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IOP Publishing

Quantum Statistical MechanicsEquilibrium and non-equilibrium theory from first principles

Phil Attard

Chapter 1

Probability operator and statistical averages

The goal of this book is to formulate quantum statistical mechanics for non-equilibrium systems. To achieve this it is necessary to go back to first principles, toestablish and explain a number of fundamental concepts that occur in equilibriumquantum statistical mechanics and to introduce some new ideas that are applicableto equilibrium systems. It is also necessary to link the quantum results toprobabilistic and thermodynamic concepts.

Over a number of years the author has addressed the non-equilibrium problemin the context of classical statistical mechanics, and the results of that researchprogram, together with already known results, have been published in book form(Attard 2012a). Since then, work has progressed on developing an analogousquantum theory. There is one immediate obstacle to such a program, namely thatclassical statistical mechanics is formulated in terms of the positions and momentaof the molecules in phase space, based on Hamilton’s equations of motion. Inquantum mechanics, the analogous quantities are the wave function in Hilbertspace and Schrödinger’s equation. The issue is that the conventional formulation ofquantum statistical mechanics is cast in terms of the quantum states rather than thewave function, and so it is not immediately obvious how to proceed with theclassical analogy.

This first chapter begins with a summary of the essential elements of equilibriumquantum statistical mechanics as it is conventionally formulated. Section 1.1 sets outthe Maxwell–Boltzmann form for the probability operator and the von Neumannmixture formulation of the density operator and of the statistical average. In thesections that follow, these are derived from first principles—Schrödinger’s equationfor the wave function in Hilbert space—in analogy with the corresponding result inclassical statistical mechanics.

doi:10.1088/978-0-7503-1188-5ch1 1-1 ª IOP Publishing Ltd 2015

1.1 Expectation, density operator and averages1.1.1 Expectation value

A quantum mechanical system can be described either by its wave function ψ, or bythe quantum states of an operator, = …n 0, 1, . When the system is in the wave stateψ, the expectation value of the operator O is (Messiah 1961, Merzbacher 1970)

∑ψ ψ ψψ ψ ψ ψ

ψ ψ≡ˆ

= *OO

O( )1

. (1.1)m n,

m mn n

The amplitude is ψ ζ ψ= ⟨ ∣ ⟩n n and the matrix element is ζ ζ= ⟨ ∣ ˆ∣ ⟩O Omn m n , where the ζn

form a basis of orthonormal eigenfunctions. Below the norm of the wave functionwill often be denoted ψ ψ ψ ψ ψ≡ ⟨ ∣ ⟩ = ∑ *N( ) n n n.

According to Born, the square of the modulus of the amplitude has theinterpretation as being proportional to the probability, ψ℘ ∝ ∣ ∣2. The square ofthe modulus of the coefficients gives the probability that the system is in the quantumstate n, ζ ψ ψ ζ ζ ψ ψ ψ℘ ∝ ∣⟨ ∣ ⟩∣ = ⟨ ∣ ⟩⟨ ∣ ⟩ = *

n n n n n n2 . In the basis of the eigenfunctions of the

operator, ζ ζˆ∣ ⟩ = ∣ ⟩O On n nO O , the matrix is diagonal, δ=O Omn n m n

O, , and the expectation

value becomes ψ ψ ψ ψ= ∑ *O O N( ) ( )n n n nO O . In this diagonal form, the expectation

value is clearly an average value: it is the sum over states of the probability of eachstate times the value of the operator in each state.

In this context one must distinguish between the expectation value and theoutcome of a measurement. An actual outcome is a particular eigenvalue On, whichcorresponds to the system wave function being ζn

O. An expectation value is theaverage value, not an actual outcome. It includes contributions from all the statessuperposed in the wave function.

Measurement collapses the wave function or system from the superposition ofstates embodied in ψ to the pure state ζn

O. This collapse of the wave function meansthat it loses coherency, and that interference between states is precluded. Sinceinterference effects due to the superposition of states are one of the most strikingfeatures of quantum mechanics, the loss of coherency is the attribute that is mostgenerally associated with measurement-induced collapse of the wave function.However, measurement also reduces the number of possible states, which are stateswith non-zero values in the wave function. This focusses or narrows the associatedprobability distribution, ψ∣ ∣2, since the eigenfunctions that have different eigenvaluesto the measured one are eliminated from the wave function. Although wave functioncollapse is most often taken to mean the loss of coherency, it can also includethe reduction in the number of possible states. Both aspects of measurement—the reduction in possible states and the loss of interference effects—will occur in thederivation of quantum statistical mechanics below.

In the probabilistic interpretation of the wave function, the superposition of statesis not the physical occupation of the states simultaneously, but rather the probabilitythat each one of the states is the one actually occupied. The collapse of the wavefunction is also a change in probability rather than a physical change. Theinterference between states prior to collapse, although undoubtedly a probabilistic

Quantum Statistical Mechanics

1-2

effect, has no classical analogue. The probabilistic interpretation is an essentialfeature of quantum mechanics, and it is one limitation on the analogy betweenHilbert space and classical phase space.

1.1.2 Density operator

The expectation value may alternatively be written as the trace of the density andobservable operators. The density operator is defined as

ρψ

ψ ψˆ ≡N

1( )

. (1.2)

One sees that the trace of this and the observable operator is equal to the expectationvalue of the operator,

∑ρψ

ψ ψ ψˆ ˆ = =*ON

O OTR1( )

( ). (1.3)m n,

n m mn

This single wave function density operator is a rather special operator: becauseof its dyadic form, it cannot be diagonalized. To see this, suppose the contrary,that in some representation it is diagonal, and that ρn n1 1

and ρn n2 2are two distinct

non-zero diagonal elements. But this means that the off-diagonal elementsρ ρ ρ ρ= =n n n n n n n n1 2 2 1 1 1 2 2

are also non-zero, which contradicts the original assumption.

The only way to avoid the contradiction is if there is precisely one non-zero element,which can only occur if a measurement has determined that one state is occupied.

1.1.3 Statistical average

This discussion of the non-diagonal nature of the single wave function densitymatrix is directly relevant to the conventional presentation of the statistical averagein quantum statistical mechanics. It is also relevant to the statistical averageobtained via the stochastic Scrödinger equation that is derived in later chapters ofthis book. As discussed shortly, the conventional presentation of the statisticalaverage is in terms of a density operator. However, as a working hypothesis, supposethat there exists a probability operator, ℘, and that the statistical average can beexpressed as the trace of it and the observable operator

∑

∑

∑

ˆ = ℘ ˆ

= ℘

= ℘

= ℘

O O

O

O

O

TR

. (1.4)

m n

n

n

,

mn nm

nn nn

nn nn

stat

O O

S S

The penultimate equality represents the operators in the eigenstates of the observ-able operator. This form has the usual probability theory interpretation of a

Quantum Statistical Mechanics

1-3

statistical average: it is the sum over states of the measured value in the state,=O Onn n

O , times the probability of the state, ℘nnO . This intuitively appealing form for

the statistical average makes the hypothetical existence of the probability operatorplausible, pending a detailed derivation.

However, it is the final equality for the statistical average that raises questionsabout the existence of a probability operator. This form invokes the eigenstates ofthe probability operator, which, it will be shown, are the eigenstates of the entropyoperator, ζ ζ℘∣ ⟩ = ℘ ∣ ⟩n n n

S S and δ℘ = ℘mn n m nS

, . (One of the objects of this chapter is toderive this relationship between the probability and entropy operators; see equation(1.49) and, more generally, section 3.1.4.) Unlike the expectation value for a wavestate, where it is not possible to diagonalize the density matrix, in the statisticalaverage the probability operator is diagonal in the entropy representation. It is thiscontradiction between the properties of the probability operator and the densityoperator that demands a closer examination of their relationship. If it can be shownthat the probability operator exists and is diagonalizable, then one must concludethat the statistical average does not come from the density operator of a single wavefunction (unless it is the wave function of a single pure quantum state).

To put it another way, how does one go from the expectation value or densitymatrix form that includes off-diagonal contributions in every representation exceptthat of the operator itself, to the above statistical average form in which the off-diagonal contributions vanish in both the entropy and the operator representations?The difference between an expectation value and a statistical average is sketchedin figure 1.1.

As mentioned above, typical presentations of quantum statistical mechanics refernot to the probability operator, but to the density operator (von Neumann 1932,Messiah 1961, Merzbacher 1970). In order to bypass the problem that the singlewave function density operator cannot be diagonalized, the conventional approachis to imagine an ensemble of M systems, = …a M1, 2, , , each with its own wave

Figure 1.1. The contributions to the expectation value from a superposition of states (left), and to thestatistical average from a mixture of pure states (right). The wave function is sketched on the axes, with shadedregions corresponding to negative or complex values.

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1-4

function ψa. The many wave function or averaged density operator is then defined asthe ensemble average of the single wave function density operators,

∑ρψ ψ

ψ ψ¯ ≡

=M1

. (1.5)a

M

1

a a

a a

Because the sum of a product is not equal to the product of two sums, this does notfactorize into a dyadic product, which is the crucial difference between the singlewave function density operator and this averaged density operator.

