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ECE 306 Homework Set #4Spring 2005 (Due 12 noon, Wednesday, February 22, 2005) 1.Consider a particle of mass m in a potential field V(x). (a)Show that the time variation of the expectation value of the momentum is given by: mpxdtdx=(b)Prove that the time variation of the expectation value of the momentum is given by: xxFdxdVpdtd=−=, which is known as Ehrenfest’s theorem. 2.Sketch the de Broglie wavelength versus the kinetic energy up to 10 eV ( = 1.6e-18 Joules) for (a)electrons, protons and (b)neutrons and compare these results with the corresponding result for photons 3.Suppose we know that there is a free particle initially located in the range –a < x < a with a spatially uniform probability. (a)Give the normalized state function Ψ(x, t = 0) of the particle in the Schroedinger-representation. Assume the phase of the wave function is arbitrarily chosen to be zero. (b)Give the corresponding momentum representation of the particle.
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