Download Homework 3 - barnes report

Homework 3
Problem 1
In the ground state of the harmonic oscillator, what is
the probability (correct to three significant digits)
of finding the particle outside the classically allowed
region? (Hint: what are the minimum and maximum values
of the coordinates of the respective classical
oscillator with a given energy E?) Look in math tables
under “Normal distribution” or “Error function” for the
numerical value of the integral.
Problem 2
Consider the potential
(1),
which describes an elastic spring which can be extended
but not compressed. Using symmetry properties of
stationary wave functions of the harmonic oscillator,
and the boundary condition at
for the wave function
in the potential given by Eq.(1), find the stationary
wave functions and energy levels for this potential.
Hint: This requires some careful thought, but very
little actual computations
Problem 3.
a) Using properties of rising and lowering operators
calculate following quantities:
and
, where
are stationary
states of a quantum harmonic oscillator.
b) Calculate the following
and
Problem 4
Consider the double delta-function potential
where  and  are positive constants.
a) How many bound states do this potential have? Find
(graphically,
using a calculator) the energies of the


bound states for and
b) Consider scattering states in this potential and
determine the transmission coefficient. Use any kind of
graphing software available to you and plot the
transmission coefficient as a function of energy.
Problem 5
Find stationary states for a centered infinite square
well with a delta function at the center
Problem 6 Griffiths, Problem 2.38.
Problem 7 Griffiths, Problem 2. 47