class27 - Today continue Matter Waves deBroglie thought...

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Today: continue Matter Waves! deBroglie thought electrons were waves with λ =h/p el Experiments demonstrated electrons have wave behavior, with wavelength matching deBroglie’s wavelength Revisit interpretation of what these waves are: • Describing electrons (or any particle) with wave functions • Interpreting (wave amplitude) 2 as probability density • Making localized particles with wave packets • Uncertainty principle … position and momentum + Three deBroglie waves are shown for particles of equal mass. I II III x The highest speed and lowest speed are: A, 2f 2A, 2f A, f x x Warm-up question Q1 b. I and II same and highest, III is lowest c. all three have same speed d. cannot tell from figures above A: Amplitude. f: frequency Last class we introduced the “wave function” Ψ (x,y,z,t). We discussed, that we can observe its amplitude square (a ‘probability density’). Probability density = P(x,t) = | Ψ | 2 = Ψ * Ψ The likelihood of a particle being detected at specific locations (and times). Matter waves: Probability density = P(x,t) = | Ψ | 2 = Ψ * Ψ P(x,t=0) Ψ (x,t=0) x L -L Wave function = Ψ (x,t) L -L x Probability of electron being in interval dx = P(x)·dx dx More general: Probability of finding electron between x 1 and x 2 at time t: P(x,t)dx = | Ψ (x,t)| 2 dx x 1 x 2 x 1 x 2 This requires ‘normalization’ of ψ (x) Normalization Probability density = P(x,t) = | Ψ | 2 = Ψ * Ψ P(x,t=0) L -L x The probability of finding the particle “Normalized wave function” The probability of finding the particle anywhere in space (i.e. between – and + ) must be 100%! P(x,t)dx =
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class27 - Today continue Matter Waves deBroglie thought...

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