Posted by peeterjoot on March 27, 2013
Here’s my second update of my notes compilation for this course, including all of the following:
March 27, 2013 Fermi gas
March 26, 2013 Fermi gas thermodynamics
March 26, 2013 Fermi gas thermodynamics
March 23, 2013 Relativisitic generalization of statistical mechanics
March 21, 2013 Kittel Zipper problem
March 18, 2013 Pathria chapter 4 diatomic molecule problem
March 17, 2013 Gibbs sum for a two level system
March 16, 2013 open system variance of N
March 16, 2013 probability forms of entropy
March 14, 2013 Grand Canonical/Fermion-Bosons
March 13, 2013 Quantum anharmonic oscillator
March 12, 2013 Grand canonical ensemble
March 11, 2013 Heat capacity of perturbed harmonic oscillator
March 10, 2013 Langevin small approximation
March 10, 2013 Addition of two one half spins
March 10, 2013 Midterm II reflection
March 07, 2013 Thermodynamic identities
March 06, 2013 Temperature
March 05, 2013 Interacting spin
plus everything detailed in the description of my first update and before.
Posted in Math and Physics Learning. | Tagged: addition of angular momentum, addition of spin, angular momentum, anharmonic oscillator, average, average diatomic separation, average dipole moment, average energy, average number of particles, average occupancy, binomial distribution, Boltzmann distribution, Boltzmann factor, Boson, canonical ensemble, Central limit theorem, chemical potential, classical harmonic oscillator, degeneracy pressure, delta function, density, density of states, diatomic molecule gas, differential form, eigenvalue, eigenvector, electric dipole, electric field interaction, electron, energy, energy eigenstate, energy eigenvalue, entropic force, entropy, equilibrium, Fermi distribution, Fermi energy, Fermi gas, Fermi temperature, Fermion, four momentum, four vector, free energy, fugacity, Gaussian approximation, Gibbs sum, grand canonical ensemble, grand canonical partition, grand partition function, graphene, hamiltonian, harmonic oscillator perturbation, heat capacity, high temperature limit, hole, ideal gas, integral approximation to sum, low temperature limit, magnetic field, magnetization, mean energy, microstate, moment of inertia, momentum, multiple paired spin, nuclear spin interaction, number of particles, number operator, occupancy, occupation number, occupation numbers, one form, orthonormal basis, partial derivative, particle in a box, Partition function, Pathria, pauli matrix, perturbation, PHY452H1S, Planck's constant, polymer, position mean value, pressure, probability, quantum anharmonic oscillator, random walk, relativistic gas, reservoir, singlet state, special relativity, specific heat, spherical harmonic, spin, spin hamiltonian, spin one half, spring constant, Statistics mechanics, subsystem, temperature, thermal average energy, thermal de Broglie wavelength, thermodynamic identity, trace, triplet states, two form, two variable Taylor expansion, variance, volume, zipper DNA model | 1 Comment »
Posted by peeterjoot on March 26, 2013
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
Peeter’s lecture notes from class. May not be entirely coherent.
Last time we found that the low temperature behaviour or the chemical potential was quadratic as in fig. 1.1.
Fig 1.1: Fermi gas chemical potential
The only change in the distribution fig. 1.2, that is of interest is over the step portion of the distribution, and over this range of interest is approximately constant as in fig. 1.3.
Fig 1.2: Fermi distribution
Fig 1.3: Fermi gas density of states
Here we’ve made a change of variables , so that we have near cancelation of the factor
Here we’ve extended the integration range to since this doesn’t change much. FIXME: justify this to myself? Taking derivatives with respect to temperature we have
With , we have for
Using eq. 1.1.4 at the Fermi energy and
This is illustrated in fig. 1.4.
Fig 1.4: Specific heat per Fermion
- Relativisitic gas
- massless Dirac Fermion
Fig 1.5: Relativisitic gas energy distribution
We can think of this state distribution in a condensed matter view, where we can have a hole to electron state transition by supplying energy to the system (i.e. shining light on the substrate). This can also be thought of in a relativisitic particle view where the same state transition can be thought of as a positron electron pair transition. A round trip transition will have to supply energy like as illustrated in fig. 1.6.
Fig 1.6: Hole to electron round trip transition energy requirement
Consider graphene, a 2D system. We want to determine the density of states ,
We’ll find a density of states distribution like fig. 1.7.
Fig 1.7: Density of states for 2D linear energy momentum distribution
Posted in Math and Physics Learning. | Tagged: chemical potential, density of states, electron, energy, Fermi distribution, Fermi gas, graphene, hole, PHY452H1S, relativisitic gas, specific heat, statistical mechanics | Leave a Comment »