# Disclaimer.

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

# 3D Scattering.

READING: section 20, and section 4.8 of our text [1].

We continue to consider scattering off of a positive potential as depicted in figure (\ref{fig:qmTwoL23:qmTwoL23fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.4\textheight]{qmTwoL23fig1}

\caption{Radially bounded potential.}

\end{figure}

Here we have for . The wave function

is found to be a solution of the free particle Schr\”{o}dinger equation.

## Seeking a post scattering solution away from the potential

What other solutions can be found for , where our potential ? We are looking for such that

What can we find?

We split our Laplacian into radial and angular components as we did for the hydrogen atom

where

Assuming a solution of

and noting that

we find that our radial equation becomes

Writing

we have

or

Writing , we have

## The radial equation and its solution.

With a last substitution of , and introducing an explicit suffix on our eigenfunction we have

We’d not have done this before with the hydrogen atom since we had only finite . Now this can be anything.

Making one final substitution, we can rewrite 2.13 as

This is the spherical Bessel equation of order and has solutions called the Bessel and Neumann functions of order , which are

\begin{subequations}

\end{subequations}

We can easily calculate

and can plug these into 2.13 to verify that they are a solution. A more general proof looks a bit trickier.

Observe that the Neumann functions are less well behaved at the origin. To calculate the first few Bessel and Neumann functions we first compute

and

so we find

Observe that our radial functions are proportional to these Bessel and Neumann functions

Or

## Limits of spherical Bessel and Neumann functions

With denoting the double factorial, like factorial but skipping every other term

we can show that in the limit as we have

\begin{subequations}

\end{subequations}

(for the case, note that by definition).

Comparing this to our explicit expansion for in 2.18 where we appear to have a dependence for small it is not obvious that this would be the case. To compute this we need to start with a power series expansion for , which is well behaved at and then the result follows (done later).

It is apparently also possible to show that as we have

\begin{subequations}

\end{subequations}

## Back to our problem.

For we can construct (for fixed ) a superposition of the spherical functions

we want outgoing waves, and as , we have

\begin{subequations}

\end{subequations}

Put for a given we have

For

Making this choice to achieve *outgoing* waves (and factoring a out of for some reason, we have another wave function that satisfies our Hamiltonian equation

The coefficients will depend on for the incident wave . Suppose we encapsulate that dependence in a helper function and write

We seek a solution

where as

Note that for in general for finite , , is much more complicated. This is the analogue of the plane wave result

# Scattering geometry and nomenclature.

We can think classically first, and imagine a scattering of a stream of particles barraging a target as in

figure (\ref{fig:qmTwoL23:qmTwoL23fig2})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{qmTwoL23fig2}

\caption{Scattering cross section.}

\end{figure}

Here we assume that is far enough away that it includes no non-scattering particles.

Write for the number density

and

We want to count the rate of particles per unit time through this solid angle and write

The factor

is called the differential cross section, and has “units” of

(recalling that steradians are radian like measures of solid angle [2]).

The total number of particles through the volume per unit time is then

where is the total cross section and has units of area. The cross section his the effective size of the area required to collect all particles, and characterizes the scattering, but isn’t necessarily entirely geometrical. For example, in photon scattering we may have frequency matching with atomic resonance, finding , something that can be much bigger than the actual total area involved.

# Appendix

## Q: Are Bessel and Neumann functions orthogonal?

**Answer:** There is an orthogonality relation, but it is not one of plain old multiplication.

Curious about this, I find an orthogonality condition in [3]

from which we find for the spherical Bessel functions

Is this a satisfactory orthogonality integral? At a glance it doesn’t appear to be well behaved for , but perhaps the limit can be taken?

## Deriving the large limit Bessel and Neumann function approximations.

For 2.22 we are referred to any “good book on electromagnetism” for details. I thought that perhaps the weighty [4] would be to be such a book, but it also leaves out the details. In section 16.1 the spherical Bessel and Neumann functions are related to the plain old Bessel functions with

\begin{subequations}

\end{subequations}

Referring back to section 3.7 of that text where the limiting forms of the Bessel functions are given

\begin{subequations}

\end{subequations}

This does give us our desired identities, but there’s no hint in the text how one would derive 4.41 from the power series that was computed by solving the Bessel equation.

## Deriving the small limit Bessel and Neumann function approximations.

Writing the function in series form

we can differentiate easily

Performing the derivative operation a second time we find

It appears reasonable to form the inductive hypotheses

and this proves to be correct. We find then that the spherical Bessel function has the power series expansion of

and from this the Bessel function limit of 2.21a follows immediately.

Finding the matching induction series for the Neumann functions is a bit harder. It’s not really any more difficult to write it, but it is harder to put it in a tidy form that is.

We find

The general expression, after a bit of messing around (and I got it wrong the first time), can be found to be

We really only need the lowest order term (which dominates for small ) to confirm the small limit 2.21b of the Neumann function, and this follows immediately.

For completeness, we note that the series expansion of the Neumann function is

## Verifying the solution to the spherical Bessel equation.

One way to verify that 2.15a is a solution to the Bessel equation 2.14 as claimed should be to substitute the series expression and verify that we get zero. Another way is to solve this equation directly. We have a regular singular point at the origin, so we look for solutions of the form

Writing our differential operator as

we get

Since we require this to be zero for all including non-zero values, we must have constraints on . Assuming first that is non-zero we must then have

One solution is obviously . Assuming we have another solution for some integer we find that is also a solution. Restricting attention first to , we must have since for non-negative we have . Thus for non-zero we find that our function is of the form

It doesn’t matter that we started with . If we instead start with we find that we must have , so end up with exactly the same functional form as 4.55. It ends up slightly simpler if we start with 4.55 instead, since we now know that we don’t have any odd powered ‘s to deal with. Doing so we find

We find

Proceeding recursively, we find

With and the observation that

we have as given in 4.46.

If we do the same for the case, we find

and find

Flipping signs around, we can rewrite this as

For those values of we can write this as

Comparing to the small limit 2.21b, the term, we find that we must have

After some play we find

latex l \ge k$} \\ \frac{(-1)^{k-l+1}}{ (2k)!! (2 (k-l) -1)!! } & \quad \mbox{if } \\ \end{array}\right.\end{aligned} \hspace{\stretch{1}}(4.64)$

Putting this all together we have

FIXME: check that this matches the series calculated earlier 4.51.

# References

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

[2] Wikipedia. Steradian — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 4-December-2011]. http://en.wikipedia.org/w/index.php?title=Steradian&oldid=462086182.

[3] Wikipedia. Bessel function — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 4-December-2011]. http://en.wikipedia.org/w/index.php?title=Bessel_function&oldid=461096228.

[4] JD Jackson. *Classical Electrodynamics Wiley*. John Wiley and Sons, 2nd edition, 1975.