1. Chapter2. Elements of quantum mechanics

2. Present outline

3. 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.

4. Classical macroscopic particles

5. Propagating plane wave

6. Huygens’ principle

8. Propagating plane wave : Light is an Electromagnetic wave

9. Standing wave

10. Frequency content of light

11. Quantum mechanics: when?
1 meter Classical mechanics
1 millimeter Classical mechanics
1 micrometer Classical mechanics
1 nanometer Quantum mechanics

13. 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.

17. Origin of quantization

18. 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

21. 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.

22. 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)

25. Photoelectric effect

26. 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 × 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

27. 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 ?

28. Solution a. We are given max = 420 nm. The work function is then:
 = ho = hc/max = (6.626  J s)(3.0  108 m s-1)/(420  10-9 m)
  =  J or 2.96 eV
b. Given  = 300 nm, the photon energy is then:
Eph = h = hc/ = (6.626  J s)(3.0  108 m s-1)/(300  10-9 m)
 Eph =  J = 4.14 eV
The kinetic energy KE of the emitted electron can then be found:
KE =  - Eph = 4.14 eV 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

29. 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?

32. Wave - particle duality

33. 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

34. De Broglie
: matter waves → wave-particle duality principle

35. 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.

36. 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

38. Scanning electron microscope

40. Which electron contribute to device current ?

41. Do I really need quantum mechanics?

42. Carrier density

43. 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)

44. Schrodinger equation for electron

47. Time independent Schrodinger equation

48. Solution of Schrodinger equation with electron energy

52. Case 1: solution for particle with E>>U (free electron)

54. Case 2: Bound state problem

55. 1-D particle in a box : solution guess

59. Quantum vs. macroscopic

60. 5 steps for closed system analytical solution

61. Case 2: Bound levels in finite potential well (step 1&2)

62. Case 2: Bound levels in finite potential well (step 3)

63. Case 2: Bound levels in finite potential well (step 4)

64. Case 2: Bound levels in finite potential well (step 4 graphical solution)

67. Case 2: Wave function (step 5)

68. Key summary of a finite quantum well