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PHYS 343 Assignment 3 Due Feb 8 1:30 PM

1.

Analyze the odd bound-state wavefunctions for the finite square well potentialdescribed by

>

L) that satisfy the

appropriate boundary conditions at x = 0 and x = . Enforce the proper matching

conditions at x = L to find an equation for the allowed energies in the system. Are

there conditions for which no solution in possible? Explain.

3. A particle with energy E is incident on a rectangular potential barrier with height

Vo (E < Vo) and width a. The transmitted wave function has the form

in the region x > a. Show that the probability density in the

region of the barrier (0

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4. (Serway 7-2) Consider the step potential

>

U. (a) Examine the Schrdinger equation to the left of the

step to find the form of the solution in the range x < 0. Do the same to the right ofthe step to obtain the solution form for x > 0. Complete the solution by enforcing

whatever boundary conditions may be necessary. (b) Obtain an expression for thereflection coefficient R in this case, and show that it can be written in the form

2

21

2

21

)(

)(

kk

kkR

+

=

where k1and k2are the wavenumbers for the incident and transmitted waves,

respectively. Also write an expression for the transmission factor T using the sumrule obeyed by these coefficients. (c) Evaluate R and T in the limiting cases of

. Are the results sensible? Explain. (This situation is

analogous to the partial reflection and transmission of light striking an interface

separating two different media.)

EUE and

5. (Serway 7-17) The attempt frequency of an alpha particle to escape the nucleus is

the number of times per second it collides with the nuclear barrier. Estimate the

collision frequency in the tunneling model for the alpha decay of thorium,assuming the alpha behaves like a true particle inside the nucleus with total

energy equal to the observed kinetic energy of decay. The daughter nucleus for

this case (radium) has Z=88 and a radius of 9.00 fm. Take the overall nuclearbarrier 30.0 MeV, measured from the bottom of the nuclear well to the top of the

Coulomb barrier (See Fig. 7.8).