10.8 The Partition Function for Ensembles
457
probabilities that certain values will be found for given observables. This probability
is intrinsic to quantum theory, and is not due to a lack of information about a system
before a measurement. 33 For example, if an atom is held in a volume by finite
forces and later its position is measured, there is a chance that the atom will not be
found in that volume. After many measurements for the same initial setup, one can
estimate the probability of finding the atom in the volume. Quantum theory lets us
predict this number. Moreover, because quantum probabilities are constructed from
interfering wave alternatives, constructive and destructive interference can manifest
themselves, making quantum probabilities for a particle being found at points across
space sometime display wave-like character.
The second kind of probability, P k , comes from our presumed lack of knowledge
about what state the system might be found in. The P k are determined by repeated
measurements on copies of the system prepared with the same initial conditions.
The P k cannot manifest interference effects between alternative system wave states.
If each copy of a large isolated system is given sufficient time to ‘relax’, and
become a steady state, then we are likely to lose the information contained in the
relative phases of waves between the different states described by quantum waves. 34
In quantum theory, the values one measures for observables, O obs , such as the
total energy of a system in a pure state, or its total angular momentum, can always
be expressed as
O obs ≡
ˆ
O obs
≡
ψ
∗
k (q) ˆ
O obs ψ k (q)dq ,
(10.83)
where ˆ
O obs is a linear operator associated with the observable O obs , and q stands
for the coordinates in the wave function.
By expanding into a complete set of eigenstates of a selected operator, wave
functions are equivalent to a vector of the components in the expansion. Operators
are then equivalent to a matrix operating on wave vectors.
For both mixed and pure state systems, the average result of many measurements
of an observable can be expressed in terms of the density operator. For example, the
average energy becomes,
33 The entropy of a pure quantum state is zero. If a measurement process involves acquiring
information about an observed system, then necessarily the entropy of the measuring system must
increase. Conventional quantum theory uses the language that the measurement has ‘collapsed’ the
wave function describing that system. But if one includes the observed system and the measuring
system in one larger system, quantum theory, in principle, can follow the transfer of information
between the two, and the observed system wave function does not collapse, but rather becomes
mixed with the measuring system states, with the observed state dominating the mixture.
34 The ergodic hypothesis includes the ‘randomization’ of the relative phases between the quantum
waves describing members of a Gibbs ensemble as the systems approach equilibrium. This kind of
loss of information is an element of Chaos Theory, wherein exact predictions become intractable
when initial conditions have any uncertainty.
457
probabilities that certain values will be found for given observables. This probability
is intrinsic to quantum theory, and is not due to a lack of information about a system
before a measurement. 33 For example, if an atom is held in a volume by finite
forces and later its position is measured, there is a chance that the atom will not be
found in that volume. After many measurements for the same initial setup, one can
estimate the probability of finding the atom in the volume. Quantum theory lets us
predict this number. Moreover, because quantum probabilities are constructed from
interfering wave alternatives, constructive and destructive interference can manifest
themselves, making quantum probabilities for a particle being found at points across
space sometime display wave-like character.
The second kind of probability, P k , comes from our presumed lack of knowledge
about what state the system might be found in. The P k are determined by repeated
measurements on copies of the system prepared with the same initial conditions.
The P k cannot manifest interference effects between alternative system wave states.
If each copy of a large isolated system is given sufficient time to ‘relax’, and
become a steady state, then we are likely to lose the information contained in the
relative phases of waves between the different states described by quantum waves. 34
In quantum theory, the values one measures for observables, O obs , such as the
total energy of a system in a pure state, or its total angular momentum, can always
be expressed as
O obs ≡
ˆ
O obs
≡
ψ
∗
k (q) ˆ
O obs ψ k (q)dq ,
(10.83)
where ˆ
O obs is a linear operator associated with the observable O obs , and q stands
for the coordinates in the wave function.
By expanding into a complete set of eigenstates of a selected operator, wave
functions are equivalent to a vector of the components in the expansion. Operators
are then equivalent to a matrix operating on wave vectors.
For both mixed and pure state systems, the average result of many measurements
of an observable can be expressed in terms of the density operator. For example, the
average energy becomes,
33 The entropy of a pure quantum state is zero. If a measurement process involves acquiring
information about an observed system, then necessarily the entropy of the measuring system must
increase. Conventional quantum theory uses the language that the measurement has ‘collapsed’ the
wave function describing that system. But if one includes the observed system and the measuring
system in one larger system, quantum theory, in principle, can follow the transfer of information
between the two, and the observed system wave function does not collapse, but rather becomes
mixed with the measuring system states, with the observed state dominating the mixture.
34 The ergodic hypothesis includes the ‘randomization’ of the relative phases between the quantum
waves describing members of a Gibbs ensemble as the systems approach equilibrium. This kind of
loss of information is an element of Chaos Theory, wherein exact predictions become intractable
when initial conditions have any uncertainty.