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2005AugQualQUANT - Ben Sauerwine Practice for Qualifying...

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Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU Qualification Exam Day 2 (August 2005) Consider a particle of mass m moving in a one-dimensional potential with infinitely high walls placed at a x ± = . The potential vanishes between the walls. (a) Write down the eigenenergies and the corresponding eigenstates of the system in the x basis. The eigenstates, even and odd, are ( ) x a n i n n e C x 2 π ψ = The corresponding eigen-energies are given by: ( ) ( ) ( ) x E x x V x m ψ ψ = + 2 2 2 2 h For: 2 2 2 2 = a n m E n π h (b) Suppose that at time t = 0, the particle is in a state that can be well approximated by the wave function ( ) 2 2 δ ψ x Ne x = , where a << δ . (i) Determine N and then determine the wave function at some later time . 0 t ( ) ( ) ( ) ( ) π δ π δ ψ δ π θ ψ ψ π δ δ δ δ δ 1 2 2 2 2 2 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = = = = = = = ∫∫ ∫ ∫ + N dx x rdrd e dxdy e dy e dx e dx x dx e dx x r y x y x x The wave function at a later time is given by
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