Chapter 1 Early Quantum Phenomena

Transcription

1 Chapter Early Quantum Phenomena... 8 Early Quantum Phenomena... 8 Photo- electric effect... Emission Spectrum of Hydrogen... 3 Bohr s Model of the atom... 4 De Broglie Waves... 7 Double slit experiment... 8 Chapter Early Quantum Phenomena Early Quantum Phenomena Blackbody Radiation 900 (Planck) the first evidence of quantization Tiny hole, emits radiation Body at temperature T (in equilibrium) Approximate black bodies: stars, a stove, flame or furnace ρ v( Tdv ) in J m- 3 8

2 radiant energy density (intensity) between v and v + dv - - Same for any material Maximum in distribution shifts to higher v with increasing T Wien: λ max T = m K Stefan- Boltzman (see McQuarrie): R = c 4 E v (T ) = c 4 0 ρ (T )dv ν = σt 4, σ J m - k - 4 s - Planck took experimental curves and fitted them to a formula: ρ v (T )dv = 8πh c 3 v 3 e hv/k B T dv c : speed of light ~ 3 x 0 8 m s - h : Planck s constant ~ 6.6 x 0-34 J s k B : Boltzmann ~.4 x 0-3 J k - Works Perfectly. Planck s problem: How can one derive it from known classical physics? Classical Physics ρ v (T ) = 8πk T B v c 3 Exact as v 0 Disaster as v large Something was VERY WRONG with classical physics Planck made a quantum hypothesis, a particular oscillator of frequency v could only have energies nhv quantized: n is an integer. With this ad hoc assumption, he could derive his formula. Nobody understood what this meant. Some more details: ρ(ν,t ) can be converted to ρ(λ,t )dλ v = c λ dv = c λ dλ Chapter Early Quantum Phenomena 9

3 also 8πh c 3 v v v 3 e hv/k B T dv dv v λ dλ λ λ λ dλ 8πh c 3 3 c λ e hc/k B Tλ c λ dλ > v λ < λ (exercise) ρ(λ,t )dλ = 8πhc λ 5 hc dλ λk e T B Einstein later on (95) gave an insightful derivation. Let us discuss: Consider equilibrium between any quantum levels (micro- balance) N N ~ e ~ e E kt B E kt B BN ρ(ν,t ) BN ρ(ν,t ) AN = 0 Stimulated ~ρ(v,t ) spontaneous ρ(ν,t )[ N N ] = A B N N E E hv kt B kt B = e = e A B 8π h v c 3 = Property of radiation 3 i.e. does not depend on material. [Derived much later by Dirac, Quantum electrodynamics] ρ ν (T ) = 8πh c 3 hv v 3 k e T B Black body: all quantum levels in equilibrium Chapter Early Quantum Phenomena 0

4 Black: To every frequency there is a set of levels E E hv (dense) all hv occur. Planck was first, but it is hard to see that Blackbody radiation implies quantization of energy levels. Few people understood the implications. Einstein did Photo- electric effect (Einstein 905) Light: oscillating electromagnetic field jiggles electrons provides kinetic energy, enough such that electron can leave the metal. Classically one expects: - Kinetic energy of electron depends on the intensity of the light - Electrons can be ejected for any frequency of light, if intensity is high enough. What is observed (Lenard, Millikan, later on) - Kinetic energy of electron does not depend on intensity of light - Only above threshold frequency do you see any electrons - # of ejected electrons depends on the intensity of field - Kinetic energy depends linearly on v Einstein: light is absorbed in discrete packets of energy that depend on the frequency (called photons, much later). The photon density is proportional to intensity of the light. Chapter Early Quantum Phenomena

5 E photon = hv E kin = hv φ = hv hv 0 φ : workfunction: energy to liberate electron from the metal Intensity of light ~ number of photons This picture explains all aspects of the experiment. Some aspects were predicted by Einstein, and verified later. Wave nature of light was well established. Now light looks like particles Modern version of photoelectric effect: Photoelectron Spectroscopy Molecular orbital picture Different binding energies different onsets for kinetic energy Chapter Early Quantum Phenomena

6 Emission Spectrum of Hydrogen Consists of discrete lines, sharp frequencies with a regular pattern. Balmer (a school teacher) found the first set of lines v wavenumber v = v c = λ in cm - The number of crests in per cm Balmer v n = R( n ) n = 3,4,5... Lyman v n = R( n ) n =,3, 4 Paschen v n = R( 3 n ) n = 4,5,6 R Picture: Hydrogen atom has discrete set of energy level n Chapter Early Quantum Phenomena 3

