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Unformatted text preview: A particle with mass m is trapped in a simple harmonic oscillator potential with resonance frequency ω , and is prepared at time t = 0 in the (not normalized) state ψ ( x, 0) = N (2 ψ ( x )-iψ 1 ( x )) , (3) where ψ and ψ 1 are the ground and ﬁrst excited eigenstates of the harmonic oscillator, which are given by the normalized expressions ψ ( x ) = ± mω π ¯ h ¶ 1 / 4 e-ζ 2 / 2 (4) and ψ 1 ( x ) = ± mω π ¯ h ¶ 1 / 4 √ 2 ζe-ζ 2 / 2 , (5) where ζ = q mω / ¯ h x . (a) [5 points] Find the normalization N . (b) [5 points] Find the expectation value for the energy. (c) [5 points] Work out the time dependence for ψ ( x,t ). (c) [10 points] Work out how long you have to wait for ψ ( x,t ) to return to the initial state ψ ( x, 0), ignoring any overall complex phase factors of the form e iφ ....
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