myriads of other macrostate that are almost as probable and very nearly identical in their properties statistical ensemb

1. myriads of other macrostate that are almost as probable and very nearly identical in their properties statistical ensembles: a large set of macroscopically similar system. Statistical Ensembles and Fluctuations • Microcanonical Ensemble: Each system is isolated from others by rigid, adiabatic,impermeable walls; U, V, and Nfor each system are constants. • Canonical Ensemble: Each system is closed to others by rigid, diathermal, impermeable walls; T, V, and N for each system are constants, but energy can be exchanged - it is as though the systems were immersed in an isolated bath. • Grand Canonical Ensemble: Each system is separated from others by rigid, diathermal, permeable walls; U, V, and N for each system are constants

2. Microcanonical ensemble: isolated Canonical ensemble: closed Grand Canonical ensemble: open

3. Basic Quantum Mechanics • Quantum mechanics: a set of laws describing the behavior of small particle such as the electron, nuclei of the atoms, and molecules. Rayleigh-Jeans law Based on the classical physics First failure of classical physics

4. Max Plank’s concept: quantization Assume the radiation is caused by the oscillations of the electrons. The energy of the oscillators is discrete and proportional to an integer multiple of frequency.  Quantization Energy of oscillator Reproducing the experimental finding for all frequency when

5. threshold frequency: • Einstein’s theory: photons Plank’s energy quantization concept: Einstein’s photon concept: radiation as small packet of energy

6. Compton’s experiment • The wave-particle duality The wave-particle duality: light has both particle nature and wave nature. Particle like nature: photon has finite energy and momentum. Wave like nature: wavelength, frequency, diffraction, interference De Broglie’s Hypothesis: All matter has a wave-like nature.

7. over or boundary conditions : • The Schrödinger Equation • Sturm-Liouville System : Hermitian operator • (self-adjoint operator) w(x) : weighting function Corresponding eigenvalue problem

8. boundary terms = boundary terms thus Definition of inner product :

9. boundary terms i) homogeneous boundary conditions ii) when or and can be finite. iii) when either at a or b

10. : at least piecewise continuous function : eigenfunction • Properties of Hermitian operator • eigenvalues : all real • eigenfunctions : orthogonal • eigenfunctions : a complete set orthogonality: : least square convergence

11. Example : Laplace equation y x assume then Ex) 1) in (x,y) coordinate system (Cartesian) with homogeneous boundary conditions in the x direction This is a special case of S-L system with eigenfunctions : trigonometric functions, sinx or cosx

12. r z assume , then or or This is a special case of S-L system with Remark : transform with Bessel equation : Ex) 2) in (r,z) coordinate system (cylindrical) with homogeneous boundary conditions in the r direction eigenfunctions : Bessel functions

13. r q with homogeneous boundary conditions in the q direction assume equation for or with or This is a special case of S-L system with 3) in (r,q) coordinate system (spherical) Legendre equation : Remark eigenfunctions : Legendre functions

14. boundary conditions: x 0 • Schrödinger Equation a fundamental postulate of quantum mechanics Consider the one-dimensional wave equation. u(x, t): displacement of string v: speed of disturbance along the string

16. boundary conditions

17. : phase angle : spatial amplitude of the wave temporal term spatial term

18. Using De Broglie’s formula

19. time-independent Schrödinger eq.

20. or • General formulation A wave in the x direction similarly

21. : potential function

22. Hamiltonian operator in the 3-D time-dependent Schrödinger eq. time-independent Schrödinger eq.

23. Quantum state as an eigenvalue problem • discrete sequence of standing wave function • each one of which corresponds with one of the • meaningful energy levels of the particle Separation of variables

24. or

25. e is turn out to be energy Solution for a system yields the allowable quantum energy levels

26. The probability density function for the existence of a particle at and t is the product of Y and its complex conjugate Y* Interpretation of wave function A measure of the probability of finding a particle at a particular position However, a probability must be real and nonnegative, and Y is generally complex. Schrödinger himself did not come up with an explanation for the meaning of wave function. The right explanation was given by Born, who suggested that Y itself is not an observable quantity, but YY* is the probability density function to find the matter at location r.

