HW9_Solutions - PHYS851 Quantum Mechanics I, Fall 2008...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 9: SOLUTIONS 1. Prove [ A, H ] = A and [ A , H ] = A , for the harmonic oscillator. Answer: [ A, H ] = [ A,A A + 1 / 2] = [ A,A A ] = AA A A AA = ( A A + 1) A A AA = A [ A , H ] = [ A ,A A ] = A A A A AA = A A A A ( A A + 1) = A 1 2. Verify that the transformations X X and P planckover2pi1 P transforms the Hamiltonian as H = 1 2 ( X 2 + P 2 ) . Derive the required length scale , and show that H = planckover2pi1 H . Answer: Start from H = P 2 2 m + 1 2 m 2 X 2 let X = X and P = ( planckover2pi1 / ) P , to give H = planckover2pi1 2 2 m 2 P 2 + 1 2 m 2 2 X 2 let m 2 2 = planckover2pi1 2 m 2 , which requires = radicalbigg planckover2pi1 m , to get m 2 planckover2pi1 2 H = 1 2 ( X 2 + P 2 ) Since m 2 planckover2pi1 2 = planckover2pi1 , this gives H = 1 2 ( X 2 + P 2 ) where we have defined H = planckover2pi1 H 2 3. Use the recursion relation n ( x ) = radicalbigg 2 n x n- 1 ( x ) radicalbigg n 1 n n- 2 ( x ) to compute 2 ( x ), 3 ( x ), and 4 ( x ). Make plots of the probability densities for the first five eigen- states. Answer: 2 ( x ) = x 1 ( x ) radicalbigg 1 2 ( x ) = 2 [ ] 1 / 2 x 2 2 e- x 2 2 2 1 [2 ] 1 / 2 e- x 2 2 2 = 1 [8 ] 1 / 2 parenleftbigg 4 x 2 2 2 parenrightbigg e- x 2 2 2 3 ( x ) = radicalbigg 2 3 x 2 ( x ) radicalbigg 2 3 1 ( x ) = radicalBigg 2 24 parenleftbigg 4 x 2 2 2 parenrightbigg x e- x 2 2 2 radicalBigg 4 3 x e- x 2 2 2 = 1 [48 ] 1 / 2 parenleftbigg 8 x 3 3 12 x parenrightbigg e...
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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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HW9_Solutions - PHYS851 Quantum Mechanics I, Fall 2008...

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