p137bp6sol - Physics 137B Fall 2007 Moore Problem Set 6...

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Unformatted text preview: Physics 137B, Fall 2007, Moore Problem Set 6 Solutions 1. For t < 0 the particle is in the stationary state ψ ( t,x ) = r 2 L e- iE t/ ¯ h sin πx L . Since this is a box potential, (i.e. an infinite square well), the wavefunction vanishes outside of the interval 0 ≤ x ≤ L . At t = 0 the Hamiltonian changes abruptly to the form of a shifted harmonic oscillator, H = p 2 2 m + 1 2 k x- L 2 2 . The ground state and first excited state of this Hamiltonian are φ ( x ) = mω π ¯ h 1 / 4 e- mω ( x- L/ 2) 2 / 2¯ h , φ 1 ( x ) = mω π ¯ h 1 / 4 2 mω ¯ h 1 / 2 x- L 2 e- mω ( x- L/ 2) 2 / 2¯ h . To find the probability of observing the particle in either of these states, we need to calculate the overlap with the wavefunction at t = 0. (a) For the ground state we have d ≡ h φ | ψ (0) i = Z ∞-∞ dxφ * ( x ) ψ (0 ,x ) 1 = r 2 L mω π ¯ h 1 / 4 Z L dx sin πx L e- mω ( x- L/ 2) 2 / 2¯ h = r 2 L mω π ¯ h 1 / 4 L π Z π/ 2- π/ 2 du sin( u + π 2 ) e- mωL 2 u 2 / 2 π 2 ¯ h ( u = π L ( x- L 2 ) ) = r 2 L mω π ¯ h 1 / 4 L π Z π/ 2- π/ 2 du cos ue- mωL 2 u 2 / 2 π 2 ¯ h ....
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This note was uploaded on 08/01/2008 for the course PHYSICS 137B taught by Professor Moore during the Fall '07 term at Berkeley.

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p137bp6sol - Physics 137B Fall 2007 Moore Problem Set 6...

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