Quantum Mechanics 101

1. Quantum Mechanics 101 Waves? or Particles?

2. Interference of Waves and the Double Slit Experiment • Waves spreading out from two points, such as waves passing through two slits, will interfere l Wave crest Wave trough Spot of constructive interference Spot of destructive interference d

3. The Double-slit experiment for particles • Particles do not diffract; they either go through a slit or they don’t • Particles passing through a slit hit a screen only in a small area; if they all have the same initial velocity, they will all hit at the exact same point • Particles passing through two slits will form two maxima in front of the two slits

4. What Happens if Electrons Pass Through Small Openings? What does that tell you about electrons?

5. The Plot Thickens An experiment called the “photoelectric effect” also gives unexpected results!

6. The Photoelectric Effect, Pictorially • Light shining on a material may be absorbed by electrons in that material The kinetic energy of the electron will be equal to the energy absorbed by the electron minus the energy needed to free it, provided the electron does not lose any energy in collisions If an electron absorbs enough energy to break free of its bonds, it can leave the material crest trough

7. Wave theory predicts . . . • the energy of emitted electrons should depend on the intensity of light • electrons will need to soak up energy from wave for period of time before being ejected • the frequency of light won’t affect the maximum kinetic energy of electrons

8. Work function (energy needed to remove electron) The Photoelectric Effect, Experimentally • As a given color (frequency) of light enters the black box-like photoelectric head, it falls on a plate of electron-emitting material inside • Emitted electrons are collected on another plate nearby, producing an electric potential difference between the two plates (like a capacitor) • When the capacitor is fully charged and no more electrons can be added, the potential energy of the capacitor equals the maximum kinetic energy of the electrons trying to leave the original plate • The potential difference on the capacitor at this point is called the stopping potentialVsfor the electrons, and it is proportional to the maximum kinetic energy of electrons emitted by the light: K = eVs = Eabsorbed - F

9. Do the Photoelectric Experiment Upon what does the energy of emitted electrons appear to depend?

10. Experiment sees . . . • the energy of emitted electrons does not depend on the intensity of light • electrons are ejected immediately • the frequency of light does affect the maximum kinetic energy of electrons; kinetic energy is linearly dependent on frequency • intensity of light determines number of emitted electrons (photocurrent)

11. If an electron absorbs one of these photons, it gets the entire hf of energy. Einstein to the Rescue • Einstein suggested that light was emitted or absorbed in particle-like quanta, called photons, of energy, E = hf If that energy is larger than the work function of the metal, the electron can leave; if not, it can’t: Kmax = Eabs – F = hf - F crest trough

12. Einstein’s Photoelectric Theory eVs = Kmax = hf – F • Kmaxµ f • Is this consistent with what you saw in the experiment? • Electrons are ejected as soon as a photon strikes the material. • Is this consistent with what you saw in the experiment?

13. Einstein’s Photoelectric Theory eVs = Kmax = hf – F • If hf < F, no electrons are emitted; cutoff frequency • What should the slope of a K vs. f plot yield? Is that what you got?

14. The Conflict • Wave theory accurately describes interference and diffraction, along with other behavior of light, such as dispersion and refraction • The particle theory accurately describes photoelectric effect, black body radiation, and other experimental results • Is light a particle? Or is it a wave? • Is a platypus a duck? Or is it a beaver? • Am I my mother? Or am I my father?

15. The Resolution • Light is not either a particle or a wave • Light exhibits wavelike properties when traveling • Light exhibits particlelike properties when interacting with matter • deBroglie suggested that traditional “particles”, like the electron, also exhibit wavelike properties • p=h/l, so large (macroscopic) momentum means small (undetectable) wavelength

16. The interpretation • Light and “particles” propagate through space as probability waves • I cannot say for certain where a particle is, where it was, or how it got to wherever it might have been • I can, however, say where it is most likely to be found, where it most likely was, and how likely it is that it took a particular path • This behavior is described by a wave function Y(x,y) which obeys Schrödinger’s equation

17. More interpretation • The probability of finding a particle in a particular region within a particular time interval is found by integrating the square of the wave function: • P (x,t) =  |Y(x,t)|2 dx =  |c(x)|2 dx • |c(x)|2 dx is called the “probability density; the area under a curve of probability density yields the probability the particle is in that region • When a measurement is made, we say the wave function “collapses” to a point, and a particle is detected at some particular location

18. What have we learned today? • Quantum mechanics is AWESOME, but it challenges our physical intuition • Light and “particles” behave like waves when traveling and like particles when interacting or being observed • Since they propagate like waves, both light and “particles” can produce interference patterns • We can describe this duality through the use of a wave function Y(x,t) which describes the (unobserved) propagation through space and time