1. The Final Lecture (#40): Review Chapters 1-10, Wednesday April 23 rd Announcements Homework statistics Finish review of third exam Quiz (not necessarily in this order) Review Chapters 3 to 7 Reading: Chapters 1-10 (pages 1 - 207) Final: Wed. 30th, 5:30-7:30pm in here Exam will be cumulative

2. Homework Statistics

3. Review of Chapters 3 & 4

4. Classical and statistical probability Classical probability: Consider all possible outcomes (simple events) of a process (e.g. a game). Assign an equal probability to each outcome. Let W = number of possible outcomes (ways) Assign probability p i to the i th outcome

5. Classical and statistical probability Statistical probability: Probability determined by measurement (experiment). Measure frequency of occurrence. Not all outcomes necessarily have equal probability. Make N trialsMake N trials Suppose i th outcome occurs n i timesSuppose i th outcome occurs n i times

6. Statistical fluctuations

7. The axioms of probability theory 1. p i ≥ 0, i.e. p i is positive or zero 2. p i ≤ 1, i.e. p i is less than or equal to 1 3.For mutually exclusive events, probabilities add, i.e. Compound events, (i + j): this means either event i occurs, or event j occurs, or both.Compound events, (i + j): this means either event i occurs, or event j occurs, or both. Mutually exclusive: events i and j are said to be mutually exclusive if it is impossible for both outcomes (events) to occur in a single trial.Mutually exclusive: events i and j are said to be mutually exclusive if it is impossible for both outcomes (events) to occur in a single trial. In general, for r mutually exclusive events, the probability that one of the r events occurs is given by:In general, for r mutually exclusive events, the probability that one of the r events occurs is given by:

8. Independent events Example: What is the probability of rolling two sixes? Classical probabilities: Two sixes: Truly independent events always satisfy this property. In general, probability of occurrence of r independent events is:

9. nini xixi Statistical distributions 879106 Mean:

10. Statistical distributions nini xixi 16 Mean:

11. Statistical distributions nini xixi 16 Standard deviation

12. Statistical distributions Gaussian distribution (Bell curve)

13. Statistical Mechanics – ideas and definitions A quantum state, or microstate A unique configuration.A unique configuration. To know that it is unique, we must specify it as completely as possible...To know that it is unique, we must specify it as completely as possible... Classical probability Cannot use statistical probability.Cannot use statistical probability. Thus, we are forced to use classical probability.Thus, we are forced to use classical probability. An ensemble A collection of separate systems prepared in precisely the same way.A collection of separate systems prepared in precisely the same way.

14. Statistical Mechanics – ideas and definitions The microcanonical ensemble: Each system has same:# of particles Total energy VolumeShape Magnetic field Electric field and so on................ These variables (parameters) specify the ‘macrostate’ of the ensemble. A macrostate is specified by ‘an equation of state’. Many, many different microstates might correspond to the same macrostate.

15. Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

16. Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

17. Entropy Boltzmann hypothesis: the entropy of a system is related to the probability of its being in a state.

18. Rubber band model d Sterling’s approximation: ln(N!) = NlnN  N

19. Chapters 5-7 Canonical ensemble and Boltzmann probability The bridge to thermodynamics through Z Equipartition of energy & example quantum systems Identical particles and quantum statistics Spin and symmetry Density of states The Maxwell distribution

20. Review of main results from lecture 15 Canonical ensemble leads to Boltzmann distribution function: Partition function: Degeneracy: g j

21. Entropy in the Canonical Ensemble M systems n i in state  i Entropy per system:

22. The bridge to thermodynamics through Z j s represent different configurations

23. A single particle in a one-dimensional box V(x)V(x) V = ∞V = 0 V = ∞ x x = L

24. The three-dimensional, time-independent Schrödinger equation: A single particle in a three-dimensional box

25. Factorizing the partition function

26. Equipartition theorem If the energy can be written as a sum of independent terms, then the partition function can be written as a product of the partition functions due to each contribution to the energy. free energy may be written as a sum. It is in this way that each degree of freedom ends up contributing 1 / 2 k B to the heat capacity. Also,

27. Rotational energy levels for diatomic molecules I = moment of inertia l = 0, 1, 2... is angular momentum quantum number CO 2 I 2 HIHClH 2  R (K)0.560.0539.415.388

28. Vibrational energy levels for diatomic molecules  = natural frequency of vibration n = 0, 1, 2... (harmonic quantum number) I 2 F 2 HClH 2  V (K)309128043006330 

29. Specific heat at constant pressure for H 2 C P (J.mol  1.K  1 )  H 2 boils Translation C P = C V + nR

30. Examples of degrees of freedom:

31. Bosons Wavefunction symmetric with respect to exchange. There are N! terms. Another way to describe an N particle system: The set of numbers, n i, represent the occupation numbers associated with each single-particle state with wavefunction  i. For bosons, occupation numbers can be zero or ANY positive integer.

32. Fermions Alternatively the N particle wavefunction can be written as the determinant of a matrix, e.g.: The determinant of such a matrix has certain crucial properties: 1.It changes sign if you switch any two labels, i.e. any two rows. It is antisymmetric with respect to exchange 2.It is ZERO if any two columns are the same. Thus, you cannot put two Fermions in the same single-particle state!

33. Fermions As with bosons, there is another way to describe N particle system: For Fermions, these occupation numbers can be ONLY zero or one. 0  

34. Bosons For bosons, these occupation numbers can be zero or ANY positive integer.

35. A more general expression for Z What if we divide by 2 (actually, 2!): Terms due to double occupancy – under counted. Terms due to single occupancy – correctly counted. SO: we fixed one problem, but created another. Which is worse? Consider the relative importance of these terms....

36. Dense versus dilute gases Either low-density, high temperature or high mass de Broglie wave- length Low probability of multiple occupancy Either high-density, low temperature or low mass de Broglie wave- length High probability of multiple occupancy Dilute: classical, particle-likeDense: quantum, wave-like D D  (mT )  1/2

37. A more general expression for Z Therefore, for N particles in a dilute gas: and VERY IMPORTANT: this is completely incorrect if the gas is dense. If the gas is dense, then it matters whether the particles are bosonic or fermionic, and we must fix the error associated with the doubly occupied terms in the expression for the partition function.If the gas is dense, then it matters whether the particles are bosonic or fermionic, and we must fix the error associated with the doubly occupied terms in the expression for the partition function. Problem 8 and Chapter 10.Problem 8 and Chapter 10.

38. Identical particles on a lattice Localized → Distinguishable Delocalized → Indistinguishable

39. Spin Symmetric Antisymmetric } Fermions:

40. LxLx LyLy LzLz Particle (standing wave) in a box Boltzmann probability:

41. kykykyky kxkxkxkx kzkzkzkz Density of states in k-space

42. The Maxwell distribution The Maxwell distribution In 3D:V/   3 is the density of states in k space density of states per unit k interval D(k)dk gives the # of states in the range k to k + dk Number of occupied states in the range k to k + dk Distribution function f (k):

43. Maxwell speed distribution function Maxwell speed distribution function

44. Density of states in lower dimensions Density of states in lower dimensions In 2D:A/   2 is the density of states in k space density of states per unit k interval D(k)dk gives the # of states in the range k to k + dk In 1D: L/   is the density of states per unit k interval

45. Density of states in energy Density of states in energy In 3D:

46. Useful relations involving f (k) Useful relations involving f (k)

47. The molecular speed distribution function

48. Molecular Flux Flux: n umber of molecules striking a unit area of the container walls per unit time.

49. The Maxwell velocity distribution function