DEPARTMENT OF PHYSICS
UNIVERSITY OF PERADENIYA
PH205 - Statistical and Thermal Physics
Note 11
Quantum Statistics II
• Bose-Einstein Statistics
The particles in Bose-Einstein statistics are identical, indistinguishable bosons (integral spin par-
ticles) that do not obey Pauli exclusion principle. There are no limit to the number of bosons
that can be in the same quantum state. To compute the probability of a partition of a system
of bosons, as in the Fermi-Dirac statistics, gi gives the degeneracy of each energy state.
To determine the different and distinguishable ways in which a system of bosons can be
arranged to produce a given partition, first we should find the number distinguishable arrange-
ments of ni bosons among the gi states corresponds to the energy Ei . Note that any number of
particles can be occupied in a particular state.
Therefore total number of different ways of arranging the ni particles among gi states with Ei
is given by
(ni + gi − 1)!
(gi − 1)!
Since the particles are identical and indistinguishable there is no different if the ni particles are
reshuffled among the states they occupied in the energy level Ei . Thus total number of different
and distinguishable arrangements of ni bosons among the gi states of energy Ei is given by
(ni + gi − 1)!
ni !(gi − 1)!
Consider some partition of a boson system such that n1 , n2 , n3 , . . . . . . number of bosons are
among the energy levels E1 , E2 , E3 , . . . . . . having degeneracies g1 , g2 , g3 , . . . . . . respectively. Total
number of different and distinguishable ways of obtaining this partition can be then given by
Q (ni + gi − 1)!
P = .................................. (c)
i ni !(gi − 1)!
This is called probability of partition for bosons.
Bose-Einstein Distribution Law (The most probable partition)
To find the most probable partition of a given system of bosons, we should maximize the
probability of the partition subjected to the conditions
P
ni = N = constant ............................. (i)
i
P
ni Ei = U = constant ............................. (ii)
i
Using Stirling approximation the partition probability in equation (c) can be simplified as
1
UNIVERSITY OF PERADENIYA
PH205 - Statistical and Thermal Physics
Note 11
Quantum Statistics II
• Bose-Einstein Statistics
The particles in Bose-Einstein statistics are identical, indistinguishable bosons (integral spin par-
ticles) that do not obey Pauli exclusion principle. There are no limit to the number of bosons
that can be in the same quantum state. To compute the probability of a partition of a system
of bosons, as in the Fermi-Dirac statistics, gi gives the degeneracy of each energy state.
To determine the different and distinguishable ways in which a system of bosons can be
arranged to produce a given partition, first we should find the number distinguishable arrange-
ments of ni bosons among the gi states corresponds to the energy Ei . Note that any number of
particles can be occupied in a particular state.
Therefore total number of different ways of arranging the ni particles among gi states with Ei
is given by
(ni + gi − 1)!
(gi − 1)!
Since the particles are identical and indistinguishable there is no different if the ni particles are
reshuffled among the states they occupied in the energy level Ei . Thus total number of different
and distinguishable arrangements of ni bosons among the gi states of energy Ei is given by
(ni + gi − 1)!
ni !(gi − 1)!
Consider some partition of a boson system such that n1 , n2 , n3 , . . . . . . number of bosons are
among the energy levels E1 , E2 , E3 , . . . . . . having degeneracies g1 , g2 , g3 , . . . . . . respectively. Total
number of different and distinguishable ways of obtaining this partition can be then given by
Q (ni + gi − 1)!
P = .................................. (c)
i ni !(gi − 1)!
This is called probability of partition for bosons.
Bose-Einstein Distribution Law (The most probable partition)
To find the most probable partition of a given system of bosons, we should maximize the
probability of the partition subjected to the conditions
P
ni = N = constant ............................. (i)
i
P
ni Ei = U = constant ............................. (ii)
i
Using Stirling approximation the partition probability in equation (c) can be simplified as
1
