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# 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 | Π | x ′ ) and use it to simplify the expression Π = integraltext dxdx ′ | x )( x | Π | x ′ )( x ′ | . Use this to compute Π 2 , Π 3 , and Π n . From these results find an expression for S ( u ) = exp[Π u ] cosh u in the form f ( u )+ g ( u )Π. What is ( x | S ( u ) | ψ ) ? Express your answer in terms of ψ even ( x ) = 1 2 ( ψ ( x ) + ψ ( − x )) and ψ odd ( x ) = 1 2 ( ψ ( x ) − ψ ( − x )). Compute ( x | S (0) | ψ ) , and the limits lim u →∞ ( x | S ( u ) | ψ ) and lim u →−∞ ( x | S ( u ) | ψ ) . 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 mω 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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