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University of California at San Diego – Department of Physics – TA: Shauna Kravec Quantum Mechanics C (Physics 130C) Winter 2014 Worksheet 4 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! • Remember my office hour is tomorrow from 1100-1200! Collect your homework! Inform me if things are wrong! Problems 1. Quis Custodiet Ipsos Custodes? Consider a two state system with basis vectors {|0i, |1i}. We are going to evolve the 0 −i system according the Hamiltonian H = ω2 Y where Y is the Pauli matrix . i 0 (a) What is the unitary operator associated with time evolution? Given an initial prepared state of |ψ0 i = |0i. Write an expression for |ψ(t)i. (b) What is the probability of measuring |1i? Consider a time interval δt ω1 so that the Hamiltonian barely has time to respond. Expand the probability to O(t3 ). Hint: You may wish to use sin2 (x) = 12 (1 − cos(2x)). (c) Consider measuring the system repeatedly over the interval [0, T ] where T δt. Space these measurements evenly as Tn , 2T , · · · What is the probability of n measuring |1i at any given time? (d) If I fail to measure |1i I need to start the evolution over again. The chance of measuring |1i at T is then given by PT [|1i] = (Pδt [|1i])n Evaluate this probability in the limit of n → ∞. This is called the quantum Zeno effect. 1 (e) Let me contextulaize the result above. We’ve made some sketchy arguments above in: our ability to measure arbitrarily quickly, and the assumption of nocorrelations between measurements. One can consider a more careful analysis where measurement is done by coupling the system to a device and one finds random ’jumps’ to the state |1i. This is an important problem in the field of continuous quantum measurement. 2. C-NOT Evil In quantum mechanics we use tensor products to describe Hilbert spaces of combined systems. Let HA and HB be Hilbert spaces spanned by {|0A i, |1A i} and {|0B i, |1B i} respectively. Let us construct the following operator on their tensor product space HAB : UAB ≡ |0A ih0A | ⊗ 1 B + |1A ih1A | ⊗ XB Where XB can be written as XB = |0B ih1B | + |1B ih0B |. It’s a Pauli X acting on HB . Write a matrix representation of UAB in the following basis for HAB : {|0A i ⊗ |0B i, |0A i ⊗ |1B i, |1A i ⊗ |0B i, |1A i ⊗ |1B i} ≡ {|0A 0B i, |0A 1B i, |1A 0B i, |1A 1B i} What are the eigenstates of this operator? In which of these states are A and B entangled?1 1 Can you write them as a product of states from HA and HB independently? 2