The Final Lecture (#40): Review Chapters 1-10, Wednesday April 23 rd

1. The Final Lecture (#40): Review Chapters 1-10, Wednesday April 23rd • 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 Let W = number of possible outcomes (ways) Assign probability pi to the ith outcome Classical probability: • Consider all possible outcomes (simple events) of a process (e.g. a game). • Assign an equal probability to each outcome.

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

6. Statistical fluctuations

7. The axioms of probability theory • pi ≥ 0, i.e. pi is positive or zero • pi≤ 1, i.e. pi is less than or equal to 1 • For mutually exclusive events, probabilities add, i.e. • In general, for r mutually exclusive events, the probability that one of the r events occurs is given by: • 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.

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

9. Statistical distributions ni xi 6 7 8 9 10 Mean:

10. Statistical distributions ni xi 16 Mean:

11. Statistical distributions ni xi 16 Standard deviation

12. Statistical distributions Gaussian distribution (Bell curve)

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

14. Statistical Mechanics – ideas and definitions The microcanonical ensemble: Each system has same: # of particles Total energy Volume Shape 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) 10 particles, 36 cells Volume V Cell volume, DV

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

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

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

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 Partition function: Canonical ensemble leads to Boltzmann distribution function: Degeneracy: gj

21. Entropy in the Canonical Ensemble M systems ni in state yi Entropy per system:

22. The bridge to thermodynamics through Z js represent different configurations

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

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

25. Factorizing the partition function

26. Equipartition theorem Also, free energy may be written as a sum. It is in this way that each degree of freedom ends up contributing 1/2kB to the heat capacity. 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.

27. Rotational energy levels for diatomic molecules l = 0, 1, 2... is angular momentum quantum number I = moment of inertia CO2 I2 HI HCl H2 qR(K) 0.56 0.053 9.4 15.3 88

28. Vibrational energy levels for diatomic molecules n = 0, 1, 2... (harmonic quantum number) w w = natural frequency of vibration I2 F2 HCl H2 qV(K) 309 1280 4300 6330

29. Specific heat at constant pressure for H2 CP = CV + nR H2 boils w CP (J.mol-1.K-1) Translation

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, ni, represent the occupation numbers associated with each single-particle state with wavefunction fi. • 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: • It changes sign if you switch any two labels, i.e. any two rows. • It is antisymmetricwith respect to exchange • It is ZERO if any two columns are the same. • Thus, you cannot put two Fermions in the same single-particle state!

33. Fermions 2e e 0 • As with bosons, there is another way to describe N particle system: • For Fermions, these occupation numbers can be ONLY zero or one.

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

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

36. Dense versus dilute gases Dilute: classical, particle-like Dense: quantum, wave-like lD • 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 lD (mT)-1/2 lD (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. • Problem 8 and Chapter 10.

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

39. Spin } Symmetric Antisymmetric Fermions:

40. Particle (standing wave) in a box Lz Ly Lx Boltzmann probability:

41. Density of states in k-space kz ky kx

42. The Maxwell distribution Number of occupied states in the range k to k + dk Distribution function f(k): In 3D: V/p3 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

43. Maxwell speed distribution function

44. Density of states in lower dimensions In 1D: L/p is the density of states per unit k interval In 2D: A/p2 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

45. Density of states in energy In 3D:

46. Useful relations involving f(k)

47. The molecular speed distribution function

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

49. The Maxwell velocity distribution function