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851HW8_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 8 Topics Covered: Algebraic approach to the quantized harmonic oscillator, coherent states. Some Key Concepts: Oscillator length, creation and annihilation operators, the phonon number oper- ator. 1. Start from the harmonic oscillator Hamiltonian H = 1 2 M P 2 + 1 2 2 X 2 . Make the change of variables X λ ¯ X , P planckover2pi1 λ ¯ P , and H planckover2pi1 2 2 ¯ H . Find the value of λ for which ¯ H = 1 2 ( ¯ X 2 + ¯ P 2 ) . 2. Write down the harmonic oscillator Hamiltonian in terms of ω , A , and A , and then write the com- mutation relation between A and A . Use these to derive the equation of motion for the expectation value a ( t ) = ( ψ ( t ) | A | ψ ( t ) ) . Solve this equation for the general case a (0) = a 0 . Prove that a * ( t ) := ( A ) = [ a ( t )] * . 3. Starting from ( x | X | n 1 ) = n - 1 ( x ), express X in terms of A and A , to derive a recursion relation of the form: φ n ( x ) = f n ( x ) φ n - 1 ( x ) + g n ( x ) φ n - 2 . (1) Starting from φ 0 ( x ) = [ πλ ] - 1 / 2 e - 1 2 ( x/λ ) 2
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