AMATHProblemSet2-2010

AMATHProblemSet2-2010 - Advanced Quantum Theory Amath 473...

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Unformatted text preview: Advanced Quantum Theory Amath 473, 673 / Phys 454 Problem Set 2 This assignment is due at the beginning of class on Wednesday, Oct 20, 2010. 1. Linear Algebra. a) Look up and write down the commutation relations for the three Pauli operators ( σ x , σ y , σ z ). b) Write down the standard 2x2 matrix representation of the Pauli operators. c) Show that the Pauli matrices are Hermitian (or symmetric to use the mathematical term). d) Give an explicit expression for the trace Tr( A ) of an operator A acting on a d-dimensional Hilbert space in terms of an O.N. basis {| φ k i} for that Hilbert space. How does this expression simplify in terms of the eigenvalues λ k of A ? e) For arbitrary linear operators A and B , prove the cyclic property of trace, i.e., show that Tr( ˆ A ˆ B ) = Tr( ˆ B ˆ A ) . 2. Infinite Dimensional Matrix Representations. Consider the lowering operator ˆ a which has an infinite dimensional matrix representation with matrix elements ( a ) lk = δ l ( k- 1) √ l where l,k ∈ { 1 , 2 , 3 ,... } . The adjoint of a , called the raising operator ˆ a † , has a matrix representation given by the transpose of the matrix for ˆ a . Define operators ˆ q = L (ˆ a...
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This note was uploaded on 01/17/2011 for the course AMATH 454 taught by Professor Joseph during the Fall '10 term at Waterloo.

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AMATHProblemSet2-2010 - Advanced Quantum Theory Amath 473...

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