University of California at San Diego – Department of Physics – TA: Shauna Kravec
Quantum Mechanics C (Physics 130C) Winter 2014
Worksheet 6
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!
• This week’s discussion will serve as review for your midterm. Remember my office
hours are 1100-1230 tomorrow. Good luck; have fun!
Problems
1. Test of Mettle
(a) Write the operator Z in terms of its spectral decomposition.
(b) Write the operator X, using bra-ket notation, in the Z basis.
What is the action of X on {| ↑i, | ↓i}?
Consider the state |φi = 21 (| ↑↑i − | ↓↓i + i| ↓↑i − i| ↑↓i) in H = HA ⊗ HB
(c) Construct the density matrix ρφ in the basis of {| ↑↑i, | ↑↓i, | ↓↑i, | ↓↓i}. Write it
out in matrix form. Careful that | ↑↓i† = (| ↑iA | ↓iB )† = h ↓ |B h ↑ |A = h ↓↑ |
(d) Is the spin in HA entangled with the spin in HB ? Argue this in two ways. (Hint:
You’ll probably want to compute the reduced density matrix)
2. Harmonic Oscillator
A particle of mass m is in a one dimensional harmonic potential and has been prepared
.
(at t = 0) in an equal superposition state of energies ω2 and 3ω
2
(a) What is the most general expression for the wavefunction at t = 0? (Hint: One
should include also the possibility of a relative phase between states)
1
p mω
(b) Suppose I find that the average value of the momentum hp̂i at t = 0 is
2
What constraint does this impose on my general |ψ0 i?
√
p
√
(↠− â) where ↠|ni = n + 1|n + 1i and â|ni = n|n − 1i
Recall that p = i mω
2
(c) What is the time evolved |ψt i? What about hp̂it ?
p
(d) Compute the uncertainty ∆pt = hp̂2 it − hp̂i2t
2