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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 | 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 )). Com- pute ( x | S (0) | ψ ) , and the limits lim u →∞ ( x | S ( u ) | ψ ) and lim u →−∞ ( x | S ( u ) | ψ ) . Answer: ( x | Π | x ′ ) = ( x | − x ′ ) = δ ( x + x ′ ) Π = integraldisplay dxdx ′ | x ) δ ( x + x ′ ) ( x ′ | = integraldisplay dx | x )(− x | { i 2 = integraldisplay dxdx ′ | x )(− x | x ′ )(− x ′ | = integraldisplay dxdx ′ | x )(− x ′ | δ ( − x − x ′ ) (− x ′ | = integraldisplay dx | x )( x | = 1 So Π 3 = Π 2 · Π = Π, which generalizes to Π 2 = braceleftbigg 1; n = even Π; n = odd Now we have S ( u ) = e Π u cosh u = 1 + Π u + Π 2 u 2 2 + Π 3 u 3 3! + ... cosh u = 1 + u 2 2 + u 4 4! + ... cosh u + Π u + u 2 3! + u 5 5! + ... cosh u = cosh u + Πsinh u cosh u = 1 + Πtanh u ( x | S ( u ) | ψ ) = ( x | ψ ) + tanh u (− x | ψ ) = ψ ( x ) + tanh uψ ( − x ) = ψ...
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HW13_Solutions - PHYS851 Quantum Mechanics I Fall 2008...

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