5.5 Simple Harmonic Oscillator Ensembles
237
this expression can be obtained from the first equality in expression (5.5.5) by multiplying the numerator and denominator by e ) and recalling the mathematical
expressions defining the hyperbolic functions. This form is often utilized in the
statistical physics of two-level systems and in solid state physics.
Let us examine the SHO problem from a different perspective. Thus far, we
have been primarily interested in the partition function for a single SHO in thermal
equilibrium at temperature T with an unspecified reservoir (the surroundings). As
a SHO (located at a fixed lattice site) could exchange energy with its surroundings,
we obtained an expression for the canonical partition function z int ≡ z int (T ; N = 1).
For a canonical ensemble of N such SHOs, the internal state partition function
Z SHO
int (T ; N) is then simply given by
Z
SHO
int (T ; N) = z
N
int (T ).
If, however, we focus on the means by which energy may be transferred via
‘thermal contact’ between the SHO and its surroundings, we may visualize that
a SHO can change the number of units of excitation energy, hν osc , either via
vibrationally inelastic collisions with individual members of its surroundings (such
as an ideal gas of such SHOs) or by exchanging photons carrying energy mhν osc (m,
an integer) with a thermal radiation reservoir. From the latter point of view, it makes
sense to focus on the degree of excitation of the SHO by expressing the energy of
the SHO as
n = hν osc
n +
1
2
, n = 0, 1, 2, · · · ,
with n representing the number of units of energy hν osc acquired from the
surroundings. In this context, we should like to consider the average energy of the
ensemble of eigenstates of an isolated SHO, as given, for example, by
(T ) =
1
2 hν osc +
hν osc
e βhν osc − 1
,
in terms of the average degree of excitation n(T ) of the SHO.
To see the connection between these two interpretations, it suffices to consider
the ensemble average energy (T ) as
(T ) = =(n +
1
2 )hν osc (T )
=
1
2 hν osc + hν osc n(T ).
Comparison of the two expressions for (T ) then shows that the average degree
of excitation for a SHO in thermal equilibrium at temperature T is thus
n(T ) =
1
e βhν osc − 1
.
237
this expression can be obtained from the first equality in expression (5.5.5) by multiplying the numerator and denominator by e ) and recalling the mathematical
expressions defining the hyperbolic functions. This form is often utilized in the
statistical physics of two-level systems and in solid state physics.
Let us examine the SHO problem from a different perspective. Thus far, we
have been primarily interested in the partition function for a single SHO in thermal
equilibrium at temperature T with an unspecified reservoir (the surroundings). As
a SHO (located at a fixed lattice site) could exchange energy with its surroundings,
we obtained an expression for the canonical partition function z int ≡ z int (T ; N = 1).
For a canonical ensemble of N such SHOs, the internal state partition function
Z SHO
int (T ; N) is then simply given by
Z
SHO
int (T ; N) = z
N
int (T ).
If, however, we focus on the means by which energy may be transferred via
‘thermal contact’ between the SHO and its surroundings, we may visualize that
a SHO can change the number of units of excitation energy, hν osc , either via
vibrationally inelastic collisions with individual members of its surroundings (such
as an ideal gas of such SHOs) or by exchanging photons carrying energy mhν osc (m,
an integer) with a thermal radiation reservoir. From the latter point of view, it makes
sense to focus on the degree of excitation of the SHO by expressing the energy of
the SHO as
n = hν osc
n +
1
2
, n = 0, 1, 2, · · · ,
with n representing the number of units of energy hν osc acquired from the
surroundings. In this context, we should like to consider the average energy of the
ensemble of eigenstates of an isolated SHO, as given, for example, by
(T ) =
1
2 hν osc +
hν osc
e βhν osc − 1
,
in terms of the average degree of excitation n(T ) of the SHO.
To see the connection between these two interpretations, it suffices to consider
the ensemble average energy (T ) as
(T ) = =(n +
1
2 )hν osc (T )
=
1
2 hν osc + hν osc n(T ).
Comparison of the two expressions for (T ) then shows that the average degree
of excitation for a SHO in thermal equilibrium at temperature T is thus
n(T ) =
1
e βhν osc − 1
.