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**Unformatted text preview: **Introduction to Quantum and Statistical Mechanics Homework 2 Solution 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) (5 pts) ∑ ∑ ∑ ∑ ∑ = = = = = = = = ≡ n i i i n i i i i n i i i n i i i n i i i a c a c c c A c A A 1 2 1 1 * * 1 1 * * | | | | * φ φ φ φ ψ ψ ∑ ∑ ∑ ∑ ∑ = = = = = = = = ≡ n i i i n i i i i n i i i n i i i n i i i a c a c c c A c A A 1 2 2 1 2 1 * * 1 2 1 * * 2 2 | | | | * φ φ φ φ ψ ψ We have used the orthogonal properties of φ i in the above derivation. Since c i is properly normalized, we also know: 1 1 2 = ∑ = n i i c ( 29 2 1 2 1 2 2 2 2 2 Δ - =- = ∑ ∑ = = n i i i n i i i a c a c A A A (b) (10 pts) If A is Hermitian, then a i is real. | c i | 2 is always real even if c i is complex. From the Cauchy inequality, we can assign x i =|c i | and y i = |c i |a i , remembering that 1 1 2 = ∑ = n i i c , we can then observe that the Cauchy inequality means ∆ A ≥ . (c) (5 pts) The only case when the Cauchy inequality has the equal sign is n = 1. This implies that ) , ( t x ψ is at one of the eigenfunction....

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