Stumbling Towards Quantum Physics

1. Stumbling Towards Quantum Physics e- e+ e- Metal e- e- e-

2. – + Things come in chunks Faraday’s experiment (1833) • Dissolve one mole of some substance in water • Let an electric current run through it • Measure how much charge runs through before it stops Na+ Cl- • All ions have the same charge (or simple multiples of that charge) • Avogadro’s number was not known at this time

3. J.J. Thomson discovers electron: 1897 – + • Charged particle, bends in presence of magnetic field • Relativity not discovered until 1905 e- • Velocity measured with help of electric field • Ratio of charge to mass now known

4. The Plum Pudding Model: 1904 • Electrons have only tiny fraction of an atom’s mass • Atoms have no net charge • 1904: J.J. Thomson proposes the “Plum Pudding” Model • Electrons “imbedded” in the rest of the atom’s charge • Rest of charge is spread throughout the atom

5. Millikan measures charge e: 1909 • Atomizer produced tiny drops of oil; gravity pulls them down • Atomizer also induces small charges • Electric field opposes gravity • If electric field is right, drop stops falling +

6. Millikan measures charge e: 1909 • Millikan always found the charge was an integer multiple of e The atom in 1909: • Strong evidence for atoms had been found • Avogadro’s number, and hence the mass of atoms, was now known • Electron mass and charge were known • Atoms contained negatively charged electrons • The electrons had only a tiny fraction of the mass of the atom • Distribution and nature of the positive charge was unknown Meanwhile . . .

7. Statistical Mechanics The application of statistics to the properties of systems containing a large number of objects gravity • Mid – late 1900’s, Statistical Mechanics successfully explains many of the properties of gases and other materials • Kinetic theory of gases • Thermodynamics Gas molecules in a tall box: • The techniques of statistical mechanics: • When there are many possibilities, energy will be distributed among all of them • The probability of a single “item” being in a given “state” depends on temperature and energy

8. Announcements ASSIGNMENTS DayReadQuizHomework Today Sec. 3-2 Quiz HHwk. H Friday Study For Test none MondaySec. 3-3 & 3-4Quiz I Hwk. I • Equations for Test: • Force and Work Equations added • Lorentz boost demoted • Test Friday: • Pencil(s) • Paper • Calculator 9/16

9. Black Body Radiation: Light in a box Consider a nearly enclosed container at uniform temperature: u() = energy/ volume /nm • Light gets produced in hot interior • Bounces around randomly inside before escaping • Should be completely random by the time it comes out • Pringheim measures spectrum, 1899

10. Black Body Radiation Goal - Predict: • Finding n() • How many waves can you fit in a given volume? • Leads to a factor of 1/4 • What are all the directions light can go? • Leads to a factor of 4 • How many polarizations? • Leads to a factor of 2 Can statistical mechanics predict the outcome? • Find effects of all possible electromagnetic waves that can exist in a volume • Two factors must be calculated: • n(): Number of “states” with wavelength  • E: Average energy

11. How to find E What do we do with these sums over energy? Example: Suppose you roll a fair die. If you roll 1 you win $3, if you roll 2 or 3 you win $1, but if you roll 4, 5, or 6, you lose $2. What is the expectation value of the amount of money you win? • What does E mean? • It is an expectation value Sum of all probabilities must be 1

12. What do we do with the sums? Waves of varying strengths with the same wavelength • Energy can be anything • Replace sums by integrals? The ultraviolet catastrophe

13. Comparison Theory vs. Experiment: Theory • What went wrong? • Not truly in thermal equilibrium? • Possible state counting done wrong? • Sum  Integral not really valid? Experiment • Max Planck’s strategy (1900): • Assume energy E must always be an integer multiple of frequency f times a constant h • E = nhf, where n = 0, 1, 2, … • Perform all calculations with h finite • Take limit h 0 at the end

14. Math Interlude: Take d/dx of this expression . . . Multiply by x . . . Some math notation:

15. Planck’s computation: From waves:

16. Planck’s Black Body Law • Max Planck’s strategy (1900): • Take limit h 0 at the end • Except, it fit the curve with finite h! Planck Constant “When doing statistical mechanics, this is how you count states”

17. Total Energy Density Let  = hc/xkBT

18. Wien’s Law For what wavelength is this maximum?

19. Planck constant Often, when describing things oscillating, it is more useful to work in terms of angular frequency instead of frequency f This ratio comes up so often, it is given its own name and symbol. It is called the reduced Planck constant, and is read as h-bar Units of Planck constant • h and h-bar have units of kg*m^2/s – same as angular momentum

20. Photoelectric Effect: Hertz, 1887 e- • Metal is hit by light • Electrons pop off • Must exceed minimum frequency • Depends on the metal • Brighter light, more electrons • They start coming off immediately • Even in low intensity Metal e- e- e- • Einstein, 1905 • It takes a minimum amount of energy to free an electron • Light really comes in chunks of energy hf • If hf < , the light cannot release any electrons from the metal • If hf > , the light can liberate electrons • The energy of each electron released will be Ekin = hf – 

21. Photoelectric Effect – + + – • Will the electron pass through a charged plate that repels electrons? • Must have enough energy • Makes it if: Metal e- V Vmax slope = h/e Nobel Prize, 1921 f

22. Sample Problem • When ultraviolet light of wavelength 227 nm strikes calcium metal, electrons are observed to come off which can penetrate a barrier of potential up to Vmax = 2.57 V. • What is the work function for calcium? • What is the longest wavelength that can free electrons from calcium? • If light of wavelength 312 nm were used instead, what would be the energy of the emitted electrons? We need the frequency: Continued . . .

23. Sample Problem continued 2. What is the longest wavelength that can free electrons from calcium? 3. If light of wavelength 312 nm were used instead, what would be the energy of the emitted electrons? • The lowest frequency comes from Vmax = 0 • Now we get the wavelength: • Need frequency for last part:

24. X-rays • Mysterious rays were discovered by Röntgen in 1895 • Suspected to be short-wavelength EM waves • Order 1-0.1 nm wavelength • Scattered very weakly off of atoms • Bragg, 1912, measured wavelength accurately   d dcos dcos • Scattering strong only if waves are in phase • Must be integer multiple of wavelength

25. The Compton Effect • By 1920’s X-rays were clearly light waves • 1922 Arthur Compton showed they carried momentum Atom Photon in e-  Photons carry energy and momentum, just like any other particle Photon out e- e- • Conservation of momentum and energy implies a change in wavelength Meanwhile . . .