## Homework 3

1. A general one dimensional scattering problem could be characterized by an (arbitrary) potential which is localized by the requirement that for . Assume that the wave-function is

Relating the outgoing'' waves to the incoming'' waves by the matrix equation

show that

Use this to show that the matrix is unitary.

2. Calculate the matrix for the potential

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 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

and that the particle has mass and energy . Estimate the lifetime of the particle inside this potential.

Jim Branson 2013-04-22