Homework 3

1. A general one dimensional scattering problem could be characterized by an (arbitrary) potential $V(x)$ which is localized by the requirement that $V(x)=0$ for $\vert x\vert>a$. Assume that the wave-function is
   
   $$\psi(x)=\left\{\matrix{Ae^{ikx}+Be^{-ikx}\qquad x<-a \cr Ce^{ikx}+De^{-ikx}\qquad x>a}\right.$$
   
   
   Relating the ``outgoing'' waves to the ``incoming'' waves by the matrix equation
   
   $$\pmatrix{C\cr B}=\pmatrix{S_{11} & S_{12}\cr S_{21} & S_{22}}\pmatrix{A\cr D}$$
   
   
   show that
   $$\begin{eqnarray*} \vert S_{11}\vert^2+\vert S_{21}\vert^2=1 \\ \vert S_{12}\vert^2+\vert S_{22}\vert^2=1 \\ S_{11}S_{12}^*+S_{21}S_{22}^*=0\\ \end{eqnarray*}$$
   
   
   
   
   Use this to show that the $S$ matrix is unitary.

2. Calculate the $S$ matrix for the potential
   
   $$V(x)=\left\{\matrix{V_0\qquad \vert x\vert<a \cr 0 \qquad \vert x\vert>a}\right.$$
   
   
   and show that the above conditions are satisfied.

3. The odd bound state solution to the potential well problem bears many similarities to the zero angular momentum solution to the 3D spherical potential well. Assume the range of the potential is $2.3\times 10^{-13}$ cm, the binding energy is -2.9 MeV, and the mass of the particle is 940 MeV. Find the depth of the potential in MeV. (The equation to solve is transcendental.)

4. Find the three lowest energy wave-functions for the harmonic oscillator.

5. Assume the potential for particle bound inside a nucleus is given by
   
   $$V(x)=\left\{\matrix{-V_0\qquad x<R \cr {\hbar^2\ell(\ell+1)\over 2mx^2}\qquad x>R}\right.$$
   
   
   and that the particle has mass $m$ and energy $e>0$. Estimate the lifetime of the particle inside this potential.