Problem Set 2
PHY 465 - Spring 2015
Assigned: Friday, Jan. 15 Due: Friday, Jan. 22
Reading: Shankar Ch.15.3-4,
Problems 1-5: Irreducible Tensor Operators
Shankar 15.3.1-5
Problem 6: Quantum Runge-Lenz Vector
For the hydrogen atom, the Hamiltonian is given by
e2
p2
− ,
H=
2m
r
and the quantum mechanical operator corresponding to the Runge-Lenz vector is
L×p−p×L
r
A=
+ m e2 .
2
r
Note that in order to obtain a Hermitian operator from the classical expression for A we
must make the substitution:
L×p−p×L
L×p→
.
2
a) Compute the following commutation relations:
[Li , Aj ] = ???
[Ai , Aj ] = ??? .
b) Show that
A · A = 2mE (L2 + h̄2 ) + m2 e4 .
This agrees with the classical result in the limit h̄ → 0, the term proportional to h̄2 is a
quantum correction.
Problem 7: Rotations in Four Dimensions
Consider a 4-dimensional space with coordinates (x, y, z, w).
a) Show that the operators
Li = ijk xj pk
Ki = w pi − xi pw ,
generate rotations in this space by showing that the transformations generated by these
operators leave the four dimensional radius, defined by R2 = x2 + y 2 + z 2 + w2 , invariant.
b) Compute the commutators
[Li , Kj ] = ???
[Ki , Kj ] = ??? .
and show they same as those found in part b) of Problem 9, up to an overall rescaling of Ai .
Please write down how many hours you spent on this problem set.