# Radial Wave Function for Spherical Squared Well Potential and $S$-Matrix

I have a problem with this exercise because I really don't know how to proceed. It's related with the "S-matrix".

In class we saw this example:

Consider the spherically symmetric potential:

$$V(r)=\begin{cases} -V_{0}, & 0\leq r\leq a\\ 0, & r>a \end{cases}$$

We know that for $r>a$ and $l=0$ we have

$$U_>''(r)+k^{2}U_>(r)=0 \qquad (1)$$

where $U(r):=rR(r)$, and $R(r)$ satisfies the radial part of the Schrödinger equation in spherical coordinates (for $l=0$):

$$-\frac{\hbar^{2}}{2m}\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{dR}{dr}\right)+\left[V(r)+\frac{\hbar^{2}l(l+1)}{2mr^{2}}\right]R=ER$$

In this case ($l=0$), from (1) we have the solution (for $r>a$)

$$U_>(r)=A e^{ikr}+Be^{-ikr} \qquad (2)$$

Or

$$U_{>}(r)=\frac{i}{2}\left[e^{-ikr}-S(k)e^{ikr}\right] \qquad (3)$$

Since we saw this really quick in class, I have 3 questions:

1 How do I manipulate (2) in order to obtain (3)?

After writing this, my teacher said the classical "it can be shown that:"

$$S(k)=\frac{-ik\sin qa-q\cos qa}{ik\sin qa-q\cos qa}e^{-2ika} \qquad (4)$$

then, my second question is:

2 How do I obtain (4)? I've been searching for this topic in my books (Griffiths, Schaum's Outlines) but I don't find anything related. Our reacher said that the function $S(k)$ is related with the "S-Matrix".

3 Could you suggest me a book where I can study this, please?

• The difference between $(2)$ and $(3)$ is just a global constant, so this will not change the physical state, if the wavefunction is normalisable, the value of this constant will be given by stating that the total probability of finding the particle is $1$. You obtain $(4)$ by considering that $U_<(r)$ and $U_>(r)$, and their derivatives, are continuous at $r=a$. A "S-Matrix" makes the link between "incoming" particles and "outcoming" particles. In your case, this is just the equation $(3)$ – Trimok Nov 27 '13 at 10:52
• @Trimok you are right! Could you please write this as an answer so I can choose it like the best answer. Also, could you suggest me a basic book (bachelors level) where I can find more about the $S$-Matrix (or "$S$-Function") in this context, please? – Ana S. H. Nov 27 '13 at 20:32
• For the S-matrix in this context, I have no precise book reference, but try this paper, pages $20-24$ – Trimok Nov 27 '13 at 20:45