Because this is diagonalizable, the averaged density operator can be, indeed mustbe, equated to the probability operator hypothesised above. This means that in theentropy representation it must be diagonal with entries equal to the stateprobabilities,

∑ρψ ψ

ψ ψδ¯ = = ℘

=

*

M1

. (1.6)a

M

1mn

a m a n

a a

n m nS ,

S,

S

,

If the phase angles of the coefficients are uncorrelated, which corresponds to adecoherent system, then the off-diagonal terms indeed average to zero. The phaseangles of the diagonal terms cancel exactly so that the average of the diagonal termsis non-zero. It is this averaging process that differentiates this from the single wavefunction, dyadic density operator.

This condition means that the averaged density operator is equal to theprobability operator, ρ = ℘. As an operator relationship, this holds in anyrepresentation, not just the entropy representation in which the condition wasimposed. (However, the phase cancelation only occurs in the entropy representation,which means that the averaged density operator is not diagonal in the basis of anyoperator that does not commute with the entropy operator.) It follows that one canwrite the statistical average as the trace of the averaged density operator and theobservable operator,

∑ρψ ψ

ψ ψˆ = ¯ ˆ =

ˆ

=

O OM

OTR

1, (1.7)

a

M

1

a a

a astat

provided that equation (1.6) holds.In conventional expositions, the wave functions ψa are pure quantum states, and

the ensemble is said to comprise a mixture of pure quantum states, which is to saythat each system in the ensemble has a fully collapsed wave function. In thepresentation here, it is necessary that the pure quantum states are entropy states, andit is sufficient that equation (1.6), or equation (1.9) below, hold (i.e. it is notnecessary that each member of the ensemble be in a pure quantum state).

The above explanation of the density operator is rather unsatisfactory because itinvokes an ensemble, which is a collection of independent replica systems. Suchensembles have no physical reality in the sense that any actual measurement is madeon a single system. Although long experience suggests that the probability operator

Quantum Statistical Mechanics

1-5

is the correct approach to statistical averages, it is difficult to accept the ensemblepicture as representing the true origin of the averaged density operator or provingthe need for it to replace the probability operator.

An alternative interpretation is that a time average gives rise to the decoherentsystem embodied by the average density operator. One could argue that a measure-ment is not an instantaneous process, but instead extends over a brief time interval,say τ, during which the wave function evolves, ψ t( ). In this view, the averageddensity operator is

∫ρτ

ψ ψ¯ ≡τ

Nt t t

1d ( ) ( ) . (1.8)

0 0

This assumes that the evolution of the wave function is unitary,ψ ψ ψ≡ ⟨ ∣ ⟩ =N t t t N( ( )) ( ) ( ) 0. Since the averaged density operator must be equal

to the probability operator, ρ = ℘, in the entropy representation one must have

∫ρτ

ψ ψ δ¯ = = ℘τ

*N

t t t1

d ( ) ( ) . (1.9)ln l n n l nS

0 0

S S,

The mechanism of diagonalization in the entropy representation is that the phasefactors of the entropymodes are effectively random (i.e. evolve independently), whichmeans that the off-diagonal elements average to zero over the measurement time.

Both of these interpretations of the averaged density operator are a little artificial.Perhaps it would be more honest to proceed directly from equation (1.4), taking theexistence of the probability operator as a given, and this equation as the definition ofthe statistical average. Alternatively, equation (1.4) and the explicit form for theprobability operator should both be derived from first principles. In either case thereis no need to introduce the averaged density matrix.

At the conclusion of this survey of conventional quantum statistical mechanicsone is left with two main questions: what is the explicit form for the probabilityoperator, and how does one get from the wave function formulation of quantummechanics, including the expectation value, to the statistical average as a weightedsum over quantum states? In the remaining sections of this chapter these questionsare addressed.

1.2 Uniform weight density of wave spaceThe first attempt at deriving the probability operator might begin at the level ofquantum states, seeking an expression for the probability matrix in some represen-tation. In order to derive the required quantum state probabilities, one would needto begin with the weight associated with microstates for an isolated system. It wouldbe simplest to proceed by analogy with the ergodic theorem of classical statisticalmechanics, which would mean starting with the axiom that

all quantum states of an isolated system have equal weight. (1.10)

It is important to note that this refers to an isolated system and not to a sub-systemof a larger total system. Also, it will be seen that the quantum states are the

Quantum Statistical Mechanics

1-6

eigenstates of any complete operator, or, equivalently, the diagonal elements in anyrepresentation.

Although this is a true statement, it is neither the starting point nor the goal of thepresent section. Instead, this result will be derived from deeper considerations,equation (1.31) below, so that it becomes a theorem rather than an assertion, axiomor hypothesis. The real aim of the present section is to obtain the probability densityof wave space, which is the precursor to obtaining an explicit expression for theMaxwell–Boltzmann probability operator of the canonical equilibrium system andthe expression for the statistical average.

1.2.1 Probability flux and trajectory uniformity

The starting point is Schrödinger’s equation,

ψ ψℏ ˙ = ˆi , (1.11)

where the over dot signifies the time derivative. This of course holds only for anisolated system, in which case the velocity and trajectory are adiabatic. In most ofthis book a superscript 0 will be used to signify the adiabatic part of the timederivative, or the adiabatic trajectory, which distinguishes it from the total timederivative when a reservoir is present. However, in this section only the total isolatedsystem appears, and so such a superscript is redundant.

One can construct a trajectory by integrating this

ψ ψ ψ ψ≡ = ˆ −( ) ( )t t t U t t( ) , . (1.12)00 0

00 0

Here and below the adiabatic time propagator is ˆ = ˆ ℏU t( ) et0 /i .Associated with each point in wave space is a real, non-negative weight density

ω ψ t( , ). When normalized this is the probability density ψ℘ t( , ). The relationship ofthese to the probability operator and to the statistical average is addressed in thenext section. For the present the probability density will be taken to exist and itsproperties obtained.

The quantity ψ ψ℘ td ( , ) gives the probability of the system being withinψ∣ ∣d of ψ. As a probability the product is a real non-negative number:ψ ψ ψ ψ ψ ψ℘ = ∣ ℘ ∣ = ∣ ∣∣℘ ∣t t td ( , ) d ( , ) d ( , ) . Since it is always the product that

occurs, without loss of generality one may take each to be individually real. Thenormalization is

∫ ∫ψ ψ ψ ψ℘ = ℘ =( )t td ( , ) d , 1, (1.13)

where in some representation, ψ ψ = { }n , = …n 1, 2, . Since the coefficients are

complex, ψ ψ ψ= + in n nr i, one can write the infinitesimal volume element as

ψ ψ ψ ψ ψ ψ ψ = ≡ …d d d d d d d (1.14)r i1r

1i

2r

2i

With this the integrations are over the real line, ψ ∈ −∞ ∞[ , ]nr , and ψ ∈ −∞ ∞[ , ]n

i ,= …n 1, 2, .

Quantum Statistical Mechanics

1-7

The compressibility of the trajectory is

ψψ

ψ ψ

ψ ψ

˙ = ∂ · + ∂ ·

= ∂ · + ∂ ·

=ℏ

−ℏ

=

ψ ψ

ψ ψ* *

dd

1i

Tr1i

Tr

0. (1.15)

r ri

i

The first two equalities are general and also hold for the total time derivative whenthe sub-system is open to a reservoir, heat bath or environment. The final twoequalities only hold for Schrödinger’s equation.

The compressibility gives the logarithmic rate of change of the volume element.Hence from the vanishing of the compressibility for the adiabatic Schrödingerequation one sees that the volume element is a constant of the motion of theisolated system,

ψ ψ=td ( ) d . (1.16)0

The total time derivative of the probability density is

⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

ψ ψ ψ ψ ψ ψ

ψ ψ ψ ψ ψ

ψ ψ ψ ψ ψ

℘ = ∂℘∂

+ · ∂ ℘ + · ∂ ℘

= ∂℘∂

+ · ∂ ℘ + · ∂ ℘

= ∂℘∂

+ ∂ · ℘ + ∂ · ℘

ψ ψ

ψ ψ

ψ ψ

* *

* *

tt

tt

t t

tt

t t

tt

t t

d ( , )d

( , )( , ) ( , )

( , )( , ) ( , )

( , )( , ) ( , ) . (1.17)

r r ii

The first and second equalities are definitions that hold in general. The final equality,which uses the vanishing of the compressibility, is valid for an isolated system thatevolves via Schrödinger’s equation. One can identify from this the probability flux,

ψ ψ ψ≡ ˙℘℘J t t( , ) ( , ).The evolution of the probability density is given by

∫ψ ψ ψ δ ψ ψ ψ℘ = ℘ −( )( ) ( ) ( )t t t t, d , , . (1.18)1 1 0 0 0 1 1 0 0

This is just a fundamental statement of probability: the unconditional pairprobability is the product of the conditional transition probability and the singletprobability, ψ ψ ψ ψ ψ℘ = ℘ ∣ ℘t t t t t( , ; , ) ( , , ) ( , )1 1 0 0 1 1 0 0 0 0 , which is Bayes’ theorem,together with the fact that the joint probability reduces to the singlet probabilityupon integrating out one of the states, ∫ ψ ψ ψ ψ℘ = ℘t t td ( , ; , ) ( , )0 1 1 0 0 1 1 . Thedeterministic nature of Schrödinger’s equation enters via the δ-function that is theconditional transition probability, ψ ψ δ ψ ψ ψ℘ ∣ = − ∣t t t t( , , ) ( ( , ))1 1 0 0 1 1 0 0 .

