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?
Free Particle Solutions Looks a lot like wave equation for sound or light guess sinusoidal solution and see what happens GUESS
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
Another Solution Both and Are solutions to TDSE What’s the diff?
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
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
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
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
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.
Superposition Solves Problem For some functions A(k) (x,t) is localized and normalizable.
Wavepacket Motion The synthesized wave function will have a non-sinusoidal shape. How does the shape move? “Group velocity” vgvg
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
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
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
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.
Complex Algebra In general, with c and d real
Complex Example Given | | 2 = ¼ show that works as well as ½.