7 R( n i n ) = R( f n f n ) i n i =,,3 etc. nf > ni Simple convincing picture Why do energy levels go as n?? What do integers have to do with it?? Bohr s Model of the atom (93) A first attempt. The concept was not quite right, but the numbers came out very well. Let us see what he did: Electron moves in circle around nucleus Chapter Early Quantum Phenomena 4

8 r = x(t) = Rcosωt v = dr dt = ω Rsinωt yt ( ) = Rsinωt + ωrcosωt a = dv dt = ω Rcosωt ω Rsinωt F = m a directed inwards F = mrω = mv R F = v = Rω e 4πε 0 R Coulomb Attraction Bohr s vital insight concerned the angular momentum L = r p = r m v L z = m(r x v y r y v x ) = mω R (cos ωt + sin ωt) = mω R = mvr ( v = ω R ) Bohr postulated that the angular momentum is quantized L z = mvr = n = n h π...and that electrons can move in stationary orbits. [glaring conflict with classical physics: accelerating charge radiates and loses energy Falls into the nucleus] Let us put it together.. mv R = e 4πε 0 R () mvr = n () Chapter Early Quantum Phenomena 5

9 E n = mv e 4πε 0 R (kinetic energy + Coulomb potential) = e 4πε 0 R e 4πε 0 R = e 4πε 0 R (3) Use () (mvr) R mr = e 4πε 0 R (n) = e m 4πε 0 R R = e m 4πε 0 n E n = e 4 m (4πε 0 ) (use in (3)) n (E k E n ) = hv = hcv = me 4 8 ε 0 h ( k n ) R H = me cm 3 8 ε 0 ch Bohr later found that m= m the mass of the electron should be replaced by the reduced mass e µ = m e m p m e + m p then R is even closer to experiment. H Repeating the analysis for a nuclear charge of Z (change Coulomb potential term) R H = Z e 4 µ 8ε 0 ch 3 m p µ = m e m e + m p This gave excellent agreement for He +, Li + etc., using the appropriate mass m p m nucleus He + measured from the solar spectrum (was assigned using Bohr s formula) Chapter Early Quantum Phenomena 6

10 De Broglie Waves In 93 de Broglie proposed that particles have a wave length λ = h p His line of thought (more or less) of photons (light particles) is that they have no mass, but they do have momentum. Scattered light imparts momentum to electron (Compton) Relativistically (see problem assignment) E = hv = pc De Broglie: λ = c v = h p Generalize to all particles, not only photons: λ = h p We can even assign direction k is wave number π λ =, k k = π λ k = k + k + k x y z π p = k h λ = π k p = " k = h π Wavelength electron: m 3 = 90 kg v ~ 3k T B 3 from mv = k T m B 4 ~0 m s - thermal energy (translational) p = mv, λ = h mv ~ m = 60 nm Chapter Early Quantum Phenomena 7

11 One can use electrons, speed v, to diffract like a wave Electron microscopy Shorter wavelengths then visible light. Larger wavelengths than x- rays. All of this is well confirmed by experiment: - Today Zeilinger (Vienna) confirmed wave- character of C 60 Bucky balls - Electron microscopy widely used De Broglie relation sheds light on Bohr quantization rate L = mvr = pr = n = nh π Or π R = n h p = nλ How to interpret? If wavelength fits on circle π R constructive interference Otherwise mismatch not a stationary solution. De Broglie opened up the perspective to view electron as a wave. Double slit experiment (take ) a. Shine light through a small hole and detect beam. Big hole straight line Chapter Early Quantum Phenomena 8

12 a. do the same experiment with tiny bullets individual holes in screen collect # of bullets in buckets Now repeat the experiment with holes: Bullets: individual bullets sum of patterns Repeat with light: Dark and light bands. Interference effect Chapter Early Quantum Phenomena 9

13 What is the explanation see McQuarrie: Extra length d sinθ = nλ constructive interference d sinθ = (n + )λ destructive interference How do electrons behave? a) Like bullets in that they hit the screen as discrete units b) As a wave: electrons, (and neutrons, bucky balls) show dispersion and interference effect How do light/photons behave? Lower intensity of the light: photons will hit the screen as isolated specks photons (moving at the speed of light) and electrons (quantum particles) behave very alike. Even if one electron or one photon at a time goes through interference pattern builds up (no prediction for individual particles, only statistics is predicted) Chapter Early Quantum Phenomena 0

14 Electron propagates like wave, is detected like particle when visualized macroscopically. Through which slit does the electron go? Measure it Experiment works beautifully. Each electron passes through one hole, only and then hits the screen (also for photon) Correlate slit spot on the screen. Pattern builds up over time, but it is like a bullet pattern, no interference. Looking at which slit the electron takes destroys the interference pattern. Chapter Early Quantum Phenomena