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Problem Set 1

# Problem Set 1 - P6572 HW#1 Due September 2 2011 1(a A...

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P6572 HW #1 Due September 2, 2011 1. (a) A hermitian operator with the spectral decomposition A = X a a | a ih a | . where | a i is an eigenstate of A with eigenvalue a . Now consider the operator f ( A ) where f ( x ) is some function of x . Find the matrix elements h a | f ( A ) | a 0 i in terms of the eigenvalues a and a 0 , and the function f ( a ). (b) Make use of the result in part (a) to show that det A = e Tr ln A where A has a finite number of discrete eigenvalues. This expression can be generalized to the case when the eigenvalues are continuous, provided the trace in the exponent exists. (c) Show that e iA is unitary (if A is Hermitian). 2. 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. The Pauli matrices are defined by σ 1 = 0 1 1 0 , σ 2 = 0 - i i 0 , σ 3 = 1 0 0 - 1 Consider the three matrices as components of a vector , and show that (a) e i ˆ n · ~σφ = cos φ + i ˆ n · sin φ where ˆ n is a unit vector.

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