2018 Quantum Physics

Transcription

1 2018 Quantum Physics Text: Sears & Zemansky, University Physics Lecture notes at TCD JF PY1P J.B.Pethica

2 Lecture 1 Summary: Classical mechanics and waves: Particle mechanics, Basic Maths, Electromagnetism, Waves, Diffraction, Thermodynamics Quantum phenomena: 1. The Photoelectric effect

3 CLASSICAL CONCEPTS ~ 1900 (i.e. things you already know) Particle Mechanics Newton s Laws of motion Force F = mass acceleration = m a Momentum p = mv Kinetic energy E = 1 2 mv 2 Conservation of momentum. Conservation of energy in elastic collisions Velocity v = dx/dt Acceleration a = dv/dt = d 2 x/dt 2 Work done by F moving from position x 1 to x 2 = x 1 x 2 Fdx

4 Periodic motion - Simple Harmonic Motion (SHM) S&Z Ch. 14 e.g. Mass on spring Spring constant µ Restoring force F= -µx F = ma = m d 2 x/dt 2 = -µx Solution x = A cos ωt ω = µ m - the resonant frequency Total energy in SHM = 1 2 µ A2 i.e. An oscillation with amplitude A and (angular) frequency ω x = Acos(ωt + φ) More generally. Phase angle φ N.B. Euler notation e iθ = cosθ + isinθ So oscillatory motion x = A e i(ωt + φ)

5 Charged Particles S&Z Ch. 21, 23 Forces on charges F = e( E + v B) (Lorenz force) Force is in direction of electric field E, plus at right angles to the plane of velocity v and magnetic field B (so B does not change v or KE) Electric potential V ( voltage ) E = - dv/dx e.g. Potential due to point charge e = e 4πεr Work done moving charge e a dist. dx through field = F dx = ee dx i.e. moving through a potential difference V = x 2 x 1 Edx changes energy by ev e.g. = change in kinetic energy for a free electron.

6 Waves Frequency f Wavelength λ Phase velocity v = fλ = ω k Angular frequency Wavenumber Plane Waves ω = 2πf ψ = Ae i ( kx ωt ) k = 2π λ Amplitude A Intensity (energy) A 2 Non-dispersive - wave velocity is constant, independent of f, λ eg. Electromagnetic waves in vacuum speed of light c Dispersive wave velocity varies with f, λ e.g. water waves Surfing (!), pond surface Group Velocity u = dω dk

7 Diffraction S&Z Ch. 36 Maxima for path difference = nλ n = 0,1,2,3,... d dsinθ = nλ θ θ 2d sinθ = nλ d Normal incidence on plane apertures Scattering from multiple planes of atoms (Bragg)

8 Relativity S&Z Ch. 37.7, 37.8 rest mass m 0 m = γ m 0 = And E = mc 2 E 2 = p 2 c 2 + m 0 2 c 4 m 0 1 v 2 c 2 Thermal properties S&Z Ch Equipartition of energy - k B T/2 per degree of freedom (mode) e.g. 1-D oscillator - k B T (1/2 P.E. 1/2 K.E.) Free particle in 3-D 3k B T/2 Oscillator in 3-D - 3k B T

9 QUANTUM PHENOMENA Classical physics has problems explaining some experiments. The distinction between classical concepts is blurred in many important experiments. Phenomena may not be regarded as strictly wave-like or particle-like. Key observations are: Photo-electric effect, Compton effect, specific heats, black-body radiation, atomic spectra, electron diffraction. Solving these led to a revolution in thinking: photons, wave-particle duality, uncertainty principle & more.

10 A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. de Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin; P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr; I. Langmuir, M. Planck, M. Skłodowska-Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch.-E. Guye, C.T.R. Wilson, O.W. Richardson 1927 Solvay conference

11 The Photoelectric Effect Evacuated tube, 2 electrodes E: emitter, C: collector Light incident on E, electrons are emitted & travel to C Current I in external circuit depends on V Note: polarity of V impedes arrival of photo-electrons: retarding or stopping potential

12 The Photoelectric Effect: what is observed (1) I-V dependence (for a single frequency of light) V = V o gives I = 0 the stopping potential implies a range of electron kinetic energies from 0 to KE max, where KE max = ev o (2) linear dependence of I on light intensity, BUT V o is unchanged by intensity i.e. intensity of light affects number of but not energies of electrons (3) no time delay ( instant emission) (4) AND. An important light frequency dependence...

13 The frequency dependence V o depends linearly on f Write V o (f - f o ) Note: cut-off frequency (f o ) below which there is no current All these observations are incompatible with Classical Physics

14 Electrons in the emitter Electrons in metal held in a potential well Highest lying electrons at energy depth φ known as work function φ Classical view: electrons accumulate energy from incident light waves! Therefore KE should increase with light intensity cf (1) + (2) Also, should see time lag at low intensity cf (3) Should be no minimum frequency cf (4) To solve this PROBLEM, Einstein (1905) borrows from Planck..

15 Einstein model of photoelectric effect Light is not waves but energy packets (later photons ) each photon has energy hf = ω Planck s constant h Photoelectron is ejected (instantly) KE max through the complete absorption of one photon. hf = KE + (depth in well) Consider the highest-lying electrons hf = KE max + φ KE max = hf φ (recall: KE max = ev o ) ev o = hf - φ V o = (h/e) f - (φ/e) = (h/e)(f - f o ) hf φ hν hf

16 Despite then the apparently complete success of the Einstein equation, the physical theory of which it was designed to be the symbolic expression is found so untenable that Einstein himself, I believe, no longer holds to it (Millikan)

17 Summary photoelectric effect Observe: 1. Electrons only emitted for Conclude: (using ω for frequency, = h/2π ) ω > ω 0 2. Intensity of light affects the number of electrons but NOT their energy 3. Emitted electron max. KE a) Photon energy E =!ω φ =!ω 0 =!( ω ω ) 0 b) Work Function is the energy required to extract an electron from the metal.