Quantum Statistical Mechanics

1-8

Using the constancy of the volume element, ψ ψ=d d0 1, setting = + Δt t t1 0 , andexpanding to linear order in Δ → 0t , one has

∫

ψψ

ψ ψ δ ψ ψ ψ

ψ ψ

ψ ψ ψ ψ ψ

℘ + Δ∂℘

∂

= ℘ −

= ℘ − Δ ˙

= ℘ − Δ · ∂ ℘ − Δ · ∂ ℘ψ ψ* *

( )( ) ( )

( ) ( )

( ) ( ) ( )( )

tt

t

t t t

t

t t t

,,

d , ,

,

, , , . (1.19)

t

t

t t

1 01 0

0

0 0 0 1 1 0 0

1 0

1 0 0 0

Hence the partial time derivative is

ψ ψ ψ ψ ψ∂℘∂

= − · ∂ ℘ − · ∂ ℘ψ ψ* *t

tt t

( , )( , ) ( , ). (1.20)

This result is valid for an isolated system that evolves under Schrödinger’s equation.Inserting this into the second equality in equation (1.17), it may be seen that the

total time derivative of the probability density vanishes, ψ℘ =t td ( , )/d 0. Hence foran isolated system, the probability density is a constant of the motion,

ψ ψ℘ = ℘( )t t t( ( ), ) , , (1.21)0 0

where ψ ψ ψ≡ ∣t t t( ) ( , )0 0 .This result can be seen directly. The deterministic nature of Schrödinger’s

equation means that trajectories do not cross, and they are neither created nordestroyed. The probability of a volume of wave space is proportional to the numberof trajectories within it, which is conserved during the motion. But since the volumeitself is a constant of the motion, the probability density must be carried on thetrajectory unchanged.

The situation is sketched in figure 1.2. Since the boundary of the second volumeevolved from that of the first, the time a single trajectory spends in each volumeconsidered as fixed in wave space is the same. In so far as the probability of a state isproportional to the time spent in it, this implies that the probability density isuniform along a trajectory.

Figure 1.2. Sketch of the evolution of a volume of wave space under Schrödinger’s equation.

Quantum Statistical Mechanics

1-9

In an equilibrium system, (i.e. the Hamiltonian operator does not depend upontime), the probability density is not explicitly dependent on time, so that one canwrite ψ℘( ). In this case the result becomes

ψ ψ℘ = ℘t( ( )) ( ). (1.22)0

Finally, one may invoke what might be called the weak form of the classical ergodichypothesis, which says that a single trajectory passes sufficiently close to all relevantpoints of the state space. This hypothesis is problematic for a single energymacrostate, but it may possibly have some validity for a superposition of energystates adding up to a given energy expectation value. (A more rigorous derivationthat avoids this hypothesis but which still gives the same final result is given in thefollowing sub-section.) The hypothesis means that any two points in state space, ψ1

and ψ2, with the same norm and energy but otherwise arbitrary, lie on a singletrajectory. By the above, they therefore have the same probability density

ψ ψ ψ ψ ψ ψ℘ = ℘ = =E E N N( ) ( ), if ( ) ( ) and ( ) ( ). (1.23)2 1 2 1 2 1

Using this result the weight density of wave space must be of the formω ψ ω ψ= E( ) ( ( )). (For simplicity the magnitude is neglected since it only triviallyaffects the analysis.) The only time one needs to compare the weight densitiesof points with different energies is when energy exchange with a reservoir occurs.In such a case the total weight density must be of the form ω ψ =( )tot

ω ψ ω ψ−E E E( ( )) ( ( ))s tot tot s tot , which has solution ω ψ = α ψw( ) e E ( ), with w and αarbitrary constants. This means that any variation in the weight density of wavespace with the energy of the sub-system will cancel identically with the equal andopposite dependence of the weight density of the wave space of the reservoir on thesub-system energy. Hence without loss of generality one can set α = 0 and the weightdensity may be set to unity,

ω ψ =( ) 1. (1.24)

This result, which is obtained more rigorously in the next sub-section, applies to thewhole of wave space of an isolated system, not just to the energy–magnitudehypersurface to which the adiabatic Schrödinger trajectory is constrained in anygiven case. By design, this is real and positive. The logarithm of the weight densitygives the internal entropy of the wave states,

ψ ω ψ≡ =S k( ) ln ( ) 0. (1.25)B

1.2.2 Time average on the hypersurface

An arguably more satisfactory derivation is as follows. The magnitude of the wavefunction is redundant in the sense that expectation values only depend upon thenormalized wave function. For example, denoting the latter as ψ ψ ψ˜ ≡ −N( ) 1/2 , withthe magnitude being ψ ψ ψ≡ ⟨ ∣ ⟩N( ) , the energy is ψ ψ ψ= ⟨ ˜ ∣ ˆ ∣ ˜⟩E ( ) . One can denoteby χ a wave function in that particular Hilbert sub-space that is a hypersurface of

Quantum Statistical Mechanics

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constant magnitude N and energy E. In this notation the normalized wave functionis ψ ψ χ˜ = ˜ E( , ) and the full wave function is ψ ψ ψ ψ χ= ˜ =N N E( , ) ( , , ).

Now the weight density on the hypersurface, ω χ∣N E( , ), will be derived, and thenthis will be transformed to the weight density on the full wave space. The derivationis the quantum analogue of that given for classical statistical mechanics in section5.1.3 of Attard (2002). The axiomatic starting point is that the fundamentalstatistical average is a simple time average. The implication of this is that theweight is uniform in time, which is to say that it must be inversely proportional tothe speed,

ω χ ψψ ψ

∝ ˙= ˆ ˆ

−

−

N E( , )

, (1.26)

1

1 2

since the Hamiltonian operator is Hermitian. This is just the time that the systemspends in a volume element χ∣ ∣d . The proportionality factor is an immaterialconstant.

This weight density is now successively transformed to the full wave space. Forthis one requires in turn the Jacobian for the transformation χ ψ⇒ ˜ ,

⎡⎣⎢

⎤⎦⎥

ψψ

ψψ

ψ ψ∇ = ∂∂ ˜

∂∂ ˜

= ˜ ˆ ˆ ˜EE E( ) ( )

, (1.27)1 2

1 2

and for the transformation ψ ψ˜ ⇒ ,

⎡⎣⎢

⎤⎦⎥

ψψ

ψψ

ψ ψ∇ = ∂∂

∂∂

=NN N( ) ( )

. (1.28)1 2

1 2

With these the full weight density is

ω ψ ω χψ ψ ψ ψ

ψ ψ

= ∇ ∇

∝ ˜ ˆ ˆ ˜ˆ ˆ

=

N E E N( ) ( , )

1. (1.29)

1 2 1 2

1 2

The interpretation of this is that the weight density is inversely proportional to thespeed of the trajectory (i.e. large speed means less time per unit volume) andlinearly proportional to the number of hypersurfaces that pass through eachvolume of wave space (i.e. for fixed spacing between discrete hypersurfaces, ΔE

and ΔN, large gradients correspond to more hypersurfaces per unit wave spacevolume, see figure 1.3).

Just as in the classical case for Hamilton’s equations of motion, it is a remarkableconsequence of Schrödinger’s equation that the speed is identical to the magnitudeof the gradient of the hypersurface, so that these two cancel to give a weight densitythat is uniform in wave space.

Both derivations (the uniformity of the probability density along a trajectory andthe proportionality of weight to time and gradients) yield the same result, namely

Quantum Statistical Mechanics

1-11

that the weight density is uniform in wave space. One may suppose that there is aweight operator whose expectation value gives the weight density,

ω ψ ψ ω ψψ ψ

≡ ˆ =( ) 1. (1.30)

Since this must hold for all wave states, the weight operator must be the identityoperator and one must have

ω ω δˆ = ˆ =I, or . (1.31)mn m n,

The result holds in any representation.One may draw two conclusions from this result. The first is that the quantum

states or microstates of the system are the diagonal elements in any representation.Equivalently, they are the eigenstates of any complete operator. Hence this resultimplies that these microstates all have equal weight, which is the axiom given at thestart of this section.

