Quantum states: wave function N particles: Identical
particles
e.g. 2 particles:
Define:
exchange particles 1 & 2:So:
Non-interating system
N-particales:
Essential information: single particle state
So accumpation number of single particle state: Fork basis
boson:
fermion:
e.g. Free particlesPartition function:Grand partition
function:boson:fermion:N:So:In the limit:Classical
Maxwell-Boltzman distribution:
Ideal boson gas
Ideal boson gas
Non-relative free particles: (dispersion relation) Wavefunction:
Periodic boundary condition: ,
So
Prescription: In d-dim: 1. N Define dimensionaless quantity: Define Poly logarithm： So:
1. E 1. lnQ So:
Non-degenerate limit:
Replace z with n: - order : -
order :
So:
>[!caution] >Bose statistics, no interaction!
Classical limit:
The degenerate Bose gas:
[[Bosons.pdf]]
Bose-Einstein condensation
,
and increase for . So
When temperature is low,
,
so it's impossible to keep N fixed. Hit : when : When
, , can not accumulate all
particles. Consider the :
So a huge number (Macroscope
large) of particles populate the lowest energy state. in which means number of paricles at ground
state.
[!abstract] Bose-Einstein Condensation Macroscope number of particles
condense into a single quantum state.
Consequece: , isn't calculate. So finite :
When , , so: Thermodynamic
limit:
Thermodynamic property
&
When , :
Define , According to
Clausius-Clapeyron Equation: ##### Energy & Capacity When : When : And
Experiment
observation of B.E.C: superfluidity in
Differences between B.E.C. and Superfluidity Helium 1. Helium is
liquid, imcompressible. 2. 3. Heat capacity
Black-body Rediation
Black body: perfect absorber of light. Rediation: EM fields at finite
T in thermal equilbrium.
Quamtized EM field:
harmonic oscillator
in
which is wave vector, and
is polarization. There are 2
descriptions: 1. a collection of harmonic oscillator, with energies
2. a gas of photons, of energies
photons
number isn't conserved:
According to Boson-Einstein distribution:
Photon despersion
Energy flux
Rate of energy flow per unit area: Stofan-Boltzman Law
Plank distibution
Wein displacement Law: 1. (high frequency):
Wein 1896 2. (classical limit):
Rayleigh-Jeans So number of
modes in :
Average numbers of photons:
Heat capacity of solid
Solid: atom form a periodic lattice, viberating around its potential
minimum. Kinetic energy： Potential:
Equipartition:
Quantized:
Einstein model
Debye model
In fact, there are "cheaper" excitations:
Quantized sound wave: phonon
Ideal phonon gas: : density
of state And
need a cutoff: 1. (low temperature): 2. (high
temperature): >[!note] >
Ideal Fermi gas
And in 3D space
So
in which is
the spin degeneracy. Define :
Non-degenerate limit:
Expand in power series
of : Equation of state:
The degenerate Fermi gas
For boson, .
For fermions, no such restriction.
Fermions successively occupy each energy
level until . So is the last filled energy level :
is step function. So : Fermi
wavevector. Fermi surface
In general, fermi surface are not spherical, non-isotropic So
>[!note] > at : degeneracy pressure
Another way We can use the density of states to calculate
the : can be seen in [[Density
of states]] consider the
spins.
differs significantly from
only within a window
. Only particles near Fermi surface can be thermally excited.
Sommerfeld
expension: First
order correction:
Electrons in
a magnetic field: Pauli paramagnelism
Electron spin
Electron spin, or, magnetic moment couples to external :
in which is Bohr
magetism.
Magnetization
The area means the number
of particles N
magnetization:
Susceptibility
1. ：paramagnetism 2. : saturates to a finite value. 3. Hign
temperature: Curie's Law 4. ### Landan
diamagnetism orbited motion
couples to : In
classical physics, we can define . No magnetism in classical
physics. Bohr-Van Leeuwen Theorem
