problems9 - terms of k x and other relevant quantum...

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Physics 4021: Introduction to Quantum Mechanics I Homework Assignment 9 Due in class: Thursday, Nov. 29, 2007 1. Consider a 3-dimensional box with infinite potential outside of the box: = otherwise L z L y L x z y x V z y x ) , 0 , 0 , 0 ( 0 ) , , ( (a) Assume that the this box is shaped as a pizza box, such that L x = L y = L and L z = d where d << L . Find the suitable quantum numbers, energy, and number of degeneracy of each energy level from the ground state to the fourth excited sate. (b) Assume that this box is shaped as a rod, such that L x = L and L z = L y = d where d << L . Find the suitable quantum numbers, energy, and number of degeneracy of each energy level from the ground state to the fourth excited sate. (c) In (b), let us take the limit that L . Following the argument in the particle-in-a-big- box problem we discussed in class, the quantum number n x can be replaced by a quasi continuous variable x x n L k π = , where n x is a positive integer. Express your energy in
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Unformatted text preview: terms of k x and other relevant quantum numbers. (d) For an electron in the conduction band of Silicon, the Hamiltonian that describes the quantum mechanical motion of electron described by ) ~ ~ ~ ( 2 1 ~ 2 2 2 z y x eff p p p m H + + = where m eff is the effective mass that is about 20% of the real electron mass. Now we put an electron in a Silicon nanowire whose cross section is 10 x 10 nm 2 . Assume that this nanowire can be reasonably described by the model in (c). Now at a temperature T , the electron has a total energy k B T where k B is Boltzmann constant. Find the upper limit of the temperatures that the electron behaves as if it is confined in 1-dimensional box (i.e., the energy is small enough that the quantum numbers other than k x is fixed to their lowest value). 2. Problem 4.38 3. Problem 4.19 4. Problem 4.24 5. Problem 4.9 6. Problem 4.13 1...
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