27. : an operator of h Average or expectation value of any physical quantity h : position operator: momentum operator: energy operator:

28. The particle restricted in : amplitude of the particle • A Particle in a Potential Well or Box m v free particle perfect reflection at the wall Schrödinger equation for a free particle in a one-dimensional box x 0 L Intensity of the particle is proportional to (because the intensity of a wave is the square of the magnitude of the amplitude) boundary conditions:

29. boundary condition normalization

30. energy of the particle in a box The energy state is discrete. The quantization arises from the boundary conditions. n changes by unity, the corresponding change in e is very small because n itself is exceedingly large. Ex) gaseous helium at 300 K, L = 10 cm The change of energy when n changes by unity is so small that, for most practical purposes, the energy may be assumed to vary continuously. → A sum may be replaced by an integral.

31. The specification of an integer for each is a specification of a quantum state of a particle. All states characterized by values of the n’s such that Ex) In general, For a free particle in a 3 dimensional cubic box of volume V = L3, Degeneracy of an actual energy level is extremely large. 12 quantum states associated with the same energy level. degeneracy of 12

32. two-body problem equivalent one-body problem • A Rigid Rotor momentum of the inertia Since the center of the mass is the origin center of mass (reduced mass)

33. total energy of a rigid rotor kinetic energy: potential energy: no potential energy no external force energy of the particle independent on its orientation

34. Laplacian operator in spherical coordinate For fixed r0,

35. periodic boundary condition: normalization

36. Let Associated Legendre equation Legendre equation: Solution to the associate Legendre’s equation

37. For each l, there are 2l +1 quantum states corresponding to each individual m, because m can take up to Associated Legendre polynomials After normalization When two atoms are identical, the atoms are indistinguishable when they switch positions. The degeneracy is reduced by a symmetry number.

38. Position of the electron in the spherical coordinates nucleus Potential field due to Coulomb’s force electron orbits e0: dielectric constant = 8.854 10-12 F/m • Atomic Emission and Bohr Radius Hydrogen atom: a proton and an electron

39. eigenvalues of rotational energy : Let total energy = rotational energy + energy associated with r associate Laguerre equation

40. Ionization energy energy eigenvalues the first line in the Balmer series the second line in the Lyman series

41. Lyman series 656.3 nm 486.1 nm 434.0 nm Balmer series

42. radius of a particular electron orbit innermost orbit : Bohr radius

43. A: amplitude, f0: initial phase, : angular resonance frequency • Harmonic Oscillator m F x

44. Let transformation by using boundary conditions: at Hermite equation

45. normalized wavefunctions eigenvalue equation eigenfunction: Hermite polynomials

47. Emission and Absorption of Photons Stimulated (Induced) Emission As radiative energy passes through a medium, it is not only absorbed, but there is an additional phenomenon in that its presence stimulates some of the atoms or molecules to emit energy. Suppose that a photon of a certain frequency from the radiation field encounters a particle, such as an atom or molecule, that is presently in an excited energy state above the ground state. There exists a certain probability that the incident photon will trigger a return of the particle to a lower energy state. If this occurs, the particle will emit a photon at the same frequency and in the same direction as the incident photon. Thus, the incident photon is not absorbed but is joined by a second identical photon.

48. stimulated absorption e2 hn e1 Spontaneous emission: the result of the excited energy state of the medium being unstable and decaying spontaneously to a lower energy state. Stimulated or induced emission: emission resulting from the presence of the radiation field. Negative absorption in a sense The incident photon of energy is necessary for the process to occur.

49. spontaneous emission e2 hn e1 stimulated emission e2 hn e1 After being excited to a high energy level, the atom will emit a photon of energy and return to the lower level. As with the absorption process, the emission process can be stimulated . The light waves associated with the two photons are in phase and have the same state of polarization thereby increasing the amplitude. light amplification by stimulated emission of radiation

50. Einstein Relations Relation among the three processes of stimulated absorption, stimulated emission and spontaneous emission Requirement: for a system of atoms in thermal equilibrium with its own radiation, the rate of upward transitions must equal to the rate of downward transitions N1 atoms in the assembly with energy e1 N2 atoms in the assembly with energy e2 stimulated absorption rate u(n,T): energy density of photons n: number of photons with the appropriate frequency n per unit volume B12: a constant for a given pair of energy levels