12_580_hw_8 - Physics 580 Handout 12 19 October 2010...

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Physics 580 Handout 12 19 October 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Homework 8 Prof. P. M. Goldbart 3135 (& 2115) ESB University of Illinois 1) Free particles – optional : a) Write down, in the x -basis, the time-independent Schr¨odinger equation for a particle of mass m moving in one dimension in the presence of a potential V ( x )? b) Show that, when V ( x ) = 0, the wave function ψ E,α ( x ) = N exp( 2 mEx/ ¯ h ) describes a particle with energy eigenvalue E and momentum eigenvalue α 2 mE , where E 0 and α = ± 1. c) The wave function in part (a) is x | E,α ( i.e. , the projection on to the x -basis of the energy and momentum eigenket | ). Derive the normalisation N such that the eigenkets {| ⟩} are orthonormal ( i.e. , | E = δ ( E E ) δ α,α ). d) A particle is in the state ψ ( x ) = N + e ipx/ ¯ h + N e ipx/ ¯ h . Give the possible outcome of the observation of the energy? What possible results fol- low from observation of the momentum? What is the probability of finding momentum p ? What is the expectation value of the current operator at the position x ( i.e. , the current density at x ) in this state? Briefly discuss the major difference between this state and a classical state of the same energy. 2) Expansion of a box (after Shankar, 5.2.1) : A particle is in the ground state of a one-dimensional box. Suddenly the box (symmetrically) doubles its length, leaving the state undisturbed. Determine the probability of finding the particle in the new ground state. 3) Expansion of a box (after Shankar, 5.2.4) : A particle is in the n th excited state | n of a one-dimensional box of length L . Determine the force encountered as the walls are slowly pushed in. Assume adiabaticity , i.e. , that the quantum number(s) of the state do not vary as the properties (here, the length) of the system are adjusted. Compare your result for the force with that obtained for a classical particle of the same energy. [Hint: You may compute the latter via the frequency of collisions and the momentum transfer per collision.] 4) Delta function potential (after Shankar, 5.2.3) : Consider the attractive potential V ( x ) = ( x ). Show that it admits a bound state, and determine the the corresponding energy eigenvalue. Sketch the wave function, indicating its essential features.
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12_580_hw_8 - Physics 580 Handout 12 19 October 2010...

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