08wave - Developing Wave Equations Need wave equation and...

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P460 - dev. wave eqn. 1 Developing Wave Equations • Need wave equation and wave function for particles. Schrodinger, Klein-Gordon, Dirac • not derived. Instead forms were guessed at, then solved, and found where applicable • So Dirac equation applicable for spin 1/2 relativistic particles • Start from 1924 DeBroglie hypothesis: “particles” (those with mass as photon also a particle…) have wavelength λ = h/p
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P460 - dev. wave eqn. 2 Wave Functions • Particle wave functions are similar to amplitudes for EM waves…gives interference (which was used to discover wave properties of electrons) • probability to observe =|wave amplitude| 2 =| ψ( x,t)| 2 • particles are now described by wave packets • if ψ = A+B then |ψ| 2 = |A| 2 + |B| 2 + AB* + A*B giving interference. Also leads to indistinguishibility of identical particles t1 t2 vel=<x(t2)>-<x(t1)> (t2-t1) merge Can’t tell apart
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P460 - dev. wave eqn. 3 Wave Functions • Describe particles with wave functions • ψ( x) = Σ a n sin(k n x) Fourier series (for example) • Fourier transforms go from x-space to k-space where k=wave number= 2 π/λ . Or p=hbar*k and Fourier transforms go from x-space to p-space • position space and λ/ k/momentum space are conjugate • the spatial function implies “something” about the function in momentum space = dk e k x ikx ) ( 2 / 1 ) ( φ π ψ = dx e x k ikx ) ( 2 / 1 ) (
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P460 - dev. wave eqn. 4 Wave Functions (time) • If a wave is moving in the x-direction (or -x) with wave number k can have • kx- ω t = constant gives motion of wave packet • the sin/cos often used for a bound state while the exponential for a right or left traveling wave ) ( ) ( ) sin( ) cos( ) , ( t kx i t kx i k Be Ae or t kx D t kx C t x ω ψ + + = ν λ πν π h h = = = = = = = = h E k h p f T k / 2 2 / 2 / 2
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P460 - dev. wave eqn. 5 Wave Functions (time) • Can redo Transform from wave number space (momentum space) to position space • normalization factors 2 π float around in Fourier transforms • the A(k) are the amplitudes and their squares give the relative probability to have wavenumber k (think of Fourier series) • could be A(k,t) though mostly not in our book • as different k have different velocities, such a wave
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.

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08wave - Developing Wave Equations Need wave equation and...

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