## PHY456H1F: Quantum Mechanics II. Lecture L24 (Taught by Prof J.E. Sipe). 3D Scattering cross sections (cont.)

Posted by peeterjoot on December 5, 2011

# Disclaimer.

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

# Scattering cross sections.

READING: section 20 [1]

Recall that we are studing the case of a potential that is zero outside of a fixed bound, for , as in figure (\ref{fig:qmTwoL24:qmTwoL22fig5})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{qmTwoL22fig5}

\caption{Bounded potential.}

\end{figure}

and were looking for solutions to Schr\”{o}dinger’s equation

in regions of space, where is very large. We found

For this will be something much more complicated.

To study scattering we’ll use the concept of probability flux as in electromagnetism

Using

we find

when

In a fashion similar to what we did in the 1D case, let’s suppose that we can write our wave function

and treat the scattering as the scattering of a plane wave front (idealizing a set of wave packets) off of the object of interest as depicted in figure (\ref{fig:qmTwoL24:qmTwoL24fig3})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{qmTwoL24fig3}

\caption{plane wave front incident on particle}

\end{figure}

We assume that our incoming particles are sufficiently localized in space as depicted in the idealized representation of figure (\ref{fig:qmTwoL24:qmTwoL24fig4})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{qmTwoL24fig4}

\caption{k space localized wave packet}

\end{figure}

we assume that is localized.

We suppose that

where this is chosen ( is built in this fashion) so that this is non-zero for large in magnitude and negative.

This last integral can be approximated

This is very much like the 1D case where we found no reflected component for our initial time.

We’ll normally look in a locality well away from the wave front as indicted in figure (\ref{fig:qmTwoL24:qmTwoL24fig5})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{qmTwoL24fig5}

\caption{point of measurement of scattering cross section}

\end{figure}

There are situations where we do look in the locality of the wave front that has been scattered.

Our income wave is of the form

Here we’ve made the approximation that . We can calculate the probability current

(notice the like term above, with ).

For the scattered wave (dropping factor)

We find that the radial portion of the current density is

and the flux through our element of solid angle is

We identify the scattering cross section above

We’ve been somewhat unrealistic here since we’ve used a plane wave approximation, and can as in figure (\ref{fig:qmTwoL24:qmTwoL24fig6})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{qmTwoL24fig6}

\caption{Plane wave vs packet wave front}

\end{figure}

will actually produce the same answer. For details we are referred to [2] and [3].

## Working towards a solution

We’ve done a bunch of stuff here but are not much closer to a real solution because we don’t actually know what is.

Let’s write Schr\”{o}dinger

instead as

where

where is really the particular solution to this differential problem. We want

and

# References

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

[2] A. Messiah, G.M. Temmer, and J. Potter. *Quantum mechanics: two volumes bound as one*. Dover Publications New York, 1999.

[3] JR Taylor. {\em Scattering Theory: the Quantum Theory of Nonrelativistic Scattering}, volume 1. 1972.

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