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PH 401 Dr. Cecilia Vogel Lecture 3

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Review Outline Requirements on wavefunctions TDSE Normalization Free Particle matter waves probability, uncertainty

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Free Particle Simplest Case for TDSE is if V=0 everywhere “free particle” classically, this particle at rest would remain at rest, or in motion would continue in uniform motion what does it do in quantum world?

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Free Particle Solutions Looks a lot like wave equation for sound or light guess sinusoidal solution and see what happens GUESS

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Free Particle Solutions Plug guess into TDSE GUESS NO! imaginary cannot equal real!

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Second Guess Try a complex sinusoidal function: Plug guess into TDSE Yeah! 2 k 2 /2m = p 2 /2m = E KE = E since V=0 everywhere

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Another Solution Both and Are solutions to TDSE What’s the diff?

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Momentum Direction and What’s the diff? Momentum: One on the left has p = k. One on right has p = - k. it moves in the negative direction. kinetic energy is the same

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Free Particle Simple Solution Problem: The probability density for this simple solution: this particle is as likely to be 20 light-years away as it is to be here this wavefunction is not normalizable Simple sinusoidal is solution to TDSE but not realistic particle wavefunction. Note: often used anyway as an approximation for beams that have large extent. We need a LOCALIZED wavefunction

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Superposition Localized wavefunctions can be concocted by summing (actually integrating) sinusoidal functions of different wavelength and frequency Superposition Principle: Sum/integral (or superposition ) of solutions to TISE is also a solution to TISE

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Superposition Wavefunction All free-particle wavefunctions can be made by summing/integrating sinusoidal functions of different wavelength and frequency Fourier synthesis is the mathematical term for this synthesis of sinusoidal functions any well-behaved function can be synthesized in this way

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Fourier Amplitude A(k) is the amplitude of each k in the combo: depends on k, independent of x, constant in time for free particle.

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Superposition Solves Problem For some functions A(k) (x,t) is localized and normalizable.

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Wavepacket Motion The synthesized wave function will have a non-sinusoidal shape. How does the shape move? “Group velocity” vgvg

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Group Velocity If each part of the synthesis (i.e. each wavelength or each wavenumber) travels at the same phase speed, v ph, then the whole group travels at that speed, v g =v ph = / k true for light in vacuum If different wavelengths travel with different speeds, then v g =d /d k

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Appendix: Complex #’s What is the meaning of You can do algebra and calculus on it just like real exponentials; just remember i 2 = -1. It is a complex number, with real and imaginary parts. Can be rewritten as: For example but

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Complex Algebra To add or subtract complex numbers, add or subtract real parts ( a ), add or subtract imaginary parts ( b ). To multiply, use distributive law. To get the absolute square | z | 2, multiply z by its complex conjugate, z *. To get the complex conjugate of z, change the sign of all the i ’s. a and b real

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Complex Example Find the absolute square, | | 2, which is the probability density. Need the complex conjugate, *. The probability density is constant, it is the same everywhere, all the time. this particle is as likely to be a million light years away, as here. Not localized.

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Complex Algebra In general, with c and d real

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Complex Example Given | | 2 = ¼ show that works as well as ½.

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