lecture 6

# lecture 6 - EEE 352: LECTURE 6 Quantum Mechanics and the...

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EEE 352: LECTURE 6 Quantum Mechanics and the Schrödinger Equation * The Schrödinger equation Time dependent & independent forms Free electron propagation Potential energy Erwin Schrödinger Nobel Prize in Physics, 1933

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Where are we going for the next few lectures? Schrödinger equation Simple examples Band structure Electron waves in the semiconductor Bragg reflection in the crystal due to the periodic structure. Fourier transform space for E vs k Band structure describes the dispersion relation between energy and wave number, e.g. between energy and momentum , for ALLOWED wave states in the semiconductor. We need quantum mechanics to describe the electron motion within the condensed matter that we call a semiconductor!
Schrödinger equation Simple examples Band structure Electron waves in the semiconductor We are going to begin with some simple examples to illustrate how we solve the equation: Free particle An infinite barrier A finite barrier (so we can go over the top)

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Solving The Schrödinger Equation A free particle We wish to solve the time-independent form of the Schrödinger equation: If we assume a solution of the form then −α 2 + k 2 ( ) e i α x = 0 = ± k ψ ( x ) = Ae ikx + Be ikx
The Schrödinger Equation These GENERAL solutions contain contributions from TWO spatially varying waves * That propagate in OPPOSITE directions to each other! * The constants A & B are determined by suitable BOUNDARY CONDITIONS Such as the INITIAL direction of the particle Ψ ( x , t ) = A exp ikx ( ) + B exp ikx ( ) [ ] e i ω t FREELY MOVING PARTICLE THAT MOVES ALONG THE x-AXIS IN THE +x DIRECTION FREELY MOVING PARTICLE THAT MOVES ALONG THE x-AXIS IN THE -x DIRECTION

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The Schrödinger Equation FREELY MOVING PARTICLE THAT MOVES ALONG THE x-AXIS IN THE +x DIRECTION FREELY MOVING PARTICLE THAT MOVES ALONG THE x-AXIS IN THE -x DIRECTION
Classically a freely moving particle should propagate with CONSTANT velocity * What do the QUANTUM solutions tell us about the motion of the particle? * We have already found the connection between energy and wavenumber Given this last result we may compute the MOMENTUM of the moving particle * And we find this to be CONSTANT as expected classically We also obtain the de Broglie relation! SINCE E IS

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## This note was uploaded on 09/28/2009 for the course EEE 352/333 taught by Professor Allee during the Fall '09 term at ASU.

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lecture 6 - EEE 352: LECTURE 6 Quantum Mechanics and the...

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