# Discussion6 - Schr Ã dinger equation are ALSO eigenstates...

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Physics 486 Fall 2007 Discussion Problems 6 1) This problem is taken from Griffiths 2.2.1 and ought to be useful for HW5. A free particle has the initial wavefunction of the old familiar form | | ( , 0) a x x Ae - Ψ = , where A and a are real positive constants. (a) Normalize Ψ ( x, 0) . (b) Find its wave-vector representation φ(k) (i.e, its inverse Fourier transform). (c) Discuss the limiting forms of ( x, 0) and φ(k) ( a very large, and very small). (d) Construct ( x, t) in the form of an integral.

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Physics 486 Fall 2007 2). In this problem, you will revisit what you have learned in the last lectures and identify the class of potentials V(x) for which the solutions to the time-dependent
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Unformatted text preview: Schr Ã¶ dinger equation are ALSO eigenstates of the momentum operator (i.e., for which one can simultaneously measure states of both well-defined energy AND momentum): (a) Find the commutator between the Hamiltonian H and the momentum operator p. (b) For what potentials V(x) do H and p commute? What kinds of particle states are associated with this class of potentials? (c) Find the rate of change of the momentum expectation value, i.e., d<p>/dt for this class of potentials, and explain how your result is consistent with Ehrenfestâ€™s theorem....
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Discussion6 - Schr Ã dinger equation are ALSO eigenstates...

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