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hw04sol_9_23

# hw04sol_9_23 - Introduction to Quantum and Statistical...

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Introduction to Quantum and Statistical Mechanics Homework 4 Solution Prob. 4.1 (a) (5 pts) ( 29 = = = * * * * * d b c a A d c b a A T , i.e., a = a* and d = d* where a and d are real, and b = c*. (b) (5 pts) The characteristic function is: 0 det = - - λ λ d c b a , ( 29 ( 29 0 2 = - - - c b a λ λ The eigenvalues are: ( 29 ( 29 2 4 2 2 c d a d a + - ± + = λ , and we can see that the eigenvalues are real since the term in the square root is always positive. (c) (10 pts) The corresponding eigenvectors are: (-c*, a - λ 1 ) and (-c, a - λ 2 ) which are orthogonal since the inner product is zero. Prob. 4.2 (a) (5 pts) ( 29 t x m k t x H , 2 ) , ( ˆ 2 0 2 ψ ψ = , and therefore, it is an eigenfunction of H ˆ . ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t x C t x k i B k t x k i A k t x p , exp exp , ˆ 0 0 0 0 0 0 ψ ϖ ϖ ψ - - - - = , and therefore, it is not an eigenfunction for p ˆ . (b) (5 pts) ψ (x, t) is the linear combination of two degenerate states, and is therefore stationary. (c) (10 pts) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 x ik AB x ik AB B A k t x k i B k t x k i A k t x k i B t x k i A t x p t x p 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 2 exp 2 exp exp exp exp exp , ˆ , * - - + - = - - - - × - - - + - - = = ϖ ϖ ϖ ϖ ψ ψ , 1

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which does not have explicit time dependence, i.e., the momentum is conserved for stationary states of the Hamiltonian. Prob. 4.3 (a) (5 pts) 2 2 cos 1 2 sin 2 0 2 0 2 2 L C dx L x n C dx L x n C L L = - = π π . L C 2 = for all n . (b) (5 pts) + = = 4 1 1 2 1 2 0 * L A dx L ψ ψ .
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