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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† − â) where â† |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