## Sample Test Problems

1. A beam of 100 eV (kinetic energy) electrons is incident upon a potential step of height eV. Calculate the probability to be transmitted. Get a numerical answer.
2. * Find the energy eigenstates (and energy eigenvalues) of a particle of mass bound in the 1D potential . Assume is a positive real number. (Don't assume that has the units of energy.) You need not normalize the state.
Answer

3. * A beam of particles of wave-number (this means ) is incident upon a one dimensional potential . Calculate the probability to be transmitted. Graph it as a function of .
Answer
To the left of the origin the solution is . To the right of the origin the solution is . Continuity of at the origin implies . The discontinuity in the first derivative is

Transmission probability starts at zero for then approaches asymptotically for .
4. * A beam of particles of energy coming from is incident upon a 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.
5. * The Schrödinger equation for the one dimensional harmonic ocillator is reduced to the following equation for the polynomial :

a)
Assume and find the recursion relation for the coefficients .
b)
Use the requirement that this polynomial series must terminate to find the allowed energies in terms of .
c)
Find for the ground state and second excited state.
6. A beam of particles of energy coming from is incident upon a potential step in one dimension. That is for and for where is a positive real number.
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 step and the probability to be transmitted.
7. * A particle is in the ground state ( .) of a harmonic oscillator potential. Suddenly the potential is removed without affecting the particle's state. Find the probability distribution for the particle's momentum after the potential has been removed.
8. * A particle is in the third excited state (n=3) of the one dimensional harmonic oscillator potential.
a)
Calculate this energy eigenfunction, up to a normalization factor, from the recursion relations given on the front of the exam.
b)
Give, but do not evaluate, the expression for the normalization factor.
c)
At the potential is suddenly removed so that the particle is free. Assume that the wave function of the particle is unchanged by removing the potential. Write an expression for the probability that the particle has momentum in the range for . You need not evaluate the integral.
9. * The Schrödinger equation for the one dimensional harmonic oscillator is reduced to the following equation for the polynomial :

a)
Assume and find the recursion relation for the coefficients .
b)
Use the requirement that this polynomial series must terminate to find the allowed energies in terms of .
c)
Find for the ground state and second excited state.
10. * Find the energy eigenstates (and energy eigenvalues) of a particle of mass bound in the 1D potential .

Jim Branson 2013-04-22