142
3 Ensembles: Systems of Particles
in which H the Hamiltonian operator and Tr designates a summation over the
diagonal elements of e −βH : in the energy representation (i.e., the representation
consisting of the energy eigenstates of the Hamiltonian H), this sum reduces
immediately to Eq. (3.2.12b).
What about the constant β? From its definition (3.2.7) we see that β has
dimensions of reciprocal energy. Notice that in our expression for the canonical
distribution, there is no reference to the system A (bath, reservoir) except that we
know that A is in contact with a large system A that is characterized by β (through
C = C (E ∗ )). In thermodynamics, two systems that are allowed to exchange
energy are said to be in thermal contact: moreover, if two systems may be brought
into thermal contact with no exchange of energy taking place, they are said to be
at the same temperature. This occurs at equilibrium, which is what we have been
describing. Thus, we may associate β with the reciprocal of the energy characteristic
of two systems in thermal equilibrium, i.e., k B T , or
β =
1
k B T
.
(3.2.13)
We obtained p r = e −βE r /z for a single available state for our subsystem A in
contact with reservoir A . What if the system A has more than one available state at
energy E r ? If the degeneracy of the energy level designated by quantum number i
is i , then
p r = r e
−βE r /
i
i e
−βE i
≡ r e
−βE r /z .
(3.2.14)
We have seen that for an equilibrium partition function z(T ) having the form z(T ) =
i i e −βE i the relative probability p r (T ) associated with a specific energy level
E r is given by p r (T ) = r e −βE r /z(T ). These probabilities tell us how a set of
particles will be distributed, or partitioned, amongst the available energy levels, so
that if we have a set of N such particles, then N r = p r N of them will be found
to have energy E r at thermal equilibrium for temperature T . The constraint that the
total number of particles is fixed at N, i.e.,
i N i = N, then allows us to express p r
in the form p r = N r /N, so that p r can equivalently be interpreted as the fractional
population associated with energy level E r . It will often turn out to be convenient to
employ this interpretation of p r .
Example 3.1 Role of (energy) degeneracy.
Let us consider two simple systems, which we shall refer to as systems A and B,
both having the same (nondegenerate) ground state energy 0 and a single excited
energy level with energy 1 = 0 + 250 cm −1 , but with the difference that 1 for
system A is nondegenerate, while 1 for system B is three-fold degenerate. We
have seen that if A and B are both in thermal equilibrium with a common bath
(i.e., a much larger system with which they may exchange energy but not mass)
3 Ensembles: Systems of Particles
in which H the Hamiltonian operator and Tr designates a summation over the
diagonal elements of e −βH : in the energy representation (i.e., the representation
consisting of the energy eigenstates of the Hamiltonian H), this sum reduces
immediately to Eq. (3.2.12b).
What about the constant β? From its definition (3.2.7) we see that β has
dimensions of reciprocal energy. Notice that in our expression for the canonical
distribution, there is no reference to the system A (bath, reservoir) except that we
know that A is in contact with a large system A that is characterized by β (through
C = C (E ∗ )). In thermodynamics, two systems that are allowed to exchange
energy are said to be in thermal contact: moreover, if two systems may be brought
into thermal contact with no exchange of energy taking place, they are said to be
at the same temperature. This occurs at equilibrium, which is what we have been
describing. Thus, we may associate β with the reciprocal of the energy characteristic
of two systems in thermal equilibrium, i.e., k B T , or
β =
1
k B T
.
(3.2.13)
We obtained p r = e −βE r /z for a single available state for our subsystem A in
contact with reservoir A . What if the system A has more than one available state at
energy E r ? If the degeneracy of the energy level designated by quantum number i
is i , then
p r = r e
−βE r /
i
i e
−βE i
≡ r e
−βE r /z .
(3.2.14)
We have seen that for an equilibrium partition function z(T ) having the form z(T ) =
i i e −βE i the relative probability p r (T ) associated with a specific energy level
E r is given by p r (T ) = r e −βE r /z(T ). These probabilities tell us how a set of
particles will be distributed, or partitioned, amongst the available energy levels, so
that if we have a set of N such particles, then N r = p r N of them will be found
to have energy E r at thermal equilibrium for temperature T . The constraint that the
total number of particles is fixed at N, i.e.,
i N i = N, then allows us to express p r
in the form p r = N r /N, so that p r can equivalently be interpreted as the fractional
population associated with energy level E r . It will often turn out to be convenient to
employ this interpretation of p r .
Example 3.1 Role of (energy) degeneracy.
Let us consider two simple systems, which we shall refer to as systems A and B,
both having the same (nondegenerate) ground state energy 0 and a single excited
energy level with energy 1 = 0 + 250 cm −1 , but with the difference that 1 for
system A is nondegenerate, while 1 for system B is three-fold degenerate. We
have seen that if A and B are both in thermal equilibrium with a common bath
(i.e., a much larger system with which they may exchange energy but not mass)