University of California at San Diego – Department of Physics – TA: Shauna Kravec
Quantum Mechanics C (Physics 130C) Winter 2014
Worksheet 3
Please read and work on the following problems in groups of 3 to 4.
Solutions will be posted on the course webpage.
Announcements
• The 130C web site is:
http://physics.ucsd.edu/∼mcgreevy/w14/ .
Please check it regularly! It contains relevant course information!
• If you guys have any specific interests in topics please let me know! Remember my
office hour is tomorrow from 1100-1200.
Problems
1. Relative Not Global Phases
Consider a two state system with basis vectors {|0i, |1i} prepared in the state |ψi =
√1 (|0i + eiθ |1i).
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(a) What are the probabilities associated with measuring in the {|0i, |1i} basis?
(b) Consider the states |+i ≡ √12 (|0i + |1i) and |−i ≡ √12 (|0i − |1i).
What are the probabilities associated with measuring in this basis?
2. Building Bloch’s Theorem
Consider a 1D Hamiltonian with a periodic potential V (x) = V (x + na) for n ∈ Z and
a the lattice spacing.
(a) Define the operator T n by T n |xi = |x + nai. Show this is a symmetry of the
system.
(b) Show that any eigenfunctions of this systems can be chosen to obey
ψk (x − a) = eika ψk (x)
Recall that T |ki = eika |ki.
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(1)
(c) What does this imply about the probability distribution || ψ(x) ||2 ?
Infer from this that ψ(x) = eikx u(x) where u(x) = u(x + a).
3. A Theorem of Kramer
Most symmetries are unitary. Some are anti -unitary. Time reversal is one of the latter.
Denote this operator with T .
Something one might expect classically is that T xT −1 = x but T pT −1 = −p. It makes
things run backwards.
A similar story is true for angular and spin momentum. T ST −1 = −S
(a) Consider the action of T on a spin- 21 : T |+i where Sz |+i = 12 |+i.
Show that −iY K is a suitable representation
T where K implements complex
for 0
−i
conjugation 1 and Y is the Pauli matrix
.
i 0
What is T 2 in this case?
(b) Consider a system whose Hamiltonian H is time reversal symmetric. Show that
if |ψi is an eigenstate then T |ψi is as well.
Does this change the energy of the state?
(c) Imagine this is a spin- 12 system such that |ψi is an eigenstate of Sz as well.
How are |ψi and T |ψi related? Can they be the same?
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This is to make it anti-unitary, don’t worry about it
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