- The 1D model of a crystal puts the following constraint on the wave number .

Assume that and plot the constraint as a function of . Plot the allowed energy bands on an energy axis assuming eV and the spacing between atoms is 5 Angstroms. - In a 1D square well, there is always at least one bound state.
Assume the width of the square well is .
By the uncertainty principle, the kinetic energy of an electron localized to that width is
.
How can there be a bound state even for small values of ?
- At a particle is in the one dimensional Harmonic Oscillator state
.
Is correctly normalized?
Compute the expected value of as a function of time
by doing the integrals in the representation.
- Prove the Schwartz inequality
.
(Start from the fact that
for any .
- The hyper-parity operator has the property that for any state .
Find the eigenvalues of for the case that it is not Hermitian and the case that it is Hermitian.
- Find the correctly normalized energy eigenfunction for the 1D harmonic oscillator.
- A beam of particles of energy
coming from is incident upon a double delta function potential
in one dimension. That is
.
- a)
- Find the solution to the Schrödinger equation for this problem.
- b)
- Determine the coefficients needed to satisfy the boundary conditions.
- c)
- Calculate the probability for a particle in the beam to be reflected by the potential and the probability to be transmitted.

Jim Branson 2013-04-22