## PHY456H1F: Quantum Mechanics II. Lecture 12 (Taught by Mr. Federico Duque Gomez). WKB Method

Posted by peeterjoot on October 21, 2011

# Disclaimer.

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

# WKB (Wentzel-Kramers-Brillouin) Method.

This is covered in section 24 in the text [1]. Also section 8 of [2].

We start with the 1D time independent Schr\”{o}dinger equation

which we can write as

Consider a finite well potential as in figure (\ref{fig:qmTwoL13:qmTwoL12fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL12fig1}

\caption{Finite well potential}

\end{figure}

With

we have for a bound state within the well

and for that state outside the well

In general we can hope for something similar. Let’s look for that something, but allow the constants and to be functions of position

In terms of Schr\”{o}dinger’s equation is just

We use the trial solution

allowing to be complex

We need second derivatives

and plug back into our Schr\”{o}dinger equation to obtain

For the first round of approximation we assume

and obtain

or

A second round of approximation we use 2.15 and obtain

Plugging back into 2.12 we have

or

If is small compared to

we have

Integrating

Going back to our wavefunction, if we have

or

FIXME: Question: the on the real exponential got absorbed here, but would not also be a solution? If so, why is that one excluded?

Similarly for the case we can find

**Validity**

\begin{enumerate}

\item V(x) changes very slowly small, and .

\item very far away from the potential .

\end{enumerate}

# Examples

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL12fig2}

\caption{Example of a general potential}

\end{figure}

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL12fig3}

\caption{Turning points where WKB won’t work}

\end{figure}

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL12fig4}

\caption{Diagram for patching method discussion}

\end{figure}

WKB won’t work at the turning points in this figure since our main assumption was that

so we get into trouble where . There are some methods for dealing with this. Our text as well as Griffiths give some examples, but they require Bessel functions and more complex mathematics.

The idea is that one finds the WKB solution in the regions of validity, and then looks for a polynomial solution in the patching region where we are closer to the turning point, probably requiring lookup of various special functions.

This power series method is also outlined in [3], where solutions to connect the regions are expressed in terms of Airy functions.

# References

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

[2] D.J. Griffiths. *Introduction to quantum mechanics*, volume 1. Pearson Prentice Hall, 2005.

[3] Wikipedia. Wkb approximation — wikipedia, the free encyclopedia, 2011. [Online; accessed 19-October-2011]. http://en.wikipedia.org/w/index.php?title=WKB_approximation&oldid=453833635.

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