The second point of note is that the off-diagonal elements have zero weight. Ofcourse the off-diagonal elements of the wave function dyadic, ψ ψ*m n, ≠m n, docontribute to the expectation value of an operator in any basis other than that of theoperator itself. Hence the weight operator is not used directly in an expectationvalue. But, as will be seen, for the case of the direct product of two systems, theaverage or expectation value of an operator on one system has to be weighted by thestates of the other system, and for this the weight operator is required.

1.3 Canonical equilibrium systemThis section is concerned with the canonical equilibrium system, in which a sub-system can exchange energy with a heat reservoir. The total system, which isisolated, comprises the sub-system and the reservoir. The reservoir can also be calledthe bath or the environment. Some workers use the phrase ‘the system’ to mean whatis here called ‘the sub-system’. The only terms and labels that will be used here aretotal system (tot), sub-system (s), and reservoir (r).

Figure 1.3. Sketch of hypersurfaces highlighting the relationship between gradient and density.

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1.3.1 Entropy of energy states

At this point it is useful to distinguish between microstates and macrostates. Here amicrostate is labeled by a single lower case roman letter and it corresponds either toan eigenstate of a complete set of commuting operators, in which case the eigenvalueis unique, or else to an eigenstate of an incomplete operator, in which case differentmicrostates can share the same eigenvalue. A macrostate is labeled by a lower caseGreek letter and it corresponds to the principal quantum number of an operator.Each macrostate has a unique label. The pairing of a Greek and roman lettersignifies a microstate, with the Greek letter labeling the principal quantum numberand the roman letter labeling the degenerate quantum states.

In the explicit terms of an operator, the eigenfunctions may be written with asingle microstate label, ζ ζˆ∣ ⟩ = ∣ ⟩O On n n

O O , in which case different values of n may yieldthe same eigenvalue (assuming an incomplete operator). Alternatively, the eigen-functions may be written as the combination of principal and degenerate state labels,

ζ ζˆ∣ ⟩ = ∣ ⟩α α αO Og gO O , in which case different values of α necessarily yield different

eigenvalues.Because it is energy that is exchanged in the canonical equilibrium system, the

focus here is on energy states. From the above theorem that microstates have equalunit weight, the weight of the energy microstates is =αw 1g

E , and they have no internal

entropy, = =α αS k wln 0g gE

BE .

As is discussed in detail in chapter 3, in general the weight of a macrostate is thesum total of the weight of microstates that it contains. Hence the weight of an energymacrostate is

∑= ≡αα

α αw w N . (1.32)g

gE (E, ) E E

This is just the number of degenerate states with energy Eα.In general, the entropy of any state is the logarithm of its weight. Hence the

entropy of an energy macrostate is the logarithm of the degeneracy,

=α αS k Nln . (1.33)EB

E

The entropy of the reservoir when the sub-system is in a particular energymicrostate will be required. Since energy is conserved one must have

= +E E E . (1.34)tot s r

Here and below tot refers to the total system, s refers to the sub-system and r refersto the reservoir.

With these there are two forms for the reservoir entropy. The first is the one basedupon the energy degeneracy, namely

∑= ≡α α αα( )S E k N Nln , with . (1.35)

g

r rB

Er Er (Er, )

(For simplicity, the energy superscript may be dropped below.) The second is basedupon the thermodynamic definition of temperature, ≡ ∂ ∂−T S E E( )/1 . Using the

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conservation of energy and the fact that the reservoir is much larger than the sub-system, a Taylor expansion of the entropy may be truncated at the linear term togive the reservoir entropy as a function of the sub-system energy,

= − = −( ) ( )S E S E EET

const. . (1.36)r r r tot ss

r

Henceforth the superscript on the reservoir temperature will be dropped as it is theonly temperature that will appear. These two expressions will be equated to eachother below.

1.3.2 Wave function entanglement

The wave function of the total system lies in the Hilbert that is the direct product ofthat of the sub-system and the reservoir, = ⊗H H Htot s r. The reason that it is theproduct is that the sub-system and the reservoir only weakly interact, and so theymay be treated as quasi-independent: for any state of one all permitted states of theother are available, so that the total number of states is essentially the product of thetwo individual totals, modified by the requirements of the conservation laws.

With ζ∣ ⟩{ }ns a basis for the sub-system and ζ∣ ⟩{ }m

r a basis for the reservoir, the mostgeneral wave state of the total system is

∑ ∑ψ ζ ζ ζ ζ= ⊗ ≡c c , . (1.37)n m n m, ,

nm n m nm n mtots r s r

If the coefficient matrix is dyadic, =c c cnm n ms r , then the wave state is separable with

ψ ζ∣ ⟩ = ∑ ∣ ⟩cn n ns(s) s s and ψ ζ∣ ⟩ = ∑ ∣ ⟩cm m mr

(r) r r . Alternatively, if the coefficients cannot bewritten in dyadic form, then the wave state is inseparable. This is called an entangledstate (Messiah 1961, Merzbacher 1970).

In general, conservation laws give rise to entangled states. This may be seenexplicitly for the present case of energy exchange, with = +E E Etot s r fixed. Usingthe energy eigenfunctions described above as a basis for the sub-system and thereservoir, the most general expansion of the total wave function is

∑ψ ζ ζ=α β

α β α βc , . (1.38)g h,

g h g htot ,Es Er

In view of energy conservation one has

= + ≠α β α βc E E E0 if . (1.39)g h,s r

tot

It is not possible to satisfy this if α βc g h, is dyadic. If it is dyadic, then if at least one ofthe sub-system energy macrostates is occupied, say α1, ≠αc 0g

s1

, then, because themacrostate labels are distinct, there can be only one non-zero reservoir macrostate,say β1, that conserves energy, =βc 0h

r , β β≠ 1. (If there were another occupiedreservoir macrostate, β2, then the coefficient α βc cg h

s r1 2

would be non-zero but wouldviolate energy conservation.) But if there is only one occupied reservoir energymacrostate, then there cannot be more than one occupied sub-system energy

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macrostate. Hence one must conclude that if the total wave function is in productform, then the sub-system and reservoir each individually have a single fixed energy,which means that neither wave function, ψs or ψr, can contain a superposition ofprincipal energy states. This means that the sub-system, for example, can neverduring its evolution change energy by exchange with the reservoir, because duringthe exchange it would be in a superposition of the initial and final energy states.Hence a dyadic form implies that the sub-system and the reservoir are effectivelyisolated from each other, which contradicts the whole point of the canonicalequilibrium system. Conversely, if the sub-system and the reservoir can exchangeenergy, then the total wave function must be entangled.

Accordingly, the total wave function does not factorize, but instead it must berepresented as

∑ ∑ψ ζ ζ ζ ζ= ⊗ ≡α α

α β α β α α βα α αc c c , . (1.40)

g h g h, , , ,g h g h g h g htot

s r s r,

Es Er

Here the reservoir energy label is defined implicitly by + =α βαE E Es r

tot. For brevity, the

coefficients are written ≡α α βαc c cg h g h,

s r . Because of the common label, this is clearlyentangled (i.e. the sum of the products is not equal to the product of the sums).

The most widely known case of entanglement is that of particles with spin thatresult from radioactive decay (figure 1.4). Like energy, it is the conservation of spinthat is responsible for the entanglement. If it were not for spin conservation, thedaughter wave function would be the superposition of four possible wave functions,which are the independent (i.e. unentangled) product of two spinors for eachparticle. Instead conservation (i.e. entanglement) means that only two distinctwave functions are superposed.

For future reference, one can project this entangled total wave function onto thesub-system,

∑ ∑ψ ζ ζ= = ¯α α

α α α αc c . (1.41)g h g, , ,

g h g g gs ,Es Es

Figure 1.4. Two possible wave functions (upper and lower, each comprising two spin half particles) for thedecay of a spin zero particle (center). The wave function after the decay is the equal superposition of twoentangled wave functions.

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In this form of a standard expansion over basis functions, it appears that the openquantum system (i.e. the sub-system that can exchange with a reservoir, environmentor bath) has a well-defined wave function that is not entangled. This appearance isdeceptive, because this projected wave function has implicit restrictions on it. Forexample, the superposition of states that appear in it must be discarded in taking theexpectation values of operators, since it should really represent a collapsed wavefunction and a mixture of pure quantum states, as will now be shown.

1.3.3 Expectation values and wave function collapse

It is now shown how the entanglement of the sub-system and reservoir wavefunctions leads to the collapse of the principal energy states of the sub-system. Thesignificance of this is in the context of the discussion in section 1.1, namely thatthe single wave function density operator could not be diagonalized, whereas theprobability operator, if it exists, could be diagonalized. The collapse of the principalenergy states is one of the two essential conditions for the existence of adiagonalizable probability operator and the subsequent mixture formulation ofthe statistical average. (The other essential conditions are the collapse of thedegenerate reservoir energy states and the collapse of the degenerate sub-systemenergy states. The former is shown here and the latter is shown in the second part ofthis section and, more rigorously, in section 1.3.4.)

