Chapter2. Elements of quantum mechanics
Present outline
Classical mechanics is “everyday life” mechanics. 1. An object in motion tends to stay in motion. 2. Force = mass times acceleration 3. For every action there is an equal and opposite reaction. Newton Classical mechanics is “everyday life” mechanics.
Classical macroscopic particles
Propagating plane wave
Huygens’ principle
Propagating plane wave : Light is an Electromagnetic wave
Standing wave
Frequency content of light
Quantum mechanics: when? 1 meter Classical mechanics 1 millimeter Classical mechanics 1 micrometer Classical mechanics 1 nanometer Quantum mechanics
Black body radiation A solid object will glow or give off light if it is heated to a sufficiently high temperature. Figure 2.1 Wavelength dependence of the radiation emitted by a blackbody heated to 300K, 1000K, and 2000K. Note that the visible portion of the spectrum is confined to wave lengths 0.4㎛≤λ≤0.7㎛. The dashed line is the predicted dependence for T=2000K based on classical considerations.
Origin of quantization
The Bohr atom Postulation by Bohr 1. Electrons exist in certain stable, circular orbits about the nucleus. 2. The electron may shift to an orbit of higher or lower energy, thereby gaining or losing energy equal to the difference in the energy levels. 3. The angular momentum Pθ of the electron in an orbit is always an integral multiple of Planck’s constant divided by 2π
The energy difference between orbits n1 and n2 Figure 2.2 Hydrogen energy levels as predicted by the Bohr theory and the transitions corresponding to prominent, experimentally observed, spectral lines.
Atomic Spectra The analysis of absorption and emission of light by atoms A series of sharp lines rather than a continuous distribution of wavelengths Photon energy h is related to wavelength by E = h = ∵c = λ Lyman Balmer Paschen λ (thousands of angstroms)
Photoelectric effect
An observation by Plank : radiation from a heated sample is emitted in discrete units of energy, called quanta ; the energy units were described by h, where is the frequency of the radiation, and h is a quantity called Plank’s constant En=nhν=nћω h = 6.63 × 10-34 J·s, ћ=h/2π Quantization of light by Einstein → photoelectric effect Em : a maximum energy for the emitted electrons Em = hν - qΦ ( Φ : workfunction ) Workfunction : the minimum energy required for an electron to escape from the metal into a vacuum
Photoelectric effect A photoelectric experiment indicates that violet light of wavelength 420 nm is the longest wavelength radiation that can cause photoemission of electrons from a particular multialkali photocathode surface. a. What is the work function of the photocathode surface, in eV? b. If a UV radiation of wavelength 300 nm is incident upon the photocathode surface, what will be the maximum kinetic energy of the photoemitted electrons, in eV? c. Given that the UV light of wavelength 300 nm has an intensity of 20 mW/cm2, if the emitted electrons are collected by applying a positive bias to the opposite electrode, what will be the photoelectric current density in mA cm-2 ?
Solution a. We are given max = 420 nm. The work function is then: = ho = hc/max = (6.626 10-34 J s)(3.0 108 m s-1)/(420 10-9 m) = 4.733 10-19 J or 2.96 eV b. Given = 300 nm, the photon energy is then: Eph = h = hc/ = (6.626 10-34 J s)(3.0 108 m s-1)/(300 10-9 m) Eph = 6.626 10-19 J = 4.14 eV The kinetic energy KE of the emitted electron can then be found: KE = - Eph = 4.14 eV - 2.96 eV = 1.18 eV c. The photon flux ph is the number of photons arriving per unit time per unit area. If Ilight is the light intensity (light energy flowing through unit area per unit time) then, ph =Ilight/Eph Suppose that each photon creates a single electron, then J = Charge flowing per unit area per unit time = Charge Photon Flux = 48.4 A m-2 = 4.84 mA cm-2
Electron impact excitation a. A projectile electron of kinetic energy 12.2 eV collides with a hydrogen atom in a gas discharge tube. Find the n-th energy level to which the electron in the hydrogen atom gets excited. b. Calculate the possible wavelengths of radiation (in nm) that will be emitted from the excited H atom in part (a) as the electron returns to its ground state. Which one of these wavelengths will be in the visible spectrum?
Wave - particle duality
Wave-particle Duality Compton effect E=hν=mc2, P=mc=hν/c=h/λ The change in frequency and the angle corresponds exactly to the results of a “billiard ball” collision between photon and an electron
De Broglie : matter waves → wave-particle duality principle
We will use wave theory to describe the behavior of electrons in a crystal. Figure 2.3 Constructive interference of waves scattered by the periodic atoms.
De Broglie’s Insight de Broglie postulated the existence of matter waves. He suggested that since waves exhibit particle-like behavior, then particles should be expected to show wave-like properties. de Broglie suggested that the wavelength of a particle is expressed as = h /p, where p is the momentum of a particle
Scanning electron microscope
Which electron contribute to device current ?
Do I really need quantum mechanics?
Carrier density
Erwin Schrödinger Erwin Schrödinger Austrian Theoretical Physicist (1887–1961) Schrödinger is best known as one of the creators of quantum mechanics. His approach to quantum mechanics was demonstrated to be mathematically equivalent to the more abstract matrix mechanics developed by Heisenberg. Schrödinger also produced important papers in the fields of statistical mechanics, color vision, and general relativity. (AIP Emilio Segré Visual Archives)
Schrodinger equation for electron
Time independent Schrodinger equation
Solution of Schrodinger equation with electron energy
Case 1: solution for particle with E>>U (free electron)
Case 2: Bound state problem
1-D particle in a box : solution guess
Quantum vs. macroscopic
5 steps for closed system analytical solution
Case 2: Bound levels in finite potential well (step 1&2)
Case 2: Bound levels in finite potential well (step 3)
Case 2: Bound levels in finite potential well (step 4)
Case 2: Bound levels in finite potential well (step 4 graphical solution)
Case 2: Wave function (step 5)
Key summary of a finite quantum well