Lecture 26 — Review for Exam II Chapters 5-7, Monday March 17th 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 Reading: All of chapters 5 to 7 (pages 91 - 159) Exam 2 on Wednesday, in class Next homework due next week

Review of main results from lecture 15 Canonical ensemble leads to Boltzmann distribution function: Partition function: Degeneracy: gj

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

The bridge to thermodynamics through Z js represent different configurations

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

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

Factorizing the partition function

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 1/2kB to the heat capacity. Also,

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

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

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

Examples of degrees of freedom:

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.

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 antisymmetric with 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!

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

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

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....

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

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.

Identical particles on a lattice Localized → Distinguishable Delocalized → Indistinguishable

Spin Symmetric Antisymmetric } Fermions:

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

Density of states in k-space kz ky kx

The Maxwell distribution In 3D: V/p 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):

Maxwell speed distribution function

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

Density of states in energy In 3D:

Useful relations involving f (k)

The molecular speed distribution function

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

The Maxwell velocity distribution function