cohen - Quantum Mechanics 1 Homework 5 Ben Sauerwine Due...

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Quantum Mechanics 1 Homework 5 Ben Sauerwine Due Friday October 14, 2005 1) (Cohen-Tannoudji) In a two-dimensional vector space, consider the operator whose matrix in an orthonormal basis { } 2 , 1 is written = 0 0 i i y σ a) Is y Hermitian? Calculate its eigenvalues and eigenvectors, giving their normalized expansion in terms of the { } 2 , 1 basis. y y i i i i = = = + + 0 0 0 0 So this operator is Hermitian. Its eigenvalues are the roots of {} 1 , 1 : 1 det 2 = λλ λ i i And so its eigenvectors are () 2 1 2 1 1 2 1 0 0 1 1 1 : 1 2 1 2 1 1 2 1 0 0 1 1 1 : 1 + = = = + = ⎡− = = i i i Null i i Null i i i Null i i Null b) Calculate the matrices that represent the projectors onto these eigenvectors and verify that they satisfy the orthogonality and closure relations. [] = + = + = = = = = ⎡− = 1 0 0 1 1 1 2 1 1 1 2 1 0 0 0 0 1 1 1 1 2 1 2 1 1 1 2 1 1 1 2 1 ; 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 i i i i e e e e closed i i i i e e e e orthogonal i i i i e e i i i i e e
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c) Repeat parts a and b for these matrices: = 3 2 2 2 i i M eigenvalues: {} 4 , 1 : 4 5 3 2 2 2 det 2 + = λλ λ i i eigenvectors: () 2 2 1 3 1 2 3 1 0 0 2 1 1 2 2 2 : 4 2 1 2 3 1 1 2 3 1 0 0 2 1 2 2 2 1 : 1 + = = = + = = = i i i Null i i Null i i i Null i i Null projectors: [] = + = + = = = = = = 1 0 0 1 2 2 2 1 3 1 1 2 2 2 3 1 0 0 0 0 2 2 2 1 1 2 2 2 3 1 3 1 2 2 2 1 3 1 2 2 3 1 1 2 2 2 3 1 1 2 1 2 3 1 4 4 1 1 4 4 1 1 4 4 1 1 i i i i e e e e closed i i i i e e e e orthogonal i i i i e e i i i i e e = 0 2 0 2 0 2 0 2 0 2 i L y h eigenvalues:
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() {} iC iC C C C C C C C C C i 2 , 2 , 0 : 4 2 2 2 2 0 2 2 0 2 det 2 2 2 2 2 + = + = = λλ λ h eigenvectors: 3 2 2 1 2 1 1 2 1 2 1 0 0 0 2 1 0 1 0 1 2 2 0 2 2 2 0 2 2 : 2 3 2 2 1 2 1 1 2 1 2 1 0 0 0 2 1 0 1 0 1 2 2 0 2 2 2 0 2 2 : 2 3 1 2 1 1 0 1 2 1 0 0 0 0 1 0 1 0 1 0 2 0 2 0 2 0 2 0 : 0 + + = ⎡ − = = + = = =
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cohen - Quantum Mechanics 1 Homework 5 Ben Sauerwine Due...

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