HW04 & Solutions - (c) Give the corresponding state...

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ECE 306 Homework Set #4 Spring 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: m p x dt d x = (b) Prove that the time variation of the expectation value of the momentum is given by: x x F dx dV p dt d = = , 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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Unformatted text preview: (c) Give the corresponding state function at an arbitrary later time (x, t &gt; 0); you can give the result in integral form. ECE 306 Homework Set #4 Solutions Spring 2005 1. Consider a particle of mass m in a potential field V(x) (a) On the basis of Heisenbergs equation of motion (2.49) and the commutation relation (2.11a): m p x x V m p i x dt d x x = + = ), ( 2 2 h (b) On the same basis, the time variation of the expectation value of the momentum is given by: x x x x F dx x dV p x V m p i p dt d = = + = ) ( ), ( 2 2 h , which is known as Ehrenfests theorem. 2. (a) o electron deBroglie A eV units E mE h ) : ( 3 . 12 2 ) ( = = o proton deBroglie A eV units E ) : ( 3 . ) ( = (b) o proton deBroglie neutron deBroglie A eV units E ) : ( 3 . ) ( ) ( = ) : ( ) 12 6 . 1 ( ) ( eV units E e hc photon deBroglie =...
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HW04 &amp; Solutions - (c) Give the corresponding state...

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