HW13_Solutions - PHYS851 Quantum Mechanics I, Fall 2008...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 13 1. [20 pts] Determine the matrix element ( x | | x ) and use it to simplify the expression = integraltext dxdx | x )( x | | x )( x | . Use this to compute 2 , 3 , and n . From these results find an expression for S ( u ) = exp[ u ] cosh u in the form f ( u )+ g ( u ). What is ( x | S ( u ) | ) ? Express your answer in terms of even ( x ) = 1 2 ( ( x )+ ( x )) and odd ( x ) = 1 2 ( ( x ) ( x )). Com- pute ( x | S (0) | ) , and the limits lim u ( x | S ( u ) | ) and lim u ( x | S ( u ) | ) . Answer: ( x | | x ) = ( x | x ) = ( x + x ) = integraldisplay dxdx | x ) ( x + x ) ( x | = integraldisplay dx | x )( x | { i 2 = integraldisplay dxdx | x )( x | x )( x | = integraldisplay dxdx | x )( x | ( x x ) ( x | = integraldisplay dx | x )( x | = 1 So 3 = 2 = , which generalizes to 2 = braceleftbigg 1; n = even ; n = odd Now we have S ( u ) = e u cosh u = 1 + u + 2 u 2 2 + 3 u 3 3! + ... cosh u = 1 + u 2 2 + u 4 4! + ... cosh u + u + u 2 3! + u 5 5! + ... cosh u = cosh u + sinh u cosh u = 1 + tanh u ( x | S ( u ) | ) = ( x | ) + tanh u ( x | ) = ( x ) + tanh u ( x ) =...
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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HW13_Solutions - PHYS851 Quantum Mechanics I, Fall 2008...

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