Problem Set 1

Problem Set 1 - starts off at t = 0 in an eigenstate of σ...

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PHYSICS 230 – PROBLEM SET 1 1. Linear Algebra Practice Remind yourself what a projection operator is. Show that the matrix M defined below is a projection operator by explicit diagonalization. M = 1 3 1 1 1 1 1 1 1 1 1 Show that M 2 = M as required for a projection operator. Write M in bra ket notation in the eigenvector basis 2. Consider the Walsh Hadamard operator W specified by the matrix below. Check that W is hermitian and unitary. Find its eigenvalues and eigenvectors. Imagine a series of experiments that measure σ z , then W then σ z again. What are the possible outcomes with their probabilities? Assume the initial state is an eigenstate of σ z . W = 1 2 1 1 1 - 1 ! Assume ¯ hW/t 0 is the Hamiltonian for a system. Imagine the system
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Unformatted text preview: starts off at t = 0 in an eigenstate of σ z with eigenvalue +1. Let the system evolve for time t . What is the probability for measuring the different eigenvalues of σ z ? 3. Read the derivation of the Uncertainty Principle in Shankar, pages 237-239. Take the specific case of Ω = σ z , Λ = σ x , | ψ > = c 1 | ↑ > + c 2 | ↓ >, < ψ | ψ > = 1 . (1) Evaluate the Uncertainty Principle inequality for general (normalized) c 1 and c 2 . Do the same thing for Λ = W with W defined above. Please note, we are asking that you compute both sides of (9.2.13) for this specific example and show that the inequality is satesfied....
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