PHYS 112 Ch 7 (Quantum Statistics)

T; chemical potential

considering a system in contact with a reservoir that can exchange energy and particles with it, the reservoir now provides constant — and constant — —

grand partition function

the sum over all possible states, including all possible values of N for each state

multiple

the grand partition function allows for — occupancy of states

classical; quantum; wave packets

for an ideal gas, when Z1 » N, it’s in the — regime, and particles do not have to share a — state because their — — hardly ever overlap

low; large

multiple occupancy becomes an issue at — temps because the quantum volume will be —

single definite quantum; thermal equilibrium

considering the distribution for quantum particles at multiple occupancy in state n; the — — — state is the system, and all other states (reservoir) must be in — — with the system

grand partition function

the probability that a particle state of energy ei is occupied by n particles

zero; infinity; partition

if the average energy of a single particle is — or temperature goes to —, then multiplicity is the — function

step function; fermi energy

FD: for T = 0, nFD is a — —, and mu is equal to — —

decreases; increases; higher energy states

FD: as T increases, the occupation below e = mu — while occupation above —; this reflects particles being thermally excited to — — —

converges; does not converge; diverges

BE: for e > mu, nBE converges, otherwise nBE does not —. At e = mu, nBE —

non-negative

for bosons occupation n can be any — integer

1; 0

for fermions, occupation n can be either — or —

diverges; Bose-Einstein condensation

BE: as mu goes to 0, the occupancy of the ground state — resulting in — —

N

MB: given N independent particles in total, the average distribution of particles at a given energy is — * 1/Ze^-e/kT

greater

for (e - mu)/kT is — than 1, the three distributions are equal

n(e,T)*p(e)

occupancy of energy levels D(e,T) between e and e+de is an integral over

total number of particles N

the integral over the occupancy over all energy levels is the —

e*n(e,T)*p(e)

the thermodynamic (average) energy is calculated as the integral over all energy states of —

real

for — particles, N is fixed and independent of temperature, and mu is dependent on temperature

virtual; helmholtz; 0

for — particles, N is not fixed and is determined by the value which minimizes — free energy, therefore mu is —

degenerate fermi gas

modelled by a non-interacting fermion gas in a box, which can be applied to metals, in which the lattice potential is then approximated by the inf potential well

quantum volume; volume

to determine whether electrons in the free e gas are in the quantum regime at a certain temperature, one must compare the — at that temperature with the — per unit cell

fermi energy

at T=0 for a degenerate ½ spin fermi gas, mu is equal to

fermi energy

quantity determined by highest possible energy levels occupied by a particle

2

when it comes to fermions, to compute N we must take into account — spin states

en²

for a single fermion in a gas, the energy is —

fermi temperature

temp range over which quantum properties (FD statistics) dominate

less

when mu is — than 0, at higher temperatures, classical ideal gas behavior is achieved

0

the transition from a quantum gas to a classical gas occurs at a temperature where mu =

eF/k

fermi temperature is defined as TF =

particles; density

higher number of — means higher fermi energy, since fermi energy is related to —

chemical potential

the — — is determined by requiring that the number of particles be constant

1

up to the fermi energy, at T = 0, the fermi distribution is equal to —

ground state

a gas of bosons abruptly condenses into the — — below a critical temperature Tc

ground state

when T is small, N must be small because particles prefer — —

chemical potential

for bosons, — — must be equal to the ground state energy at T = 0 and only a tiny bit less than the energy when T is small (nonzero)

2; photons

the density of states for bosons is the same as that of nonrel. electrons, but no factor of — for spin (excluding —)

negative; negative; energy

for temperatures greater than the critical temperature, particle density is kept more constant by making mu more —, since a more — mu means less — is required to add particles to the system

positive

— values of mu > e0 are not possible for a Bose gas

populated; greater; negative

for a bose gas, the ground state is — at the expense of excited states when T < Tc, and is virtually unpopulated for temperatures — than the critical temperature, therefore past the critical temperature mu is —

lower; one boson; ground state

for identical bosons, entropy is —, excited states will tend to be occupied by — -—, and the remaining particles will tend to condense to — — to minimize F

pc

for relativistic particles, energy is equal to

2 polarizations

when determining the density of states for a photon gas, we must take into account — —

0

mu is — for photons, as they are constantly being created and destroyed, and therefore N is not conserved.

photon energies

as temperature increases, peak emission of photons occurs at higher — —

low

at — temperatures, high frequency photons are not excited, limiting total energy density