Elements of Modern Physics
210
It is clear from earlier discussions that a complete description of even oneelectron atom is quite complicated. The complexity of the problems increases
rapidly as the number of particles increases, so as to make a detailed solution of
a many particle system almost impossibly difficult to obtain. However, as the
number of particles becomes very large, say of the order of 10
23
as in the case of
macroscopic bodies, the very largeness of the degrees of freedom leads to the
result that the average properties (macroscopic and in some cases, microscopic)
of the system correspond to, statistically, the most probable behaviour of the
system. This feature of many-particle systems forms the basis of the quantum
statistical description of their properties.
The statistical energy distributions of a collection of particles are discussed
first and then some important applications based on the most probable
distributions are considered.
7.1 DISTINGUISHABLE ARRANGEMENTS
The basic assumption for obtaining the most probable statistical distribution is
that every physically distinct arrangement of particles in the various available
states is equally likely to occur. This implies that the most probable distribution
is the one which has the largest number of distinguishable arrangements
associated with it. Therefore, the procedure for determining the most probable
statistical distribution involves two steps: (i) obtaining the number of
distinguishable arrangements which give rise to the same distribution, and
(ii) maximizing this number of arrangements with respect to different
distributions. In this section, expressions for the number of distinguishable
arrangements are obtained for a given distribution.
Consider a collection of N particles, which interact weakly with each other
and with the wall. Since these particles interact only weakly, each particle will
have a set of states with well-defined energies available to it. Let their possible
energies be grouped into cells of sizes ∆ε 1 , ∆ε 2 ..., ∆ε i ..., with average energies
ε 1, ε 2, ..., ε i, ..., respectively. These particles may belong to one of the following
three classes of particles.
1. There are Q identical but distinguishable particles. These are classical
particles whose trajectories may, in principle, be followed. There are f i
number of states and q i number of particles in the i-th energy cell.
2. There are R identical, indistinguishable bosons with integral spin. The
i-th energy cell has g i number of states and r i number of these bosons
(g i ≠ f i in general).
3. There are S identical, indistinguishable fermions with half-integral spin.
The ith energy cell has h i number of states and s i number of these fermions
(h i ≠ f i or g i , in general ). It should be noted that no two of these fermions
can be in the same state, so that h i ≥ s i .
210
It is clear from earlier discussions that a complete description of even oneelectron atom is quite complicated. The complexity of the problems increases
rapidly as the number of particles increases, so as to make a detailed solution of
a many particle system almost impossibly difficult to obtain. However, as the
number of particles becomes very large, say of the order of 10
23
as in the case of
macroscopic bodies, the very largeness of the degrees of freedom leads to the
result that the average properties (macroscopic and in some cases, microscopic)
of the system correspond to, statistically, the most probable behaviour of the
system. This feature of many-particle systems forms the basis of the quantum
statistical description of their properties.
The statistical energy distributions of a collection of particles are discussed
first and then some important applications based on the most probable
distributions are considered.
7.1 DISTINGUISHABLE ARRANGEMENTS
The basic assumption for obtaining the most probable statistical distribution is
that every physically distinct arrangement of particles in the various available
states is equally likely to occur. This implies that the most probable distribution
is the one which has the largest number of distinguishable arrangements
associated with it. Therefore, the procedure for determining the most probable
statistical distribution involves two steps: (i) obtaining the number of
distinguishable arrangements which give rise to the same distribution, and
(ii) maximizing this number of arrangements with respect to different
distributions. In this section, expressions for the number of distinguishable
arrangements are obtained for a given distribution.
Consider a collection of N particles, which interact weakly with each other
and with the wall. Since these particles interact only weakly, each particle will
have a set of states with well-defined energies available to it. Let their possible
energies be grouped into cells of sizes ∆ε 1 , ∆ε 2 ..., ∆ε i ..., with average energies
ε 1, ε 2, ..., ε i, ..., respectively. These particles may belong to one of the following
three classes of particles.
1. There are Q identical but distinguishable particles. These are classical
particles whose trajectories may, in principle, be followed. There are f i
number of states and q i number of particles in the i-th energy cell.
2. There are R identical, indistinguishable bosons with integral spin. The
i-th energy cell has g i number of states and r i number of these bosons
(g i ≠ f i in general).
3. There are S identical, indistinguishable fermions with half-integral spin.
The ith energy cell has h i number of states and s i number of these fermions
(h i ≠ f i or g i , in general ). It should be noted that no two of these fermions
can be in the same state, so that h i ≥ s i .