classical and quantum statistics: mb, be & fd statistics · • particles must obey...
TRANSCRIPT
Dr. Neelabh Srivastava(Assistant Professor)
Department of PhysicsMahatma Gandhi Central University
Motihari-845401, BiharE-mail: [email protected]
Programme: M.Sc. Physics
Semester: 2nd
Classical and Quantum Statistics: MB, BE & FD Statistics
Lecture NotesPart – 2 (Unit-III & IV)
PHYS4006: Thermal and Statistical Physics
Need for Quantum Statistics
2
Classical statistics due to Maxwell-Boltzmann explained the
energy and velocity distribution of the molecules of an ideal gas
but it failed to explain the observed energy distribution of electrons
in the so called ‘electron gas’ and that of photons in the ‘photon
gas’.
Moreover, M-B distribution function suffers from the following
two major objections –
i. First the particles are assumed to be distinguishable though
electrons or other elementary particles are indistinguishable.
ii. Any number of particles was allowed to occupy the same
quantum state while many particles particularly electrons,
obey Pauli’s exclusion principle which does not allow a
quantum state to accept more than one particle.
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In classical statistics,
So, the probability of a cell containing two or more particles is
negligible and the particles thus can be treated as distinguishable.
But, all natural systems obey uncertainty principle and minimum size
of a cell in phase space in such a case is h3 so that number of cells gi is
limited by the value of h3.
When occupation index , the particles cannot be treated as
distinguishable and thus classical statistics cannot therefore be applied.
These difficulties were resolved by the use of Quantum statistics and
can be divided as:
i. Bose-Einstein (BE) statistics
ii. Fermi-Dirac (FD) statistics
1i
i
g
n
1i
i
g
n
Fermi-Dirac (FD) StatisticsThe basic assumptions of FD statistics are:
• It is applicable for identical and indistinguishable particles.
• Total energy and total number of the particles of the entire
system is constant.
• Spin of the particles should be half integral spin i.e.
• Particles must obey Pauli’s exclusion principle, the number of
particles in each sublevel will be one or more i.e. gi ≥ ni.
• Functions associated with the particles should be anti-
symmetric.
• All particles which obey F.D. statistics are known as Fermions
(electron, proton, neutron, 3He, neutrino, muons, all hyperons
(Λ, Σ, Ξ, Ω) etc).
4
.........,2
3,
2
1
5
Fermi-Dirac distribution law
Consider a system of Fermions having total number of particles N.
We assume that these particles can be divided into different quantum
groups or levels having energies ε1, ε2…….. εk and the number of
particles in these levels are n1, n2, ………. nk, respectively and let gi
represent the degeneracy of the energy level εi.
Since the particles are indistinguishable and only one fermion can
accommodate in one cell.
So, the number of meaningful ways in which ni particles are arranged
in gi quantum states is given by -
i
i
n
gC
!
)1()1)((
i
iiii
n
nggg or
)!(!
!
iii
i
ngn
g
6
Thus, the total thermodynamic probability for an assembly of
fermions (or particular macrostate) is
k
i iii
ik
ngn
gnnnW
1
21)!(!
!),(
Most probable state:
taking natural logarithms
k
i
ii
k
i
ii
k
i
ngngW111
))!ln(()!ln()!ln()ln(
Using Stirling’s approximation in above equation, we get
k
i
ii
k
i
iiii
k
i
i
k
i
iii
k
i
ngngngnnngW11111
)()ln()()ln()!ln()ln(
Differentiating on both sides, we get
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k
i
iii
k
i
ii ngdnndnWd11
)ln()()ln())(ln(
k
i
i
i
ii dnn
ngWd
1
)(ln))(ln(
In addition, the system must satisfy two auxiliary conditions:
i. Conservation of total number of particles:
ii. Conservation of total energy of the system:
Applying Lagrange’s method of undetermined multipliers
0)()()(ln111
k
i
ii
k
i
i
k
i
i
i
ii dndndnn
ng
For the most probable state, 0))(ln( Wd
k
i
idndn1
0
k
i
iidndU1
0
8
0)(ln1
k
i
ii
i
ii dnn
ng
Proceeding as before
i
i
iii
i
ii
n
ng
n
ng
ln0ln
1
1)(1
i
i
eg
nfe
n
g
i
ii
i
i
Now, as we know thatTk
1
The other multiplier, α is related to the chemical potential μ
kT
E
kT
F
where EF is Fermi energy
FD distribution
function
Bose-Einstein (BE) Statistics
The basic assumptions of BE statistics are:
• It is applicable for identical and indistinguishableparticles.
• Any number of particles can occupy a single cell inthe phase space.
• The size of cell can not be less than h3 where h isPlanck’s constant.
• The number of phase space cells is comparable withthe number of particles i.e., occupation index,ni/gi=1.
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• It is applicable to particles with integral spin angular
momentum in units of h/2π. All particles which obey
B.E. statistics are known as Bosons (photon, phonon, all
mesons etc).
