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Lecture 07 - EEE 352 LECTURE 7 Quantum Mechanics and the Schrdinger Equation The Schrdinger equation Where are we going for the next few lectures

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1 EEE 352: LECTURE 7 Quantum Mechanics and the Schrödinger Equation * The Schrödinger equation Time dependent & independent forms Free electron propagation Potential energy Ψ = Ψ + Ψ E V m ) ( 2 2 2 r h Erwin Schrödiner Nobel Prize in Physics, 1933 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. Solving The Schrödinger Equation We wish to solve the time-independent form of the Schrödinger equation: If we assume a solution of the form then 2 2 2 2 2 2 2 2 2 , 0 ) ( ) ( ) ( ) ( 2 h h mE k x k dx x d x E dx x d m = = + = ψ x i e α ( ) ikx ikx x i Be Ae x k e k + = ± = = + ) ( 0 2 2 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 ( ) ( ) [ ] t i e ikx B ikx A t x ω + = Ψ exp exp ) , ( [] = Ψ = Ψ h h iEt ikx B t x iEt ikx A t x exp exp ) , ( exp exp ) , ( 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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2 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 [] 0 0 exp exp ) , ( ) ( > = = = = = = Ψ k t x v t kx Ae iEt ikx A t x phase t kx i ω φ h 0 0 exp exp ) , ( ) ( < = = + = = = = Ψ + k t x v t kx Ae iEt ikx A t x phase t kx i h The Schrödinger Equation 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! m k E 2 / 2 2 h = = = = = = = λ π h h k m k m mE p m p E 2 2 2 2 2 2 2 2 2 h h SINCE E IS CONSTANT p IS CONSTANT The de Broglie relation is recovered! The Free Particle For the free particle, we have now found that m k m k E 2 / 2 / 2 2 2 h h = = k m k m mE p h h = = = 2 2 2 2 2 The two velocities that result from this PARTICLE WAVE are particle group particle phase v m p m k k v v m p m k k v = = = = = = = = h h 2 2 2 As indicated earlier, we connect the free particle velocity with the GROUP VELOCITY of the wave.
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This note was uploaded on 09/19/2009 for the course EEE 352 taught by Professor Ferry during the Spring '08 term at ASU.

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Lecture 07 - EEE 352 LECTURE 7 Quantum Mechanics and the Schrdinger Equation The Schrdinger equation Where are we going for the next few lectures

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