PH 401 Dr. Cecilia Vogel Lecture 3

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

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

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

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.