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Wavefunctions and Uncertainty Chapter 41 Radical view of light • Planck and Einstein re-introduced the particle notion for light • a wave “becomes” a particle and still a wave … hmmm de Broglie... • If light (ie “a wave”) can be a particle then maybe a particle (electron?) can “be a wave” what are the implications of this “leap of reason”? Compare and contrast: Waves & Particles • Waves are extended • Waves are continuous • Waves conform to wave equations • Waves diffract and interfere • Waves have amplitude, frequency and velocity • Particles are points • Particles are discontinuous • Particles obey equations of mechanics • Particles “bounce” • Particles have mass, size(?) and velocity ParticleWaves, Wavicles or just Weirdness … • If de Broglie is correct then we can ascribe a wavelength to a particle: E hf p c c p mv h h mv c Schroedinger... Once at the end of a colloquium I heard Debye saying something like: “Schroedinger, you are not working right now on very important problems… why don’t you tell us some time about that thesis of de Broglie’s… in one of the next colloquia, Schroedinger gave a beautifully clear account of how de Broglie associated a wave with a particle, and how he could obtain the quantization rules… When he had finished, Debye casually remarked that he thought this way of talking was rather childish… To deal properly with waves, one had to have a wave equation. Felix Bloch, Address to the American Physical Society, 1976 What’s a Wavefunction? • Schroedinger’s Equation: Hy= E y Wavefunction “psi” solves Schroedinger’s equation and contains, in its components, all of the information we need to determine values of observables… Wavefunctions are only part of the story… • The information in the wavefunction is “coded” in its components. Actual values for observables depend on “how you ask” the wavefunction • Operators tell us what we want to know: • Example: momentum – Classical: p mv – Quantum: d p i dx This is not as strange as it seems – you got used to thinking of momentum as “em” times “vee” – this is just another way to think about it! Schroedinger’s key assumptions concerning a quantum-mechanical wave equation... 1 It must incorporate the relations: h p and f E h 2 p E 2m V 2 Since normal waves “add” linearly (principle of superposition), so too must the solutions to the qm-wave equation. This means the solutions must be linear. What wavefunctions are (and are not!) • Wavefunctions are mathematical ideas that depict probability distributions (Born interpretation) • Wavefunctions can be described using the mathematics of waves but are not “real” • Wavefunctions obey strict mathematical rules: – continuous, differentiable, finite Key Ideas… • Chp 41.1 – 41.3 – Wave Interference and how it applies to “quantons” – Probability and Probability Density – Wave Functions • Chp 41.4 – 41.6 – Normalization – Wave packets – Uncertainty