Course Level:
Fourth Year
Academic Year:
2005/2006
- Introduction: Some History. Boltzmann probabilities for energy levels. Some basics of probabilities and probability distributions.
- Independent events: Coin tossing (and/or spin-1/2, systems, distribution of gas molecules between two vessels, random walk, etc.), s 2µ N, Gaussian limit. Central Limit Theorem (merely stated), error distribution, Poisson…
- Ideal gases: (a)P=1/3 Nmv2 (Clausius). Statement (or Maxwell’s derivation) of Maxwell’s velocity distribution. Calculations. Boltzmann (translational energy) distribution. (b)Distribution of energy e of one molecule vs distribution of E calculated for N molecules. Gaussian limit (as usual). [Some Math: Integrals, x!, Stirling].
- Statistical states in Quantum Mechanics (QM): (a) Density matrix, S=1/2 example. (b) Equilibrium case: Time independenceÆr diagonal in Energy basisÆspecial role of energy levels, and statistical state characterized simply by probabilities r n, and r n = f(En).
- Canonical Distribution: Derivations: (i) from fact that r n multiply while En add. (ii) from principle of equal a priori probabilities (of energy states of the thermostat). The concept of temperature. Partition function.
- Independent particles: Permutation symmetry. Boltzmann statistics. validity. Fermi-Dirac and Bose-Einstein statistics (merely stated at this stage, except for T=0).
- Translational states: Density of states, Qtrans, "thermal wave length," e , Cv etc.
- Thermodynamic functions: E, dqrev = S Endpn, dwrev = S pndEn, meaning of adiabatic. Entropy: dS = dqrev/T® S = -kS pn1npn. Various formulas. Density W of N-molecule states, and entropy as measure of number of states in e.g. a fluctuation width. Calculations of S from partition functions. Translational entropy for ideal gas (dependence on mass, density of translational states, etc.) Other thermodynamic functions, A=-klnQ, etc.
- More on systems of independent particles: Interelations of formulas expressed in terms of one-particle level occupation numbers, nk, to those in terms of N-particle probabilities. S=klnW. Derivations of Boltzmann, FD, and BE statistics from most probable (coarse grained) distribution of nk.
- Rotations and Vibrations: Distributions, densities of states, partition functions. High T limits. Calculations of E, Cv, S, etc.
- Rotations and Spin Statistics: Diatomic molecules. Ortho and para Hydrogen (Cv and equilibrium.) First look at equilibrium constants from probabilities.
- Heat capacity of Solids: Einstein, Low T, Debye. Electronic contribution in metals. 12a) Thermal radiation: (from radiation modes as harmonic oscillators) History! nk. Planck’s law. Examples, and in particular —12b)Einstein’s A and B coefficients (and detailed balance)
- Classical Statistics and the Quantum Correspondence: Classical distributions: Maxwell’s distribution, spatial distributions in a potential Energy calculations: equipartition. Correspondence of phase space volume to numbers of QM states: classical partition functions and densities of states.Many possible applications including non-ideal gases.
- Systems of independent electric and magnetic moments: Electric (classical only,) Magnetic: classical and quantum, Curie Law, entropy considerations.
- Chemical Equilibrium (for ideal gases): Derivation of Keq in terms of partition functions, without reference to thermodynamics, from maximum probability in the Canonical ensemble. Associated concept of Chemical Potential. Fluctuations. Relation to (familiar?) thermodynamic principles. Examples.
- Chemical rates: Collision numbers and simple collision theory. Transition state theory: Derivation as an equilibrium, etc., and classical derivation. Limits, order of magnitude estimates, and realistic example (F + H2 reaction).