HW13 - PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK...

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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 )). Compute ( x | S (0) | ) , and the limits lim u ( x | S ( u ) | ) and lim u ( x | S ( u ) | ) . 2. [10 pts] Consider a general Hamiltonian of the form H = 1 2 m P 2 + V ( X ). Prove that T ( d ) HT ( d ) = P 2 2 m + V ( X d ). 3. [20 pts] Recall HW 9.6, which asked you to find the eigenstates and eigenvalues of H = 1 2 m P 2 + 1 2 m 2 X 2 + mgX a). Determine the value of d required so that T ( d ) HT ( d ) = H SHO + c , where...
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