## PHY456H1F: Quantum Mechanics II. Lecture 22 (Taught by Prof J.E. Sipe). Scattering (cont.)

Posted by peeterjoot on November 30, 2011

# Disclaimer.

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

# Scattering. Recap

READING: section 19, section 20 of the text [1].

We used a positive potential of the form of figure (\ref{fig:qmTwoL22:qmTwoL22fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL22fig1}

\caption{A bounded positive potential.}

\end{figure}

for

for

integrate these equations back to .

For

where both and are proportional to , dependent on .

There are cases where we can solve this analytically (one of these is on our problem set).

Alternatively, write as (so long as )

latex x x_2$}\end{array}\end{aligned} \hspace{\stretch{1}}(2.8)$

Now want to consider the problem of no potential in the interval of interest, and our window bounded potential as in figure (\ref{fig:qmTwoL22:qmTwoL22fig3})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL22fig3}

\caption{Wave packet in free space and with positive potential.}

\end{figure}

where we model our particle as a wave packet as we found can have the fourier transform description, for , of

Returning to the same coefficients, the solution of the Schr\”{o}dinger eqn for problem with the potential 2.8

For ,

where as illustrated in figure (\ref{fig:qmTwoL22:qmTwoL22fig4})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL22fig4}

\caption{Reflection and transmission of wave packet.}

\end{figure}

For

and

Look at

where

for , this is nonzero for .

so for

In the same way, for

What hasn’t been proved is that the wavefunction is also zero in the interval.

## Summarizing

For

latex x x_2$ (and actually also for (unproven))}\end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.19)$

for

latex x x_2$ }\end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.20)$

Probability of reflection is

If we have a sufficiently localized packet, we can form a first order approximation around the peak of (FIXME: or is this a sufficiently localized responce to the potential on reflection?)

so

Probability of transmission is

Again, assuming a small spread in , with for some

we have for

By constructing the wave packets in this fashion we get as a side effect the solution of the scattering problem.

The

are called asymptotic in states. Their physical applicability is only once we have built wave packets out of them.

# Moving to 3D

For a potential for as in figure (\ref{fig:qmTwoL22:qmTwoL22fig5})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL22fig5}

\caption{Radially bounded spherical potential.}

\end{figure}

From 1D we’ve learned to build up solutions from time independent solutions (non normalizable). Consider an incident wave

This is a solution of the time independent Schr\”{o}dinger equation

where

In the presence of a potential expect scattered waves. We’ll next be indentifying the nature of these solutions.

# References

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

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