# Disclaimer.

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

# Scattering theory.

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

Here’s (\ref{fig:qmTwoL21:qmTwoL21Fig1}) a simple classical picture of a two particle scattering collision

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig1}

\caption{classical collision of particles.}

\end{figure}

We will focus on point particle elastic collisions (no energy lost in the collision). With particles of mass and we write for the total and reduced mass respectively

so that interaction due to a potential that depends on the difference in position has, in the center of mass frame, the Hamiltonian

In the classical picture we would investigate the scattering radius associated with the impact parameter as depicted in figure (\ref{fig:qmTwoL21:qmTwoL21Fig2})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig2}

\caption{Classical scattering radius and impact parameter.}

\end{figure}

## 1D QM scattering. No potential wave packet time evolution.

Now lets move to the QM picture where we assume that we have a particle that can be represented as a wave packet as in figure (\ref{fig:qmTwoL21:qmTwoL21Fig3})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig3}

\caption{Wave packet for a particle wavefunction }

\end{figure}

First without any potential , lets consider the evolution. Our position and momentum space representations are related by

and by Fourier transform

Schr\”{o}dinger’s equation takes the form

or more simply in momentum space

Rearranging to integrate we have

and integrating

or

Time evolution in momentum space for the free particle changes only the phase of the wavefunction, the momentum probability density of that particle.

Fourier transforming, we find our position space wavefunction to be

To clean things up, write

for

where

Putting

we have

Observe that we have

## A Gaussian wave packet

Suppose that we have, as depicted in figure (\ref{fig:qmTwoL21:qmTwoL21Fig4})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig4}

\caption{Gaussian wave packet.}

\end{figure}

a Gaussian wave packet of the form

This is actually a minimum uncertainty packet with

Taking Fourier transforms we have

For our wave packet will start moving and spreading as in figure (\ref{fig:qmTwoL21:qmTwoL21Fig5})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig5}

\caption{moving spreading Gaussian packet.}

\end{figure}

## With a potential.

Now “switch on” a potential, still assuming a wave packet representation for the particle. With a positive (repulsive) potential as in figure (\ref{fig:qmTwoL21:qmTwoL21Fig6}), at a time long before the interaction of the wave packet with the potential we can visualize the packet as heading towards the barrier.

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig6}

\caption{QM wave packet prior to interaction with repulsive potential.}

\end{figure}

After some time long after the interaction, classically for this sort of potential where the particle kinetic energy is less than the barrier “height”, we would have total reflection. In the QM case, we’ve seen before that we will have a reflected and a transmitted portion of the wave packet as depicted in figure (\ref{fig:qmTwoL21:qmTwoL21Fig7})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig7}

\caption{QM wave packet long after interaction with repulsive potential.}

\end{figure}

Even if the particle kinetic energy is greater than the barrier height, as in figure (\ref{fig:qmTwoL21:qmTwoL21Fig8}), we can still have a reflected component.

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig8}

\caption{Kinetic energy greater than potential energy.}

\end{figure}

This is even true for a negative potential as depicted in figure (\ref{fig:qmTwoL21:qmTwoL21Fig9})!

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig9}

\caption{qmTwoL21Fig9}

\end{figure}

Consider the probability for the particle to be found anywhere long after the interaction, summing over the transmitted and reflected wave functions, we have

Observe that long after the interaction the cross terms in the probabilities will vanish because they are non-overlapping, leaving just the probably densities for the transmitted and reflected probably densities independently.

We define

The objective of most of our scattering problems will be the calculation of these probabilities and the comparisons of their ratios.

**Question**. Can we have more than one wave packet reflect off. Yes, we could have multiple wave packets for both the reflected and the transmitted portions. For example, if the potential has some internal structure there could be internal reflections before anything emerges on either side and things could get quite messy.

# Considering the time independent case temporarily.

We are going to work through something that is going to seem at first to be completely unrelated. We will (eventually) see that this can be applied to this problem, so a bit of patience will be required.

We will be using the time independent Schr\”{o}dinger equation

where we have added a subscript to our wave function with the intention (later) of allowing this to vary. For “future use” we define for

Consider a potential as in figure (\ref{fig:qmTwoL21:qmTwoL21Fig10}), where for and .

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL21Fig10}

\caption{potential zero outside of a specific region.}

\end{figure}

We won't have bound states here (repulsive potential). There will be many possible solutions, but we want to look for a solution that is of the form

Suppose , we have

Defining

we write Schr\”{o}dinger’s equation as a pair of coupled first order equations

At this specifically, we “know” both and and have

This allows us to find both

then proceed to numerically calculate and at neighboring points . Essentially, this allows us to numerically integrate backwards from to find the wave function at previous points for any sort of potential.

# References

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