Using the entangled form of the total wave function and the vanishing of thecoefficients for terms that violate energy conservation, the expectation value of anoperator that acts only on the sub-system is

∑ ∑

∑ ∑

∑ ∑

ψψ

ζ ζ ζ ζ

ψ

ζ ζ ζ ζ

ψ

=

× ˆ

=

× ˆ

=

α α β β

α α

α

′ ′ ′ ′

′ ′ ′

′

α β α β

α α β β

β βα α

α α β βα

βα α α α

′ ′ ′ ′*

′ ′ ′ ′

′ ′ ′*

′ ′ ′

′*

′

α α

α

α

′

′

( )

( )

ON

c c

O

Nc c

O

Nc c O

( )1

( )

1( )

1( )

. (1.42)

g g h h

g g h h

g g h

, ,

, ,

, ,

g h g h

g g h h

g h g h

g g h h

g h g h g g

stot

tot

(s) (r); ;

Es s Es Er Er

tot

(s) r, ,, ,

Es s Es Er Er

tot

(s) Er,, , ,

s,E

One sees how energy conservation and entanglement have reduced and collapsed theoriginal four sums over independent principal energy states into a single sum overprincipal energy states. The degenerate energy states of the reservoir have alsocollapsed, since the orthogonality of the reservoir basis functions have reduced thesums over h and h′ to a single sum over h. Specifically, whereas for an isolated systemthe non-diagonal entries of the operator matrix in the energy representation wouldcontribute to the expectation value, when energy exchange with a reservoir canoccur, orthogonality of the reservoir energy basis functions together with energyconservation eliminate the off-diagonal sub-system energy terms. The principal

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energy states of the sub-system have at this stage collapsed, whilst the degenerateenergy states of the sub-system remain in superposition form.

Collapse is used here in its two senses: the probability distribution has beennarrowed by reducing the number of possible states, and the interference betweensuperposition states has been reduced by the elimination of off-diagonal energy terms.

Now the statistical average that follows from this expectation value will beobtained in two ways. The first method is based on physical arguments and thequantum states. The second, which is given in the following section, is carried out inwave space and is a little more rigorous.

For the first derivation, one can take the non-zero coefficients in the entangledexpansion to have magnitude unity

=αθαc e , (1.43)g h

g h,

i ,

with the θ being real. This form is reminiscent of the so-called Einstein–Podolsky–Rosen (EPR) state, where entangled qubits are often described in similar terms (seealso figure 1.4). That all the allowed coefficients have the same magnitude is astatement of the fact that microstates of the total system have equal weight. Henceall total wave functions ψtot compatible with the energy constraint must have thesame weight. (See also the more general wave space derivation that follows.)

Different total wave functions correspond to different sets of exponents. If onewere to repeat the expectation value with many different choices of ψtot, or if onewere to take a time average over the interval in which the operator is applied, theproduct of coefficients that appears in equation (1.42), α α′

*c cg h g h, , , would average tozero unless ′ =g g. This will be shown explicitly in the statistical derivation in wavespace that follows in the next section. This means that the reservoir sum in theexpectation value becomes

∑ ∑δ

δ

δ

δ

=

=

=

= ×

βα α αβ

β

′* ′

′

′

′−

α

α

βα

α

( ) ( )c c

N

e

const. e , (1.44)

h hstat

g h g h g g

g g

g gS k

g gE k T

Er,, , ,

Er,

,Er

,

,

rB

sB

where the two forms for the reservoir entropy, equations (1.35) and (1.36), have beenused.

One effect of repeating or averaging the expectation value is to collapse thedegenerate sub-system energy states, g = g′. This is the second result that had to beestablished in order to derive the mixture form for the statistical average, equation(1.4). The interference between energy states, both principal and degenerate, had tobe eliminated (collapse) in order for the probability operator to be diagonal. (Theenergy representation is equivalent in this case to the probability or entropyrepresentation.) This signifies the transformation from a superposition of states toa mixture of states.

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The corollary of diagonalizing the probability matrix is that if the density operatoris equated to the probability operator, then the density operator cannot correspond toa single wave function but rather to an average over multiple wave functions.

The second type of degenerate energy states are those of the reservoir, andsumming over these leads to the second effect: each sub-system energy state isweighted by the number of degenerate reservoir microstates, which is just theexponential of the corresponding reservoir entropy. It is this second effect that givesthe Maxwell–Boltzmann form for the probability operator directly.

This averaging of the expectation value of the sub-system operator for the totalwave function, equation (1.42), over the coefficients αc g h, may be equated to thestatistical average of the operator. One therefore has

∑ψ =α

α α− αO

ZO( )

1e . (1.45)

g,

E k Tg g

stot stat

(s),

s,EB

This has the same functional form as the final equality in equation (1.4), and so theprobability matrix entries may be identified explicitly as the Maxwell–Boltzmannform,℘ =α α

− − αZ eg gE k T

,S 1 / B . This result is based upon the assertion that the entangled

coefficients have equal weight. A somewhat more rigorous wave space derivation isnow given.

1.3.4 Statistical average and probability operator

Instead of fixing the coefficients to lie on the unit circle in the complex plane, oneintegrates over all possible values that respect energy conservation. A uniformweight density in wave space is invoked, as derived in section 1.2. This means that

ψ ≡ ≡ ∏α α αc c cd d d dg h, g h g htot ,r

,i , with the real and imaginary parts of the coefficient

each belonging to the real line, ∈ −∞ ∞( , ). Because of entanglement, only the threeindices are required. Using the fundamental form for the expectation value, equation(1.42), one obtains for the statistical average

∫

∫

∫

∑ ∑

∑ ∑

∑

∑

ψ ψ

ψ

ψ

ˆ =′

=′

=′

=′

=

α

α

α

α

′

βα α αα α

βα α

α

β α α

α α

′′

*

−

α

α

α

( )

( )

OZ

O

ZO c

c c

N

ZO c

c

N

ZO

ZO

1d ( )

1d

( )

1d

( )

const.e

1e . (1.46)

g g h

g h

g

g

, ,

,

,

,

g gg h g h

g g

g h

S kg g

E k Tg g

s

stattot

stot

(s) Er,,

s,E , ,

tot

(s) Er,,

s,E ,2

tot

(s),

s,E

(s),

s,E

rB

sB

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1-18

The third equality follows because the terms in which the integrand is odd vanish,and so the only non-vanishing terms have g = g′. The fourth equality follows becauseall the integrations give the same value irrespective of the indices, and so this is aconstant that can be taken outside of the summations and incorporated into thenormalization factor. The fourth and fifth equalities use the two forms for thereservoir entropy, equations (1.35) and (1.36). As always, the normalizing partitionfunction ensures that ⟨ˆ⟩ =I 1stat .

This is the same as the average obtained in a more heuristic fashion above. Bothmethods reveal that the reduction and collapse of the energy macrostates is due toenergy conservation and entanglement of the total wave function, and that thecollapse of the sub-system degenerate energy states is due to the fact that the off-diagonal contributions average to zero.

This average has the form of the sum over states of the probability of a state timesthe value of the observable in the state. The only possible measured values of anoperator O on an isolated system are its eigenvalues, =α α αO O g h,

O . But one interpreta-tion of the statistical average is that the possible measured values of the operator on anopen quantum system (sub-system interacting with a reservoir) are the diagonalelements in the entropy representation, α αO g g,

S or On n,S .

It is straightforward to convert this average to the form of the trace over theprobability and observable operators, equation (1.4). Since any function ofthe Hamiltonian is diagonal in the energy basis with entries equal to the functionof the eigenvalues, one may write

δ δ =α αα α

−′ ′

− ˆ

′ ′α { }e e . (1.47)E k T

g gk T

g g, ,

,

EsB B

Here and below ˆ is the sub-system Hamiltonian operator. For the canonicalequilibrium system, the entropy operator is ˆ = − ˆS T/ (cf equation (1.36)), and sothe entropy eigenfunctions are identical to the energy eigenfunctions. Hence theaverage of a sub-system operator can be written

∑

∑

ˆ =

=

= ℘ ˆ

α

α α′ ′

α α

α αα α

−

− ˆ

′ ′′ ′

α

{ }

OZ

O

ZO

O

1e

1e

TR . (1.48)

g

g g

,

,

E k Tg g

k T

g gg g

stat

(s),

s,E

(s)

,

E

,s,E

B

B

Here the Maxwell–Boltzmann probability operator has been identified (Feynman1972),

℘ ≡ ≡ˆ − ˆ

Z Z1

e1

e . (1.49)S k k TB B

The normalizing partition function is = − ˆZ TR e k T/ B . The first equality is general;the second equality is specific to the canonical equilibrium system.

Quantum Statistical Mechanics

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The Maxwell–Boltzmann probability operator arises directly from the sum overthe degenerate energy microstates of the reservoir, which gives the exponential of thereservoir entropy for each particular sub-system energy macrostate. This convertsdirectly to the matrix representation of the Maxwell–Boltzmann probabilityoperator in the energy basis. Although the energy representation was used to derivethis result, the final expression as a trace of the product of the two operators isinvariant with respect to the representation.

