Download 1 Quantum Mechanics Problem 2, 2017 Due: Wednesday, March 15

Quantum Mechanics
Problem 2, 2017
Due: Wednesday, March 15
every day of delay results in 20% reduction of the mark
Consider a quantum system in which an electron can move freely in the xy-plane while its
motion in the z-direction is restricted by the potential
U (x, z) =
mω 2 z 2
,
2
where m is the effective electron mass and ω is a parameter of the confining potential. A
uniform magnetic field B = (B, 0, 0) directed along the x-axis is applied to the system.
Use the gauge A = (0, −Bz, 0) for the vector potential.
A. Write down the Hamiltonian of the electron.
B. Using separation of variables,
ψ(x, y, z) = eipx x/h̄ eipy y/h̄ ϕ(z) ,
solve the Schrodinger equation and find energy levels of the electron. Notice that the motion
within the xy-plane is described by a plane wave.
C. Using the result obtained in part B determine the electron dispersion
p2y
p2x
= 0 +
+
,
2m∗x 2m∗y
Hence determine
(i) the formula for 0 which depends on the transverse quantum number,
and
(ii) formulas for the effective masses m∗x and m∗y .
D. Consider the following values of the parameters
m = 0.1me
h̄ω = 10 meV
B = 10 T esla ,
where me is the free electron mass. Find the values of m∗x and m∗y in units of me .