Chap. 3. Elementary Quantum Physics 3.1 Photons - Light: e.m "waves" - interference, diffraction, refraction, reflection with y E y Velocity = c Direction of Propagation z B z Fig. 3.1: The classical view of light as an electromagnetic wave. An electromagnetic wave is a travelling wave which has time varying electric and magnetic fields which are perpendicular to each other and to the direction of propagation.
The instantaneous intensity (energy flow per unit area per second) *Light: a stream of discrete energy packets (photons: "particles" of zero rest-mass), each carrying energy and momentum. - Young's interference eperiment: Path difference = for constructive interference = for destructive interference - Bragg's law: -ray beam from "a single crystal" (diffracted patterns of "spots") or "a polycrystalline material, powered crystal" (diffracted patterns of bright "rings" - no unique orientation of crystal aes) Eistence of a diffracted beam:
Photographic film Photographic film X-rays 2 1 1 Detector 2 θ A θ Scattered X-rays d dsinθ dsinθ Scattered X-rays Single crystal Powdered crystal or polycrystalline material d B X-rays with all X-rays with single wavelengths wavelength (c) (a) (b) Fig. 3.3: Diffraction patterns obtained bypassing X-rays through crystals can only be eplained by using ideas based on the interference of waves. (a) Diffraction of X-rays from a single crystal gives a diffraction pattern of bright spots on a photographic film. (b) Diffraction of X-rays from a powdered crystalline material or a polycrystalline material gives a diffraction pattern of bright rings on a photographic film. (c) X-ray diffraction involves constructive interference of waves being "reflected" by various atomic planes in the crystal. Atomic planes Crystal
3.1.1. The Photoelectric Effect For an incident light with emitted (the current I is generated). onto a metal surface, the electrons will be where V o = the negative anode voltage at which the current I etinguishes. where h = Plank's constant. - The work function: KE m Cs K W υ 03 slope = h 0 υ υ 02 υ 01 -Φ 3 -Φ 2 -Φ 1 Fig. 3.6: The effect of varying the frequency of light and the cathode material in the photoelectric eperiment. The lines for the different materials have the same slope of h but different intercepts.
3.1.2. Compton Scattering - X-ray scattering by an electron: KE of the elelctron = momentum of the photon Recoiling electron X-ray photon c Electron φ υ, λ y θ Scattered photon υ', λ' c Fig. 3.9: Scattering of an -ray photon by a "free" electron in a conductor.
The scattered -rays are detected at various angles with respect to the original direction ( ), their wavelength is measured. Photon energy: Photon momentum: where Source of monochromatic X-rays Collimator X-ray spectrometer λ' λ 0 θ λ 0 X-ray beam Unscattered - rays Path of the spectrometer (a) A schematic diagram of the Compton eperiment. Intensity of X-rays θ = 0 Primary beam Intensity of X-rays θ = 90 Intensity of -rays θ = 135 λ 0 λ λ 0 λ' λ λ 0 λ' λ (b) Results from the Compton eperiment Fig. 3.10. The Compton eperiment and its results.
3.1.3, Blackbody Radiation - Rayleigh-Jeans law (classical): and where the spectral irradiance = the emitted radiation intensity (power per unit area) per unit wavelength, so that = the intensity in a small range of wavelength - UV catastrophe in the short wavelength range Escaping black body radiation Hot body Small hole acts as a black body I λ Spectral irradiance 0 3000 K Classical theory Planck's radiation law 2500 K λ ( µm) 1 2 3 4 5 Fig. 3.11. Schematic illustration of black body radiation and its characteristics. Spectral irradiance vs wavelength at two temperatures (3000K is about the temperature of the incandescent tungsten filament in a light bulb).
- Plank's blackbody radiation formula: "Classical" "Quantum mechanically": light quanta = photon 3.2. The Electron as a Wave 3.2.1. De Broglie Relationship - Electron: a wave of wavelength (wave-like & particle-like) - Electron diffraction eperiments: where ("real" particle) 3.2.2. Time-Independent Schroedinger Equation - A travelling em wave: where = the spatial dependence. - The average intensity
- 1926, Ma Born : a probability wave interpretation for "the wave-like behavior of the electron" : a plane traveling wavefunction for an electric field
- The wave property of the electron described by : 1) = the probability of finding the electron per unit vol. at (,y,z) at time t. or = the probability in a small vol. ddydz. 2) has physical meaning, not itself. 3) : single-valued (See Fig. 3.14) 4) : continuous (See Fig. 3.14) ψ() ψ() not continuous ψ() dψ not continuous d ψ() ψ() not single valued Fig. 3.14: Unacceptable forms of ψ()
- Total wavefunction where, the angular frequency. - Time-independent Schroedinger equation for = the spatial dependence ; in 3-dim. space. 3.3. Infinite Potential Well: A Confined Electron - For a certain region,, an electron is confined. Using the b.c. of, The energy of the electron:
V() Electron V = V = 0 V = 0 0 a Energy levels in the well ψ() sin(nπ/a) ψ Probability density ψ() 2 4 E 4 n = 4 Energy of electron E 3 E 2 E 1 0 = 0 n = 3 n = 2 n = 1 = a ψ 3 ψ 2 ψ 1 0 a0 a Fig. 3.15: Electron in a one-dimensional infinite PE well. The energy of the electron is quantized. Possible wavefunctions and the probability distributions for the electron are shown.