One sees in this derivation that it is the statistical average that causes thesuperposition of the degenerate energy microstates of the sub-system to collapse intoa mixture. The reduction and collapse of the energy macrostates occurred at the levelof the expectation value due to the energy conservation law and the entanglement ofthe reservoir and the sub-system. As has been mentioned above, both collapses arenecessary for the mixture form for the statistical average, equation (1.4). Thesuperposition entropy states, both principal and degenerate, had to collapse so thatthe probability operator was diagonal in the entropy representation. Consequently,equating the density operator to the probability operator implies that it is an averageover multiple wave functions.

It is worth mentioning that the Maxwell–Boltzmann probability operator can bewritten as a Feynman path integral for the time propagator, but with the temper-ature interpreted as an imaginary time (Feynman and Hibbs 1965). Although thisapproach has been successfully exploited in many applications (Schulman 1981,Kleinert 2009), it is not pursued here.

In later chapters, a stochastic, dissipative time propagator will be developed thatcan be used to give a wave function trajectory ψ t( ). In the entropy representation theexpansion coefficients may be written in the form ψ ψ= ∣ ∣α α

θ− αt t( ) ( ) eg gtS S i ( )g . The phases

are uncorrelated with each other and with the amplitude. The statistical average maybe written as a time average over the trajectory. Hence the averaged density matrixat time t is

∫

∫

∫

∑∑

∑∑

∑

ψ ψ ψ ψ ζ ζ

ζ ζ ψ ψ

ζ ζ

= ′ ′ ′

= ′ ′ ′

× ′

=

= ℘

α β

α β

α

α β α β

α β α β

θ θ

α α

*

′ − ′

−

α β

α

t tt

t t t

tt t t

tt

Z

( ) ( )1

d ( ) ( )

1d ( ) ( )

1d e e

e

. (1.50)

g h

g h

g

, ,

, ,

,

t

g h g h

g h

t

g h

tt t

S kg g

stat0

S S S S

S S

0

S S

0

i ( ) i ( )

1 S S

g h

B

In the second equality, since the amplitudes and phases have been taken to beuncorrelated, the average of the product has been written as the product of theaverages. The phases are uniformly distributed and the average over them givesδ δα β g h, , . The average over the amplitudes gives the Maxwell–Boltzmann factor. Thisis an explicit realization of equations (1.6) and (1.9).

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The statistical average can be written as an integral of the expectation value overthe sub-system wave space,

∫ ψ ψ ψψ ψ

ˆ = ℘ ˆO

Od . (1.51)

stat

The superscript s signifying the sub-system has been dropped here and belowbecause the reservoir has been integrated out of the problem. This may be regardedas a continuum version of the trace. In order to see the equivalence of this with theabove expression for the statistical average, one can simply expand the sub-systemwave function in any basis, ψ ψ ζ∣ ⟩ = ∑ ∣ ⟩n n n , and integrate

∫ ∫

∫

∑

∑

∑

ψ ψ ψψ ψ

ψψ ψ

ψ ψ

ψψ ψψ ψ

℘ ˆ=

℘

=″

=″

= ℘ ˆ

*

ˆ*

ˆ

{ }

{ }

O O

ZO

ZO

O

d d

1e d

const.e

TR . (1.52)

m n l

m n

m n

, ,

,

,

m l mn nl

S k

mnnm

m m

S k

mnnm

B

B

The second equality follows because only the terms l = m survive the integration,since odd powers of ψl vanish. The third equality follows because the integration isthe same for all m, and so it can be taken as a constant outside of the sums andincorporated into the partition function.

In section 1.2, the probability density in wave space for an isolated total systemwas analyzed. It was shown that this was uniform in wave space, which was the basisfor the theorem that microstates of an isolated system are equally likely. Note thedifference between the total wave space of an isolated system and the wave space ofa sub-system of a total system. The non-uniform probability density of the latteris the concern here. Indeed, the expectation value of the Maxwell–Boltzmannprobability operator for the present sub-system that can exchange with a reservoirgives explicitly the non-uniform probability density of the sub-system wave space forthe present canonical equilibrium system.

The relationship between the probability density and the probability operator isof course that the former is the expectation value of the latter,

ψ ψ ψψ ψ

℘ = ℘( ) . (1.53)

The wave function that appears here is that of the sub-system.As is discussed in detail in chapter 3, the entropy operator is related to the

probability operator by the definition ℘ ≡ ˆ−Z S kexp /1B, the entropy expectation is

defined to be ψ ψ ψ ψ ψ≡ ⟨ ∣ ˆ∣ ⟩ ⟨ ∣ ⟩<>S S( ) / and the entropy of a wave state of a sub-

system is defined to be ψ ψ ψ ψ ψ≡ ⟨ ∣ ∣ ⟩ ⟨ ∣ ⟩ˆS k( ) ln[ e / ]S kB

/ B .

Quantum Statistical Mechanics

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Returning to the probability density, it should be clear that the statistical averageis not the integral over the sub-system wave space of the probability density and theexpectation value of the operator,

∫ ψ ψ ψˆ ≠ ℘O Od ( ) ( ). (1.54)stat

The physical reason that this does not hold is that in general statistical averages arethe value of a variable in a state times the probability of the state, summed over thestates. The problem with this expression is that both ψ℘( ) and ψO ( ) are expectationvalues, which are average values. They are not actual values.

In order to see the problem mathematically, one can evaluate the integral usingentropy basis functions,

∫ ∫

∫

∫

∫

∑ ∑

∑

∑

∑

∑ ∑

∑ ∑

ψ ψ ψ ψψ

ψ ψ ψ ψ

ψψ ψ ψ ψ

ψ

ψψ ψ

ψ

ψψ ψ ψ ψ

ψ

℘ = ℘

= ℘

= ℘

+ ℘

= ℘ + ℘

= ′ ℘ + ′

* *

* *

*

≠* *

≠

( )

ON

O

NO

ON

ON

C O C O

C O C O

d ( ) ( ) d1

( )

d( )

d( )

d( )

. (1.55)

m n l

m n

n

m n

n m n

n n

,

,

,

,

m m m n l nl

m m n nm nn

n nnn n

m nm nn

m m n n

n nnm n

m nn

n nn nn

S2

S S S S S

SS S S S

2S

S S

S S 2

2

( ) S SS S S S

2

2S

1( ) S

2S

1S

This is clearly not the correct expression for the statistical average. In principle, thetwo constants that appear could be evaluated and the correct statistical average ofthe operator could be extracted from this. In contrast, equation (1.51) gives thestatistical average directly and correctly. This issue of writing the statistical averageof an operator as a weighted integral of its expectation value over wave space will berevisited in appendix A.

1.4 Environmental selectionThe technique used to derive the above results for the probability operator and forthe statistical average has some similarity to other treatments of open quantumsystems (Davies 1976, Breuer and Petruccione 2002, Weiss 2008), specifically thatknown as environmental selection or einselection (Zeh 2001, Zurek 2003).Einselection includes the influence of the environment on the system of interest,and this is one obvious point in common with the above analysis of the sub-systemand reservoir. The specific goal of einselection is to explain the apparent collapse ofthe wave function upon measurement. Wave function collapse occurred several

Quantum Statistical Mechanics

1-22

times in the derivation of equilibrium quantum statistical mechanics just given, andthis is another point of similarity.

For these reasons it is useful to give a brief summary of the einselection approach.This summary departs from the literature presentations of environmental selection(Zeh 2001, Zurek 2003) by using the entropy representation, entanglement andconservation laws. Nevertheless, it conveys the spirit of the einselection program,particularly in invoking the quasi-orthogonality of the reservoir, as is brieflyexplained below.

Consider a total system tot that consists of a sub-system s that interacts witha reservoir r. In the environmental selection literature, the reservoir is calledthe environment and the sub-system is called the system. Let ζ{ }n

Ss and ζ{ }nSr

be the entropy eigenfunctions that form a basis for the sub-system and the reservoir,respectively. For the canonical equilibrium system, entropy eigenfunctions are thesame as energy eigenfunctions. More generally, they account for all quantitiesexchanged between the sub-system and reservoir.

The interaction region of the sub-system and the reservoir is relatively smallcompared to either, and so they are quasi-independent. Hence for each state ofthe sub-system all the permitted states of the reservoir can occur, which means thatthe total number of states is the product of the allowed numbers of each. Hence theHilbert space of the total system is the direct product of that of the sub-system andthat of the reservoir. Because of this the wave function of the total system may berepresented as

∑ ∑ψ ζ ζ ζ ζ= ⊗ =c c , . (1.56)n m n m, ,

nm n m nm n mtotSs Sr Ss Sr

As discussed above in section 1.3.2, due to the conservation law the total wavefunction must be entangled and it cannot be written as the product of a sub-systemwave function and a reservoir wave function. In the entropy basis the conservationlaws are directly accounted for and so one can write

∑ ∑ψ ζ ζ ζ ζ= ⊗ ≡α α

α β α β α α βα α αc c c , . (1.57)

g h g h, , , ,g h g h g h g htot

s r Ss Sr,

Ss Sr

Because the common label precludes factorization, this is an entangled wavefunction.

