hw1 - | a a is an eigenstate of A with eigenvalue a . Now...

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HOMEWORK # 1 Physics 6572 Friday, 9/12/08; due 9/19/08 1. b) Show that U 1 f ( A ) U = f ( U 1 AU ). Therefore, if U is unitary, the unitary transform of the function of an operator A is the function of the transformed operator. 3. In this problem we will Fnd an expression for the determinant of an operator that is applicable when the operator has a continuous spectrum. a) A is a hermitian operator with the spectral decomposition A = s a a | a aA a | , (1) where
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Unformatted text preview: | a a is an eigenstate of A with eigenvalue a . Now consider the operator f ( A ) where f ( x ) is some function of x . ind the matrix elements A a | f ( A ) | a a in terms of the eigenvalues a and a , and the function f ( a ). b) Make use of the result in part (a) to show that det A = e Tr ln A , (2) where A has a Fnite number of discrete eigenvalues. This expression can be generalized to the case when the eigenvalues are continuous, provided the trace in the exponent exists....
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