1. PH 301 Dr. Cecilia Vogel Lecture

2. Review Outline  Wave-particle duality  wavefunction  probability  Photon  photoelectric effect  Compton scattering

3. When do We See Which? Wave-particle duality:  Light can show wave or particle properties, depending on the experiment.  While propagating, light acts as a wave  while interacting, light acts as a particle.

4. When do We See Which?  Two-slit experiment  Light will propagate through both slits  and waves through slits interfere with each other, but  when it strikes the screen,  it interacts with the screen one photon at a time.

5. When do We See Which?  Interference  seen if waves are coherent  Diffraction  seen if obstacle/opening about size of wavelength

6. Why is the sky blue?  The sky is blue, because more blue light is scattered by the air to our eye (than red, yellow, etc).  Blue light is more likely to scatter than red, because red is more likely to diffract instead.  Less diffraction occurs for shorter wavelengths.  Blue light has shorter wavelength, so it diffracts less and scatters more.

7. Why are the clouds white?  The water droplets are much larger than the wavelength of all visible light  (not just blue/violet)  almost no visible light is diffracted by clouds  every color of visible light is scattered by clouds  all colors scattered, so scattered light is white

8. Matter  Matter particles, like electrons, have particle properties (of course)  individual, indivisible particles  energy & momentum  (paintball)

9. Duality of Matter  Matter particles also have wave properties!  They diffract!  They interfere!  Diffract from a crystal, interference pattern depends on  crystal structure ...from a powder, pattern depends on  molecular structure

10. Duality equations  Light/photons  Matter, e.g. electrons Only for light Cue: ‘c’ Only for matter Cue: ‘m’ Same eqns

11. Example What is the wavelength of an electron which has 95 eV of kinetic energy? Note: K<<m o c 2, so we can use classical equations. Note: DO NOT USE E=hc/.

12. Wave Function  For light, the wavefunction is E( x,t )  electric field (and B( x,t ) = magnetic field).  For matter the wave function is  ( x,t )  like nothing we’ve encountered before.  Not an EM wave.  The matter itself is not oscillating.

13. Wavefunction Interpreted  For light beam, where the wave function (E-field) is large,  the light is bright  there are lots of photons  For beam of matter particles, where the wave function is large  there are lots of particles.  The bright spots in interference pattern  are where lots of photons or matter particles strike.

14. Probability Interpretation  If you have one particle, rather than a beam,  the wavefunction only gives probability  P(x,t) = |  (x,t)| 2.  there is no way to predict precisely where it will be.  Where the wave function is large  the particle is likely to be.  The bright spots in interference pattern  are where a photon or matter particle is likely to strike.

15. Probability Interpretation  P(x,t) = |  (x,t)| 2.  If we repeat an experiment many, many times, the probability tells in what fraction of the experiments, we will find the particle at position x at time t.  Do we have to do the experiment many, many times for the probability to have meaning?  NO!  With one particle, you can still determine probabilities

16. Averages and Uncertainty  P(x,t) = |  (x,t)| 2.  If you have many possibilities with known probabilities  Average = x ave =  x= probability weighted sum of possibilities  =  Uncertainty  x=rms dev = root mean square deviation   x =  Also   x =

17. Imaginary Exponentials  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

18. 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

19. Complex Algebra  In general, with c and d real

20. 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.

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

22. PAL Probability  Given the wavefunction where x is in nm and ranges from 0 to 3 nm. 1) Find the probability density as a function of x. 2) Find = the average value of x. 3)Find = the average value of x 2. 4)Find  x = the uncertainty in x.