Quantum Distributions Statistical Physics Quantum Distributions In quantum systems only certain energy values are allowed. In quantum theory particles are described by wave functions. Identical particles cannot be distinguished from one another if there is a significant overlap of their wave functions. Pauli principle has a significant impact on how energy states can be occupied and therefore on the corresponding energy distribution.

Quantum Distributions Statistical Physics Quantum Distributions Fermions: particles with half-integer spins The Fermi-Dirac distribution, which is valid for fermions:

Quantum Distributions Statistical Physics Quantum Distributions Bosons: particles with zero or integer spins The Bose-Einstein distribution, which is valid for bosons:

Statistical Physics Fermi-Dirac Statistics It provides the basis for our understanding of the behavior of a collection of fermions. Applying it to the problem of electrical conduction.

Statistical Physics Fermi-Dirac Statistics From Fermi-Dirac distribution: The factor B1 is computed by integrating n(E)dE over all allowed energies. The parameter β(=1/kT) is contained in FFD, so that B1 should be temperature dependent. B1 = exp(-βEF)

Statistical Physics Fermi-Dirac Statistics EF : Fermi energy FFD = ½ for E = EF FFD = 1 for E < EF FFD = 0 for E > EF

Statistical Physics Fermi-Dirac Statistics At T=0 fermions occupy the lowest energy levels available to them. They cannot all be in the lowest level, since that would violate the Pauli principle. Rather, fermions will fill all the available energy levels up to a particular energy ( the Fermi energy ). At T=0 there is no chance that thermal agitation will kick a fermion to an energy greater than EF.

Statistical Physics Fermi-Dirac Statistics The Fermi-Dirac factor FFD at various temperatures

Statistical Physics Fermi-Dirac Statistics As the temperature increases from T=0, more fermions may be in higher energy levels. The Fermi-Dirac factor “smears out” from the sharp step function [ Figure (a) ] to a smoother curve [ Figure (b) ]. A Fermi temperature’s defined as TF= EF /k which shown in Figure (c).

Statistical Physics Fermi-Dirac Statistics When T << TF the step function approximation for FFD is reasonably accurate. When T>> TF , FFD approaches a simple decaying exponential [ Figure (d) ]. As the temperature increases, the step is gradually rounded. Finally, at very high temperatures, the distribution approaches the simple decaying exponential of Maxwell-Boltzmann distribution.

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conduction In 1900 Paul Drude developed a theory of electrical conduction in an effort to explain the observed conductivity of metals. Drude model assumed that the electrons in a metal existed as a gas of free particles. Paul Drude (1863-1906)

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conduction In this model the metal is thought of as a lattice of positive ions with a gas of electrons free to flow through it . The electron have a thermal kinetic energy proportional to temperature. The mean speed of an electron at room temperature can be calculated to be about 105 m/s. The velocities of the particles in a gas are directed randomly. Therefore, there will be no flow of electrons unless an electric field is applied to the conductor.

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conduction When an electric field is applied, the negatively charged electrons flow in the opposite direction to the field. Drude was able to show that the current in conductor should be linearly proportional to the applied electric field.

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conduction The principal success of Drude’s theory was that it did predict Ohm’s law. Unfortunately, the numerical predictions of the theory were not so successful. One important prediction was that the electrical conductivity could be expressed as

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conduction n is the number density of conduction electrons e is the electronic charge is the average time between electron-ion collisions m is the electronic mass When combined with the other parameters in above Equation, produced a value of σ that is about one order of magnitude too small for most conductors. The Drude theory is therefore incorrect in this prediction.

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conduction Drude model, the conductivity should be proportional to T-1/2. But for most conductors the conductivity is nearly proportional to T-1 except at very low temperatures, where it no longer follows a simple relation. Clearly the classical model of Drude has failed to predict this important experimental fact.

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction How electron energies are distributed in a conductor? The real problem we face is to find g(E), the number of allowed states per unit energy. What energy values should we use? From assumption of the Drude model about “free electron” and use the results obtained in Quantum theory for a three-dimensional infinite square well potential.

The allowed energies are Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction The allowed energies are Where L is the length of side of the cube and ni are the integer quantum numbers. Above Equation can be rewritten as E = r2E1 E1 is just a constant, not the ground-state energy .

The extra factor of 2 is due to spin degeneracy. Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction The number of allowed states up to “radius” r will be directly related to The spherical “volume”(4/3)πr3 . The exact number of states up to r is The extra factor of 2 is due to spin degeneracy.

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction The factor 1/8 is necessary because we are restricted to positive quantum numbers and, therefore, to one octant of the three-dimensional number space. Nr as a function of E :

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction At T=0 the Fermi energy is the energy of the highest occupied energy level. If there are a total of N electrons, then

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction The density of states can be calculated by differentiating Equation of N with respect to energy :

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction At T = 0 we have n(E) = g(E) for E < EF , n(E) =0 for E > EF . With the distribution function n(E) the mean electronic energy can be calculated easily :

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction Therefore the internal energy (U) of the system is: The fraction of electrons capable of participating in this thermal process is on the order of kT/EF. The exact number of electrons depends on temperature, because the shape of the curve n(E) changes with temperature.

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction T = 300 K T= 0 K

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction The heat capacity

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conduction The electrical conductivity varies inversely with temperature. This is another striking success for the quantum theory.

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The intensity of the emitted radiation as a function of temperature and wavelength as:

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The electromagnetic radiation is really a collection of photons of energy hc/λ. Use the Bose-Einstein distribution to find how the photons are distributed by energy, and then use the relationship E=hc/λ to turn the energy distribution into a wavelength distribution. The desired temperature dependence should already be included in the Bose-Einstein factor.

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The energy of a photon is pc, so

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The density of states g (E) is

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The next step is to convert from a number distribution to an energy density distribution u(E). To do this it is necessary to multiply by the factor E/L3 ( that is, energy per unit volume):

Statistical Physics Bose-Einstein Statistics Blackbody Radiation Using E=hc/λ and |dE|=(hc/λ2)dλ, we find In the SI system multiplying by a constant factor c/4 is required to change the energy density to a spectral intensity:

Statistical Physics Bose-Einstein Statistics Blackbody Radiation u(λ,T) is energy per unit volume per unit wavelength inside the cavity. U(λ,T) is power per unit area per unit wavelength for radiation emitted from the cavity.

Quantum Statistics Summary Fermi-Dirac distribution Bose-Einstein distribution Function Energy Dependence Quantum Particles Fermions e.g., electrons, neutrons, protons and quarks Bosons e.g., photons, Cooper pairs and cold Rb Spins 1 / 2 integer Properties At temperature of 0 K, each energy level is occupied by two Fermi particles with opposite spins.  Pauli exclusion principle At very low temperature, large numbers of Bosons fall into lowest energy state.  Bose-Einstein condensation