851HW3_09 - 1 2 i 1 − 2 i 2 − 2 i 1 2 i 1 Find the...

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PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 3: Fundamentals of Quantum Mechanics 1. [10pts] The trace of an operator is defined as Tr { A } = m ( m | A | m ) , where {| m )} is a suitable basis set. (a) Prove that the trace is independent of the choice of basis. (b) Prove the linearity of the trace operation by proving Tr { aA + bB } = aTr { A } + bTr { B } . (c) Prove the cyclic property of the trace by proving Tr { ABC } = Tr { BCA } = Tr { CAB } . 2. Consider the system with three physical states {| 1 ) , | 2 ) , | 3 )} . In this basis, the Hamiltonian matrix is: H =
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Unformatted text preview: 1 2 i 1 − 2 i 2 − 2 i 1 2 i 1 Find the eigenvalues { ω 1 ,ω 2 ,ω 3 } and eigenvectors {| ω 1 A , | ω 2 A , | ω 3 A} of H . Assume that the initial state of the system is | ψ (0) A = | 1 A . Find the three components a 1 | ψ ( t ) A , a 2 | ψ ( t ) A , and a 3 | ψ ( t ) A . Give all of your answers in proper Dirac notation. 3. Cohen-Tannoudji: pp 203-206: problems 2.2, 2.6, 2.7 4. Cohen-Tannoudji ;pp341-350: problem 3.14...
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