PH 301

1. PH 301 Dr. Cecilia Vogel Lecture

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

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

11. Example What is the wavelength of an electron which has 95 eV of kinetic energy? Note: K<<moc2, so we can use classical equations. Note: DO NOT USE E=hc/l.

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 Y(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) = |Y(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) = |Y(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) = |Y(x,t)|2. If you have many possibilities with known probabilities Average <x> = xave=x= probability weighted sum of possibilities <x> = Uncertainty Dx=rms dev = root mean square deviation Dx = Also Dx =

17. Imaginary Exponentials What is the meaning of • You can do algebra and calculus on it just like real exponentials; • just remember i2 = -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, |Y|2, which is the probability density. Need the complex conjugate, Y*. • 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 |A|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 <x> = the average value of x. Find < x2 > = the average value of x2. Find Dx = the uncertainty in x.