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ps06 - x that a classical particle can reach iF it has the...

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Quantum Mechanics Problem Sheet 6 (Solutions are to be handed in for marking before the lecture on Monday, 28 Feb 2005) 1. Calculate the reflection and transmission probabilities for right-incident scattering from the potential V ( x ) = V 0 for x < 0 0 for x > 0 . at an energy E < V 0 . Find the probability density and the probability current density in the region x < 0. What can you say about where the reflection is taking place? (The strategy for tackling this problem is the same as in problem 2 of the previous problem sheet and in the examples of scattering from piecewise constant potentials that we looked at in the lecture.) 2. Consider a particle in the harmonic potential V ( x ) = 2 x 2 / 2. Its lowest energy eigenvalue is E 0 = ¯ hω/ 2 and the eigenfunction associated with this energy, ie the ground-state wave function, is φ 0 ( x ) = π ¯ h 1 / 4 exp - h x 2 . (a) Determine the limits of the classical motion in this potential (the “classical turning points”), ie the smallest and the largest values of
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Unformatted text preview: x that a classical particle can reach iF it has the total energy E . (b) Assume that the wave Function oF the particle is the stationary state φ ( x ) exp(-i E t/ ¯ h ). Determine the probability (in the Form oF an integral) oF ²nding the particle outside the region where classical motion can occur. By making an appropriate change oF variable in the integral you obtain, show that the answer is independent oF m , ω , and ¯ h . (c) Calculate the variance Δˆ x in the ground-state oF the system and compare it to the limits oF the classical motion. Useful integral: Z ∞-∞ d z z 2 exp(-αz 2 ) = 1 2 r π α 3 . 3. Show that the three stationary wave Functions φ 1 ( x ) = C 1 exp(-x 2 ), φ 2 ( x ) = C 2 x exp(-x 2 ), and φ 3 ( x ) = C 3 (4 x 2-1) exp(-x 2 ) are all mutually orthogonal. Claudia Eberlein, 11 Jan 2005 1...
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