fig:qmTwoL9fig2}).

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL9fig2}

\caption{Phase whipping around.}

\end{figure}

So, while is moving really slow, but our phase space portion is changing really fast. The key to the approximate solution is factoring out this quickly changing phase term.

**Note** is called the “Berry” phase [2], whereas the part is called the geometric phase, and can be shown to have a geometric interpretation.

To proceed we can introduce terms, perhaps

and

FIXME: try this, or at least check the text to see if this is done in more detail.

## Degeneracy

Suppose we have some branching of energy levels that were initially degenerate, as illustrated in figure (\ref{fig:qmTwoL9fig3})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL9fig3}

\caption{Degenerate energy level splitting.}

\end{figure}

We have a necessity to choose states properly so there is a continuous evolution in the instantaneous eigenvalues as changes.

**Question: A physical example?**

FIXME: Prof Sipe to ponder and revisit.

# Fermi’s golden rule

See section 17.2 of the text [1].

Fermi originally had two golden rules, but his first one has mostly been forgotten. This refers to his second.

This is really important, and probably the *single most* important thing to learn in this course. You’ll find this falls out of many complex calculations.

Returning to general time dependent equations with

and

where

Example:

If , then to first order

and

Assume the perturbation vanishes before time .

**Reminder**. Have considered this using 3.26 for a pulse as in figure (\ref{fig:gaussianWavePacket})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{gaussianWavePacket}

\caption{Gaussian wave packet.}

\end{figure}

Now we want to consider instead a non-terminating signal, that was zero before some initial time as illustrated in figure (\ref{fig:unitStepSine}), where the separation between two peaks is .

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{unitStepSine}

\caption{Sine only after an initial time.}

\end{figure}

Our matrix element is

t > 0$} \\ 0 & \quad \mbox{if $t < 0$} \\ \end{array}\right.\end{aligned} \hspace{\stretch{1}}(3.27)$

Here the factor of 2 has been included for consistency with the text.

Plug this into the perturbation

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL9fig6}

\caption{ illustrated.}

\end{figure}

Suppose that

then

but the exponential has essentially no contribution

so for and we have

Similarly for as in figure (\ref{fig:qmTwoL9fig7})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL9fig7}

\caption{qmTwoL9fig7}

\end{figure}

then

and we have

# References

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

[2] Wikipedia. Geometric phase — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 9-October-2011]. //en.wikipedia.org/w/index.php?title=Geometric_phase&oldid=430834614.

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