Suppose that at time =t 0 the normalized wave function of the total system isψ ζ ψ∣ ⟩ = ∣ ⟩α ,gtot

Ss0r . This is unentangled, with the sub-system having wave function ζαg

Ss

(i.e. being in the entropy state αg) and the reservoir having wave function ψ0r that is

arbitrary apart from the conservation laws. After time τ this evolves to

∑ψ ζ ψ→τ

ββ

αβ β

αc , . (1.58)h,

hg

h hg

tots( ) Ss r( )

Note that the evolved sub-system coefficient and the evolved reservoir wave functiondepend upon the initial αg and evolved βh sub-system states (shown), and upon the

Quantum Statistical Mechanics

1-23

initial reservoir wave function ψ0r and the time interval τ (not shown). This is due to

the coupling between the reservoir and the sub-system.Consider now a second initial normalized total wave function, with a different

sub-system basis function but the same reservoir initial wave function, ζ ψ∣ ⟩γ ,lSs

0r . The

inner product of this and the first wave function evolves to be

∑∑

∑

ψ ζ ζ ψ ψ ζ ζ ψ

ψ ψ δ δ

δ δ

→

=

≡

τ

β η

β

γ α ηγ

βα

ηγ

η β βα

βγ

βα

βα

βα

α γ

α α γ

*

*

c c

c c

C

, , , ,

. (1.59)

h p

h

, ,

,

l g pl

hg

pl

p h hg

hl

hg

hg

hg

l g

g l g

0r s s

0r s( ) s( ) r( ) Ss Ss r( )

s( ) s( ) r( ) r( ), ,

, ,

The second equality in this derivation invokes the orthonormality of the sub-systembasis functions and the quasi-orthogonality of the reservoir wave functions. Quasi-orthogonality means that two wave functions chosen at random are likely to beorthogonal; the likelihood of overlap decreases with increasing reservoir size.

This idea is illustrated in figure 1.5. Here the projection onto an axis representsthe overlap of two random unit vectors. The fact that the length of projectiondecreases from √1/ 2 to √1/ 3 in going from two to three dimensions shows howthe overlap decreases with increasing dimensionality. For a typical reservoir orenvironment the dimension is of the order of Avogadro’s number, and so the overlapbetween two wave functions is effectively a delta function.

Since the total system is isolated, its time propagator is the adiabatic timepropagator, which is unitary. Hence the scalar product of total wave functions ispreserved during their evolution. Taking γ α= and =l g in the above, since theleft-hand side is unity, one concludes that =αC 1g .

Now consider the evolution of the expectation value of an operator on thesub-system O. Let the initial total wave function be in superposition form,ψ ζ ψ∣ ⟩ = ∑ ∣ ⟩α α αa ,g g gtot ,

Ss0r . After time τ, which may be interpreted as the time over

Figure 1.5. The likely overlap between a unit vector and a random axis. In two dimensions (left) the projectionhas length √1/ 2 and in three dimensions (right) the projection has length √1/ 3.

Quantum Statistical Mechanics

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which the measurement is performed or, equivalently, the operator is applied, theexpectation value evolves to

∑∑

∑∑ ∑∑

∑∑

∑

ψ ψ ζ ζ

ψ ζ ζ ψ

ζ ζ ψ ψ

ˆ =

→

× ˆ

= ˆ

≡

α βτ

α β η γ

α γ

γ

α β α β

α β ηα

γβ

ηα

η γ γβ

α α γ γ γα

γα

γα

γα

γ γ γ γ

*

* *

* *

*

O a a

a a c c

O

a a O c c

A A O

, ,

, (1.60)

g h

g h p l

g l

l

, ,

, , , ,

, ,

,

g h g h

g h pg

lh

pg

p l lh

g g l l lg

lg

lg

lg

l l l l

tots

totSs Ss

s( ) s( )

r( ) Ss s Ss r( )

Ss s Ss s( ) s( ) r( ) r( )

S S,

Ss

with ψ ψ≡ ∑ ⟨ ∣ ⟩γ α α γα

γα

γαA a cl g g l

gl

gl

gS,

s( ) r( ) r( ) . The third equality follows from the quasi-

orthogonality of the reservoir wave functions. By setting ˆ = ˆO I, which shows that∑ =γ γ γ

*A A 1l l l,S S , one can see that the wave functions are correctly normalized.

This result is really the statistical average rather than the expectation value. Tosee this, note that this expectation value only involves diagonal elements of theoperator matrix in the entropy representation, and so it is not identical in form to theexpectation value derived above, equation (1.42), which has off-diagonal, degener-ate state contributions. The statistical average of the expectation value, equation(1.46), only has diagonal elements contributing. Strictly speaking, it is reservoirorthogonality plus decoherent averaging that gives the diagonal form. Overlookingthis minor difference, the statistical mechanical analysis may be used to quantify theeinselection result, namely = ℘γ γ γ

*A Al lS S .

This evolved expectation value of the operator has the appearance of being due toa mixture of entropy states of the sub-system, with the γA l

S being the coefficients ofthe wave functions. (In general, the entropy states are the eigenstates of theoperators corresponding to the exchangeable variables.) The signature of themixture of decoherent wave functions is the fact that only diagonal terms appearin the evolved expectation value. This of course differs from a coherent wavefunction that corresponds to a superposition of states and that contributes off-diagonal terms to the expectation value in all representations other than thatof the operator itself. The latter is the usual expectation value and, for a sub-systemwave function in the entropy representation, it would be ψ ψ ψ≡ ⟨ ∣ ˆ∣ ⟩ =O O( )

ψ ψ∑ * Om n m n mn,S S S , assuming that the sub-system wave function is normalized. One

sees that the off-diagonal terms contribute to this, whereas they do not in the casewhere environmental selection creates a mixture of pure entropy eigenstates.

As has been mentioned, environmental selection and the reservoir analysis of thepreceding section are closely related. Both show that in the expectation value of asub-system operator averaged over the total wave function the off-diagonal termsare suppressed (in the entropy representation). This is variously called wave function

Quantum Statistical Mechanics

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collapse, decoherence or a mixture of pure quantum states, and is thought to berelevant to the so-called measurement problem. Both show that this occurs due tothe interactions with the reservoir or environment. The advantage of the reservoirformalism is that it is exact, since it does not invoke quasi-orthogonality. Also, thereservoir formalism yields an explicit formula for the probability operator and thestatistical average.

1.5 Wave function collapse and the classical universeBy way of concluding chapter 1, the above results for quantum statistical mechanicsare now linked to the broader conceptual issues that arise in quantum mechanics.Specifically, this section addresses the superposition of states embodied in the wavefunction, the collapse of the wave function upon measurement and more generally,and the classical behavior of the macroscopic world in view of the fundamentalquantum nature that must underlie it.

The strategy employed is to detail those aspects of quantum statistical mechanicsthat are relevant to the question of the classical nature of the world, to identify thespecifically non-classical aspects of quantum systems and show that these vanish inthe statistical analysis, and to derive in detail the classical equations of motion forthe relevant states. To this end, the section is divided into several sub-sections.First, in section 1.5.1 there is a detailed summary of the mechanism that generatesthe quantum statistical average that is the focus of the chapter, with the emphasison the detailed stages of the collapse of the wave function. Second, in section 1.5.2there is a brief argument for the probabilistic interpretation of the wave function,and this is used to explain the meaning of the quantum phenomena of wavefunction superposition and collapse. Third, in section 1.5.3 it is argued thatquantum interference is the uniquely non-classical phenomenon, and that itsabsence in the statistical average signifies the quantum to classical transition ofall open quantum systems. Fourth, in section 1.5.4 Hamilton’s classical equationsof motion are derived from the collapsed wave function implied by the statisticalaverage, and it is discussed how this process gives rise to the classical behavior ofthe world at large.

The important issue of the necessary symmetries imposed upon the wave functionby bosons and fermions, which is arguably the other ‘uniquely non-classicalphenomenon’, is discussed in section 2.1.

1.5.1 Mechanism of statistical collapse

The derivation of equilibrium quantum statistical mechanics commences in section1.2 with Schrödinger’s equation, from which the uniform weight of wave space andthe consequent equal weight of microstates of an isolated system are established.From this follows in section 1.3 the explicit form for the probability operator in thecanonical equilibrium system—the Maxwell–Boltzmann form—and the expressionfor the statistical average as the trace of the probability and observable operators.The statistical average corresponds to a mixture of pure entropy quantum states andmerits further discussion.

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The most interesting aspect of the derivation is the role played by wave functioncollapse.Most commonly, it is measurement that is associated with the collapse of thewave function, since this fixes the wave function as a unique eigenfunction of theoperator (equivalently, microstate of the system). Certainly, measurement producinga unique eigenstate is the narrowest form of wave function collapse, but moregenerally one should view collapse as belonging to a continuum. Wave functioncollapse, in the broad sense, means the reduction in the number of available states,which leads to a sharpeningor focussing of the associatedprobability distribution, ψ∣ ∣2.

