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hw02 - Introduction to Quantum and Statistical Mechanics...

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1 Introduction to Quantum and Statistical Mechanics Homework 2 Prob. 2.1. We will observe the variance of an operator through the wave functions expanded by the eigenfunctions of the operator. The mathematical form is very helpful when we choose only a few eigenfunctions for expansion in the later finite-base approximation. Give an operator A with eigenfuntions of φ i and corresponding eigenvalues of a i . For a particle in an arbitrary state described by = = n i i i c t x 1 ) , ( φ ψ , where c i ’s are properly normalized, and φ i forms an orthogonal set, i.e., ij j i δ φ φ = | (a) Express the variance by summation of terms with c i and a i . (5 pts) (b) Prove that if A is Hermitian, the variance is always positive. You will need to use the Cauchy-Schwarz inequality: = = = n i i n i i n i i i y x y x 1 2 1 2 2 1 when x i and y i are real numbers. (10 pts) (c) Prove that if A is Hermitian, the variance is zero only if ψ (x,t)

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