Physics 531
Problem Set #11
Due Thursday 12/11/14 (in my office)
~ be expressed in terms of its spherical tensor components,
1. Let the vector operator U
(1)
U1 =
Ux iUy
,
2
(1)
U0 = Uz
and similarly for the vector operator V~ . Define
(K)
TQ
=
X
(1)
h11
Physics 531
Problem Set #6
Due Thursday 10/16/14
1. Suppose that a one dimensional scattering potential V (x) satisfies V (x) = 0
for |x| > a and V (x) = V (x). It follows that for |x| > a the eigenstates with
E=h
2 k 2 /2m can be written as parity eigen
Physics 531
Problem Set #7
Due Tuesday 11/11/14
1. Derive the form of the propagator K(x, t; x0 , t0 ) for a one dimensional free particle
of mass m. Then generalize your result to three dimensions.
2. A fundamental quantity in statistical mechanics is th
Physics 531
Problem Set #5
Due Thursday 10/2/14
1. Evaluate the correlation function
C(t) = hxH (t)xH (0)i,
in the groundstate of a harmonic oscillator.
2. Suppose that A and B are two operators whose commutator is a c-number.
(a) Show that [A, f (B)] = [
Physics 531
Problem Set #10
Due Tuesday 12/2/2014
1. Consider a system made up of two spin 1/2 particles. Observer A specializes in
measuring the spin components of one of the particles (s1z , s1x , and so on), while
observer B measures the spin component
Physics 531
Problem Set #8
Due Tuesday 11/18/14
1. Using the fundamental commutation relations for angular momentum show that
~ =0
(a) [J 2 , J]
(b) [Jz , J ] = hJ
(c) J+ J = J 2 Jz2 + h
Jz
(d) Using the fact that J~ is the generator of infinitesimal rot
Physics 531
Problem Set #9
Due Tuesday 11/25/14
1. A particle of mass m moves in a three dimensional square well,
(
V (r) =
0 for r < R
for r R
Find the solution to the radial Schrodinger equation for angular momentum ` and
find the energy eigenvalues En
Physics 531
Problem Set #3
Due Thursday 9/18/14
q
1. Consider the Lagrangian, L = mc2 1 v 2 /c2 , where v = |d~r/dt|.
(a) Find the conjugate momentum p~.
(b) Find the Hamiltonian H(~r, p~). Is it familiar?
2. Suppose a free particle (H = p2 /2m) moving in
Physics 531
Problem Set #1
Due Friday 9/11/14
1. Bra-Ket algebra Use bra-ket notation for the following:
(a) Prove that tr(XY ) = tr(Y X)
(b) Prove that (XY ) = Y X .
(c) A function f (A) of an operator can be defined using the power series expansion of
f
Physics 531
Problem Set #2
Due Thursday 9/11/14
1. Degenerate 3 level system Consider a three-dimensional ket space. If a certain
set of orthonomal kets - say |1i, |2i and |3i - are used as the base kets the operators
A and B are represented by
a 0
0
A =