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qmhw3 - PHY4221 Homework 3 1 Let y = Due Sep 25 2008 0 i i0...

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PHY4221: Homework 3 Due Sep. 25, 2008 1. Let σ y = 0 - i i 0 , calculate h σ y i and h σ y ) 2 i ≡ σ 2 y fi - σ y fi 2 for the state being an eigenvector of σ x = 0 1 1 0 with the positive eigenvalue. 2. The Hamiltonian of a three-level system is described by an operator H , with the matrix elements: h 11 = h 33 = g , h 22 = 0, h 12 = h 21 = g , h 23 = h 32 = g , where h ij ≡ h e i | H | e j i and | e i j are the base vectors. (a) A state vector is given by | ψ i = c 1 | e 1 i + c 2 | e 2 i + c 3 | e 3 i , find the expectation value of the energy of the system. (b) Find the eigenvalues and normalized eigenvectors of H . (c) Verify the completeness and orthogonality relation of the normalized eigenvectors. (d) Rewrite the Hamiltonian in the base vectors defined by eigenvectors of H , then find the form of the time evolution operator e - iHt/ ~ . (e) What is the general form of an operator
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