mid06w - 1 University of California Santa Barbara...

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1 University of California, Santa Barbara Department of Physics Physics 115A - Midterm - Winter 2006 A.N. Cleland Closed notes, closed book; you are allowed one sheet of formulas and a calculator. Some identities that you might find useful: Z -∞ e - u 2 du = π, Z -∞ u e - u 2 du = 0 Z -∞ u 2 e - u 2 du = π 2 Z π/ 2 0 cos z cos z dz = Z π/ 2 0 sin z sin z dz = π/ 4 , Z π/ 2 0 cos z sin 2 z dz = Z π/ 2 0 sin z sin 2 z dz = 2 / 3 , Z π/ 2 0 cos 2 z sin 2 z dz = 0 , Z π/ 2 0 sin 2 z sin 2 z dz = π/ 4 , cos 2 u = 1 2 (cos 2 u + 1) , cos 3 u = 1 4 (cos 3 u + 3 cos u ) . (1)
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2 Problem 1. [25 points] A particle with mass m is trapped in an infinite square well, extending from - a/ 2 to a/ 2, with potential function V ( x ) → ∞ for | x | > a/ 2, and V ( x ) = 0 for | x | < a/ 2. The particle’s wavefunction at t = 0 is given by Ψ( x, 0) = A fl fl fl fl sin 2 πx a ¶fl fl fl fl , (2) where | . . . | represents the absolute value. (a) [5 points] Find the normalization A. (b) [10 points] Calculate the probability the particle can be found in the ground state of the square well. (c) [10 points] Calculate the probability the particle can be found in the first excited state of the square well. Problem 2.
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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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  • Winter '03
  • Nelson
  • Physics, mechanics, cos z cos, sin z sin, Santa Barbara Department of Physics Physics, sin z dz

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