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University of California at San Diego – Department of Physics – TA: Shauna Kravec Quantum Mechanics C (Physics 130C) Winter 2015 Worksheet 5 Announcements • The 130C web site is: http://physics.ucsd.edu/∼mcgreevy/w15/ . Please check it regularly! It contains relevant course information! • Office hours are 2:30-3:30PM but I’m available upon request. Grab your homework! Problems 1. Try it out! 1 3 4 3 4 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. Purity Define again the state |ψi = √1 (|0i 2 + eiφ |1i) as well as ρβ = 12 (|0ih0| + |1ih1|) (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. 1