Normalization condition (A) : The normalized wavefunction - Quantized energy levels: (free electron case: continuous) 3.4. Heisenberg's Uncertainty Principle - For an electron trapped in a 1-dim infinite PE well in the region, The uncertainty in position = a, the uncertainty in momentum = For n = 1 (ground state), - Heisenberg's uncertainty principle: and
3.5. Tunneling Phenomena: Quantum Leak: Finite PE Well Start here from rest D A C E B V() E < V o A 1 ψ Ι () Incident (a) V o ψ ΙΙ () ψ ΙΙΙ ( ) A 2 Reflected I II III = 0 = a Transmitted Fig. 3.16 (b) (a) The roller coaster released from A can at most make to C, but not to E. Its PE at A is less than the PE at D. When the car is the bottom its energy is totally KE. CD is the energy barrier which prevents the car making to E. In quantum theory, on the other hand, there is a chance that the car could tunnel (leak) through the potential energy barrier between C and E and emerge on the other side of the hill at E. (b) The wavefunction of the electron incident on a potential energy barrier (Vo). The incident and reflected waves interfere to give ψ I (). There is no reflected wave in region III. In region II the wavefunction decays with because E < Vo.
- Three regions: I, II, and III (boundary conditions, C 2 = 0, normalization) where - Transmission coefficient T : the relative probability that the electron will tunnel from I to III. where - For a "wide" or "high" barrier, using where
Metal ψ() Vacuum Metal ψ() Vacuum Second Metal V() V o V() V o (a) E < V o I tunnel (b) Probe Scan I tunnel Material surface Image of surface (schematic sketch) (c) Fig. 3.17: (a) The wavefunction decays eponentially as we move away from the surface because the PE outside the metal is Vo and the energy of the electron, E < Vo.. (b) If we bring a second metal close to the first metal, then the wavefunction can penetrate into the second metal. The electron can tunnel from the first metal to the second. (c) The principle of the Scanning Tunneling Microscope. The tunneling current depends on ep(-αa) where a is the distance of the probe from the surface of the material and α is a constant.
3.6. Potential Bo (3-Dim. Quantum Numbers) with V= 0 in 0<<a, 0<y<b, and 0<z<c. Let, then Using b.c, - The eigenfunctions of the electron: - The energy eigenvalues: For a=b=c, with = the quantum numbers.
3.7. Hydrogen Atom 3.7.1. Electron Wavefunctions - Potential: ; the wavefunction: - Principle quantum number: n = 1,2,3,... Orbital angular momentum quantum number: l = 0,1,2,3...(n-1) < n Magnetic quantum number: m l = -l, -(l-1),...0,...(l-1), l - Labeling of various n l possibilities : n=1 (K), L, M, N ; l=0 (s), l=1 (p), l=2 (d), l=3 (f),... - The probability that the electron is in the spherical shell of thickness : 3.7.2. Quantized Electron Energy - The electron energy: where
3.7.3. Orbital Angular Momentum and Space Quantization - Orbital angular momentum: where where states - Selection rules for EM radiation: Energy 0 n l = 0 l = 1 l = 2 l = 3 l 5 5s 5p 5d 5f 4 4s 4p 4d 4f 3 3s 3p 3d 2 2s 2p Photon -13.6eV 1 1s Fig. 3.27: An illustration of the allowed photon emission processes. Photon emission involves l = ±1.
3.7.4. Electron Spin and Intrinsic Angular Momentum S - Spin: - Magnetic dipole moment of the electron: Since, B µ orbital N ω -e = i A (a) = B = S S Spin direction = S Equivalent current N µ spin Magnetic moment (b) Fig. 3.29: (a) The orbitting electron is equivalent to a current loop which behaves like a bar of magnet. (b) The spinning electron ican be imagined to be equivalent to a current loop as shown. This current loop behaves like a bar of magnet just as in orbital case.
3.7.5. Total Angular Momentum J - Total angular momentum: J = L + S B z J J z =m j h S L J S L (a) (b) Fig. 3.31 (a) The angular momentum vectors L and S precess around their resultant total angular momentum vector J. (b) The total angular momentum vector is space quantized. Vector J precesses about the z-ais along which its component must be mjh.
[Reading Assignment] 3.8. The He Atom and The Periodic Table 3.9. Stimulated Emission and Lasers 3.10. Time-Dependent Schroedinger Equation [Homework] Prob. #3.5