• The wave function of the system is symmetric under the
positional exchange of any two particles.
• Total energy and total number of particles of the entire
system is constant.
10
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Let us consider a system of bosons having total number of particles
N. Suppose these identical, indistinguishable and non-interacting
particles are to be distributed among gi different quantum states each
having energy εi.
So, the number of meaningful arrangements for which ni
indistinguishable particles are to be distributed among gi cells
without any restriction on the number of particles is given by -
Number of ways =
!1!
!1
ii
ii
gn
gn
Bose-Einstein distribution law
So, the thermodynamic probability for a macrostate is then:
k
i ii
iik
gn
gnnnnW
1
21!1!
!1),,(
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Taking natural logarithm:
k
i
i
k
i
iii
k
i
gngnW111
))!1ln(()!ln())!1(ln()ln(
Using Stirling’s approximations, we get
k
i
i
k
i
ii
k
i
i
k
i
ii
k
i
iiii
k
i
ii
gggn
nngngngnW
111
111
)1()1ln()1(
)ln()1()1ln()1()ln(
i
k
i ii
i dngn
nWd
1
ln))(ln(On further solving, we get
Using Lagrange’s method of undetermined multipliers wherein we
multiply with two constraints α and β, respectively, one get
0ln111
k
i
ii
k
i
i
k
i
i
ii
i dndndngn
n
13
Proceeding as before:0sin0ln
ii
ii
i dncegn
n
Since, ni and gi are very large therefore, ni+gi-1 = ni+gi
ii
ii
i
ii
i
n
ng
gn
n
exp
expln
1
11
i
i
eg
ne
n
g
i
i
i
i
Hence, expression for the most probable distribution of particles among
various energy levels of a system obeying B-E statistics is given by -
)(1
lawondistributiEinsteinBosee
gn
i
ii
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In general, combining all the three statistics, we can write the expression
as
kT
i
i
ieg
nf
/)(
1)(
where, constant takes values 0, -1 and 1 for M-B (classical), B-E and
F-D statistics.
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Plot of fFD versus ε for
fermions at two temperatures
T1 and T2 (> T1)
Plots of fBE versus ε for
bosons at two different
temperatures T1 and T2 (> T1)
For εi=μ, fFD=1/2 whereas fBE→∞.
At low temperatures, when the f(ε) is nearly a step function, the
distribution function is said to be strongly degenerate. At very high
temperatures, when the step like character is lost, it is said to be
nearly non-degenerate.
1. Distribution of Bosons is skewed towards lower energy states, whereas
fermions are skewed towards higher energy states, i.e. higher probability of
finding bosons in low level energy states and fermions in higher energy
states.
2. At sufficiently high energies, classical and quantum results are identical.
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at lower energies,
fBE > fMB > fFD
• Thus, it is obvious that the quantum statistics (B-E and F-D) will tend to classical one (M-B) onlywhen
then
• Under high temperature and low particle density,quantum statistics tend to classical statistics.
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1/)(
kTie
1i
i
g
n
Degeneracy
A system is said to be –
• Strongly degenerate when
• Weakly degenerate when
• Non-degenerate when
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1i
i
g
n
1i
i
g
n
1i
i
g
n
References: Further Readings
1. Statistical Mechanics by R.K. Pathria
2. Statistical Mechanics by K. Huang
3. Statistical Mechanics by B.K. Agrawal and M. Eisner
4. Thermal Physics (Kinetic theory, Thermodynamics and
Statistical Mechanics) by S.C. Garg, R.M. Bansal and C.K.
Ghosh
5. Heat, Thermodynamics and Statistical Physics by C.L. Arora
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Assignment1) Calculate the number of different arrangements of
10 indistinguishable particles in 15 cells of equal apriori probability considering that one cell containsonly one particle.
2) A system has 7 particles arranged in twocompartments. The first compartment has 8 cellsand the second has 10 cells (all cells are of equalsize). Calculate the number of microstates in themacrostate (3, 4) when particles obey F-D statistics.
3) The molar mass of lithium is 0.00694 and itsdensity is 0.53×103 kg/m3. Calculate the Fermienergy and Fermi temperature of the electrons.
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4) In a system of two particles, each particle can be in anyone of three possible quantum states. Find the ratio of theprobability that the two particles occupy the same state tothe probability that the two particles occupy differentstates for MB, BE and FD statistics.
5) Three spin-1/2 fermions are to be distributed in twolevels having energies E1 & E2, respectively. Each levelcan accommodate a maximum of two fermions withopposite spins, +1/2 and -1/2. Determine the number ofmicrostates and macrostates.
6) Show that if f is the FD distribution function
is a maximum at the Fermi level. Also, show that
is symmetric about the Fermi level.21
E
f
E
f
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For any questions/doubts/suggestions and submission of
assignment
write at E-mail: [email protected]