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 refection 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 . ±ind the probability density and the probability current density in the region x < 0. What can you say about where the refection 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 ) = ² π ¯ h ³ 1 / 4 exp ² - h x 2 ³ . (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 ( x ) exp(-i E t/ h ). Determine the probability (in the Form oF an integral) oF nding 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 , , and h . (c) Calculate the variance 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(-z 2 ) = 1 2 r 3 . 3. Show that the three stationary wave Functions 1 ( x ) = C 1 exp(-x 2 ), 2 ( x ) = C 2 x exp(-x 2 ), and 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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This note was uploaded on 08/12/2008 for the course PHYS 252 taught by Professor Pocanic during the Spring '02 term at UVA.

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