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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). 2 (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. 1 (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? 1 This is to make it anti-unitary, don’t worry about it 2