Homework 7

Problem 1:

(a) Calculate the transmission coefficient for a particle with mass m and kinetic energy E passing through the rectangular potential barrier
.

(b) Show that for E<<V0 and  the transmission coefficient can be written as   .

(c) Many heavy nuclei decay by emitting an alpha particle.  In a simple one-dimensional model, the potential barrier the alpha particles have to penetrate can be approximated by

where R0 is the radius of the nucleus and V0 is the barrier height for r0=R.  The energy E of the alpha particle can be assumed to be much smaller than V0.  For a non constant potential barrier the expression for the transmission coefficient found in part (b) can be used as a guide.  Assume that for E<<V0, we have 
   
The integration limits R1 and R2 are determined as solutions to the equation V(r)=E. Calculate the alpha transmission coefficient and the decay constant l, i.e. the decay probability per second.

 

Problem 2:

Let V(x)=¥ for x<0, V(x)=(1/2)mw2x2 for x>0.  Use the WKB approximation to find the energy levels of a particle of mass m in this potential.  Compare the WKB energies with the exact energies for this potential.

 

Problem 3:

Consider a potential field in one dimension with V(-x)=V(x).  The parity operator O has the property that Ou(x)=u(-x).  Find the eigenvalues of O and show that O is a constant of motion.