HW9 - PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 9: DUE FRIDAY!!! 1. Prove [ A, ¯ H ] = A and [ A † , ¯ H ] = − A † , for the harmonic oscillator. 2. Verify that the transformations X → λ ¯ X and P → planckover2pi1 λ ¯ P transforms the Hamiltonian as ¯ H = 1 2 ( ¯ X 2 + ¯ P 2 ) . Derive the required length scale λ , and show that H = planckover2pi1 ω ¯ H . 3. Use the recursion relation ψ n ( x ) = radicalbigg 2 n x λ ψ n- 1 ( x ) − radicalbigg n − 1 n ψ n- 2 ( x ) to compute ψ 2 ( x ), ψ 3 ( x ), and ψ 4 ( x ). Make plots of the probability densities for the first five eigen- states. 4. Using the relation ( p | P = p ( p | , derive the recursion relation for the momentum space wavefunctions ψ n ( p ) = ( p | n ) . Follow the same procedure used in lecture for ψ n ( x ). Also find the ground state and first-excited states in momentum representation ψ ( p ), and ψ 1 ( p ). Compare your results to the results for position representation, and use the similarity to deduce the general expression for the...
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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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