456
10 The Statistical Basis of Bioenergetics
called ‘phase space’, and the full ensemble can be described as a fluid in the
phase space for such a system. The dynamics of particles can be used to show
that the ensemble fluid moves in phase space without changing the density of the
systems per unit phase-space volume. The phase-space volume taken up by a given
number of adjacent systems of the ensemble does not change in time. (This is called
Liouville’s Theorem). This volume preserving property allowed Gibbs to give the
same unchanging probability for any ensemble state found within each infinitesimal
fixed phase-space volume for all those completely isolated systems with the same
macroscopically measurable values, such as the energy and volume. This is referred
to as the ‘assumption of equal a priori probability’. Quantum theory later showed
that in the semi-classical limit, the smallest phase-space volume is of size h 6N ,
where h is Planck’s constant, a result anticipated by Sackur and by Tetrode in 1912.
10.8.3 Systems Described by Quantum Theory
Quantum theory incorporates the Heisenberg uncertainty principle
q i p i ≥ ¯
h/2 ,
and so does not allow coordinates and momenta to be simultaneously known.
Rather, the results of measurement can be described in terms of a set of complex
wave functions ψ k ({q i }, t) for the system, and a set of probabilities {P k } for each
possible micro-states of the system. If only one of those probabilities, say P l , is nonzero (and therefore has a value of 1) then the system is known (by measurement)
to be in a ‘pure’ quantum state labeled by ‘l’. If more than one such probability is
non-zero, then we say the system is in a mixed state. The index k labels a full set
of simultaneously measurable quantities for the system. When we have incomplete
knowledge of what microstate the system is in, but only have probabilities that the
system will be found in one of the allowed states, it is then convenient to describe
the system in terms of the density operator 32
ˆ
ρ =
k
P k ψ k ψ
∗
k .
(10.82)
(For a pure quantum state, and only for a system in a pure state, ˆ
ρ 2 = ˆ
ρ.)
We should note that probabilities for the result of measurement of a quantum
system decompose into two distinctly different kinds. The first kind comes from
the fact that quantum theory, even for systems in pure states, only predicts the
32 The operator ˆ
ρ is also called the ‘density matrix’, introduced by John von Neumann in his
Mathematical Foundations of Quantum Mechanics (1932).
10 The Statistical Basis of Bioenergetics
called ‘phase space’, and the full ensemble can be described as a fluid in the
phase space for such a system. The dynamics of particles can be used to show
that the ensemble fluid moves in phase space without changing the density of the
systems per unit phase-space volume. The phase-space volume taken up by a given
number of adjacent systems of the ensemble does not change in time. (This is called
Liouville’s Theorem). This volume preserving property allowed Gibbs to give the
same unchanging probability for any ensemble state found within each infinitesimal
fixed phase-space volume for all those completely isolated systems with the same
macroscopically measurable values, such as the energy and volume. This is referred
to as the ‘assumption of equal a priori probability’. Quantum theory later showed
that in the semi-classical limit, the smallest phase-space volume is of size h 6N ,
where h is Planck’s constant, a result anticipated by Sackur and by Tetrode in 1912.
10.8.3 Systems Described by Quantum Theory
Quantum theory incorporates the Heisenberg uncertainty principle
q i p i ≥ ¯
h/2 ,
and so does not allow coordinates and momenta to be simultaneously known.
Rather, the results of measurement can be described in terms of a set of complex
wave functions ψ k ({q i }, t) for the system, and a set of probabilities {P k } for each
possible micro-states of the system. If only one of those probabilities, say P l , is nonzero (and therefore has a value of 1) then the system is known (by measurement)
to be in a ‘pure’ quantum state labeled by ‘l’. If more than one such probability is
non-zero, then we say the system is in a mixed state. The index k labels a full set
of simultaneously measurable quantities for the system. When we have incomplete
knowledge of what microstate the system is in, but only have probabilities that the
system will be found in one of the allowed states, it is then convenient to describe
the system in terms of the density operator 32
ˆ
ρ =
k
P k ψ k ψ
∗
k .
(10.82)
(For a pure quantum state, and only for a system in a pure state, ˆ
ρ 2 = ˆ
ρ.)
We should note that probabilities for the result of measurement of a quantum
system decompose into two distinctly different kinds. The first kind comes from
the fact that quantum theory, even for systems in pure states, only predicts the
32 The operator ˆ
ρ is also called the ‘density matrix’, introduced by John von Neumann in his
Mathematical Foundations of Quantum Mechanics (1932).