University of California at San Diego – Department of Physics – TA: Shauna Kravec
Quantum Mechanics C (Physics 130C) Winter 2014
Worksheet 5
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!
• Make sure to attempt all the homework problems! My office hours are tomorrow from
1100-1200 and I’m available longer upon request.
Problems
1. Try it out!
3
4
3
4
1
4
3
4
Consider the following operators: ρa =
ρb =
1
7
− 27
− 27
ρc =
4
7
1
2
− 34
i 43
1
2
Explain why each can’t represent a physical state.
1
4√
−i 3
4
Consider the following operators: ρ1 =
√ !
i 3
4
3
4
2
ρ2 =
7
0
0
5
7
0 0
ρ3 =
0 1
Which of these can possibly represent a pure state?
Hint: If ρ is pure it must be a projector onto some state.
2. Tracing
Recall the trace of an operator Tr [A] =
P
m hm|A|mi
for the some basis set {|mi}
Prove that this definition is independent of basis.
Prove the cycle property: Tr [ABC] = Tr [BCA] = Tr [CAB]
3. Junketsu
Define again the state |ψi =
√1 (|0i
2
+ eiφ |1i) as well as ρβ = 12 (|0ih0| + |1ih1|)
1
(a) Write the density matrix ρψ associated with |ψi
(b) Show that for both the states hZi = 0
(c) Define the purity of a state as Tr [ρ2 ]. Prove that this equal to 1 if ρ is pure.
Compute it for both ρψ and ρβ .
(d) Compute hXi with the above density matrices.
2