Problem Set 1
PHY 465 - Spring 2015
Assigned: Thursday, Jan. 8 Due: Friday, Jan. 15
Reading: Shankar Ch. 12.4-5, 15.1-2, “Notes on Levi-Civita Tensor”, posted on the course
webpage.
Problem 1: Levi-Civita Tensor
Shankar 12.4.1
Problem 2: Levi-Civita Tensor
From “Notes on Levi-Civita Tensor”, verify Eqs.(1.81)-(1.84) using the Levi-Civita tensor.
Problem 3: Angular Momentum and Generators of Rotations
a) Using the commutations for orbital angular momentum, L, and position, r, show that
eiL·Θ re−iL·Θ = r + Θ × r .
to lowest order in Θ.
b) Show that
 


x
eiLz θ  y  e−iLz θ
z


x

cos θ − y sin θ 

=  x sin θ + y cos θ  .


z
Problem 4: Commutation Relations for Tensors
The jth component of a vector operator, Vj , has the following commutation relation with
the ith component of the angular momentum operators, Ji :
[Ji , Vj ] = ih̄ ijk Vk .
What is the corresponding commutation relation for the second rank tensor
[Ji , Tjk ] = . . . ?
0
Recall that under rotations Tjk
= Rjl Rkm Tlm , where R is a rotation matrix.
Problem 5: Addition of Angular Momentum: A Simple Example
Shankar 15.1.2
Problems 6-10: Addition of Angular Momentum: The General Problem
Shankar 15.2.1-5
Please write down how many hours you spent on this problem set.