Wave function collapse also refers to the reduction in the interference betweenstates, corresponding to the evolution from a superposition to a mixture of states.Loss of interference, more technically known as decoherence, leads to only thediagonal elements contributing to the value of an operator, at least in the entropyrepresentation. The removal of the off-diagonal contributions by the statisticalaverage is a consequence of wave function collapse. Of course one could still describethe mixture of pure quantum states as a superposition of states, provided that onekept in mind that there is no longer interference between the superposed states.

For the statistical average in general there are four distinct collapses of the totalwave function. First, the principal entropy states of the sub-system and the reservoirbecome entangled due to the conservation laws. Second, the principal entropy statesof the reservoir collapse due to orthogonality, taking the sub-system principalentropy states with them. Third, the reservoir degenerate entropy states collapse dueto orthogonality. And fourth, the sub-system degenerate entropy states collapsebecause they average to zero.

After these collapses of the total wave function there remain three sums: one overthe degenerate entropy states of the reservoir, one over the principal entropy statesof the sub-system and one over the degenerate entropy states of the sub-system.

The single sum over the degenerate reservoir states does not depend upon the sub-system operator being averaged. The microstates have equal weight (section 1.2), andtheir number is the exponential of the entropy (section 1.3.1). It is this that gives theMaxwell–Boltzmann factor for the probability, equation (1.44) and equation (1.46).

The remaining two sums over principal and degenerate entropy states of the sub-system is actually a single sum over all the entropy microstates of the sub-system. Itcan be expressed in the form of the trace of the probability and observable operators,equation (1.48). This has the form of a mixture of pure entropy quantum states.The reason for interpreting it this way is that in the entropy representation theprobability operator is diagonal, and only the diagonal terms of the observableoperator contribute. The cross- or off-diagonal terms, which represent interferencefrom the superposition of states in the wave function, average to zero. This is whythe probability operator cannot be expressed as the density operator of a single wavefunction. It can be expressed, somewhat artificially, as the ensemble or time averageof the density operator provided that the off-diagonal terms in the entropyrepresentation average to zero, equation (1.6) or equation (1.9).

The statistical average subsumes the reservoir or environment into its thermody-namic parameters, which appear in the probability operator. Both the probabilityoperator and the observable operator act only on the sub-system wave function.

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However, one should be cautious in speaking of the ‘wave function’ for such acollapsed system. It is possible to project the entangled total wave function onto thesub-system, equation (1.41). But such a projected wave function when used to giveexpectation values of a sub-system operator contains a superposition of states,whereas the collapsed wave function represents a mixture of pure states. Hence thecollapsed wave function is qualitatively different to the standard wave function.There is an argument to say that the two are so different that an open quantumsystem does not possess a wave function in the ordinary sense of the word. The needto formulate an average over multiple wave functions rather than a single wavefunction density operator for an open quantum system is one such argument. Thisqualitative difference between the wave function of an open and closed quantumsystem forms the basis of the discussion of the quantum to classical transition in thefollowing sections. The unique nature of the wave function of an open quantumsystem, in so far as it exists at all, is a theme that recurs in later chapters.

1.5.2 Probabilistic nature of the wave function

The present derivation and results for quantum statistical mechanics have broadimplications for quantum theory itself and for the classical appearance of the world. Along-standing and puzzling question is why the world of common experience isclassical, whereas the underlying nature of the universe is quantum. In order to makesense of the question, one first requires a mathematical picture for the classicaluniverse. For the reasons now given, the appropriate conceptual basis and language isprovided by probability theory. The phrase ‘classical probability’ will be used todistinguish the mathematical theory of probability from the probabilistic aspects ofquantummechanics, since these are qualitatively different, as will be discussed shortly.

Since determinism is one of the hallmarks of classical mechanics, it might appearcounterintuitive to discuss the classical universe in probabilistic terms. However, thisobjection carries no weight since the relevant probability distributions are as sharplypeaked as the outcome is certain. In fact, measurement, including visual observa-tion, is generally reported in terms of a value and an error, ±a b, from which it isclear that the outcome of a classical measurement is really a Gaussian probabilitydistribution.

There are very persuasive arguments that probability theory characterizes rationalthought itself (Jaynes 2003), which of course evolved best suited to the classical world.This relationship of probability to inductive reasoning has led some to posit thatprobability is purely subjective, measuring nothing but the degree of reasonable beliefof the observer. However, one may accept the relevance of probability theory toinductive reasoning without embracing the view that probability is exclusivelysubjective. The objective interpretation of probability has a physical basis andaccounts for classical statistical mechanics (Attard 2002, 2012a). In any case, thepoint is that probability theory was developed to describe the classical world.

The goal here, therefore, is to identify in detail the sense in which quantummechanics is inconsistent with classical probability theory. Thence the aim is to usethe present derivation of quantum statistical mechanics to elucidate the processes by

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which the universe goes from the underlying quantum behavior to beingwell-described by classical probability theory.

The first stage is to isolate what makes quantum mechanics uniquely non-classical. There are of course many surprising quantum effects at the molecularlevel that can have macroscopic consequences and that would not have beenpredicted from classical mechanics. But here the focus is on the transition fromthe quantum to the classical world, and for this one needs the essential differencesbetween quantum and classical behavior. One can identify three prima facie non-classical features of quantum mechanics that the present analysis has a bearing on:the superposition of quantum states, the reduction of the wave function uponmeasurement and the interference between quantum states. (For the purposes of thepresent discussion these three features are treated as more or less distinct, whereasabove they were all treated as aspects of collapse.)

In fact, however, the superposition of states and the reduction of the wavefunction upon measurement are just as much classical as quantum phenomena. Thisfollows from the probabilistic interpretation of the wave function.

Neither the superposition of states nor the collapse of the wave function are to beunderstood in a physical sense. Non-zero values for the wave function for multiplestates mean that the probability of the system being in one of those states isnon-negligible. They do not mean that the system is actually in all those states. Thisis the same as the meaning of classical probability. Similarly, the probabilisticinterpretation of the wave function means that reduction upon measurement issimply a collapse in the probability rather than in the physical state of the system.That is, the original wave function is an unconditional probability that reflects theindeterminate state of the system prior to the measurement, and this collapses into aconditional probability, more narrowly distributed than the original, as a result ofthe state of the system being determined by the measurement. Again, this is identicalto the behavior of classical probability.

The collapse of a wave function upon experimental measurement is illustratedschematically in figure 1.6. When a spin zero particle decays, the wave function is thesuperposition of two entangled wave functions, figure 1.4, both contributing with

Figure 1.6. Measurement of the spin of a particle.

Quantum Statistical Mechanics

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equal weight. This is a probabilistic statement about the state of the system.Measuring the spin of one of the particles determines which of the two equallyprobable entangled wave functions actually exists. In common parlance it is saidthat the superposed wave function has collapsed into a pure state. This collapse,however, is not a physical change to the system but rather the identification of thestate of the system that was previously known probabilistically.

1.5.3 Quantum interference is uniquely non-classical

In the light of the probabilistic interpretation of the wave function, it is theinterference between states that is the distinctly quantum phenomenon that has noclassical analogue. (But see section 2.1, for a discussion of particle statistics andwave function symmetries.) Although quantum interference has specific physicalmanifestations, such as in the well-known double slit experiment, the concern here iswith its generic occurrence as the signature of a quantum rather than a classicalsystem.

Interference effects are manifest in the off-diagonal parts of the expectation value,

∑ ∑ψ ψ ψ ψˆ = +* ≠ *O O O , (1.61)n m n,

n n nnm n

m n mn( )

assuming a normalized wave function. The first sum has the form of a classicalaverage: the value in a state times the probability of the state. The second sum showshow the interference between states contributes to the quantum average.

Interference effects do not satisfy the axiomatic basis of classical probabilitytheory. The latter is predicated upon mutually exclusive states, otherwise known asoutcomes: the system possesses a complete set of disjoint states, functions of thesystem have a value in each state, and each state has a real, non-negative weight thatgives its probability. (For technical reasons ‘state’ here really means ‘microstate’,which is indivisible and defined without respect to system functions. Macrostates aresets of microstates and each set generally refers to a specific system function.)

That the quantum mechanical value of the operator is required simultaneously intwo states, Omn, violates these fundamental axioms of probability theory. If one triesto circumvent the problem by regarding the ‘twinned’ state mn as the label of the setof states, one faces the problem that the weight ψ ψ*m n is complex. One could retain theoriginal definition of the single labeled states by focussing solely on the diagonalelements, since in the operator representation one only needs ≡O On nn

O with theweight ψ ψ*n n

O O being real and non-negative. However, this requires a different set ofstates for each operator. In contrast, probability theory is predicated on the axiomthat at any one time the system is in ‘a’ microstate in which every function of thesystem has ‘a’ value (i.e. the set of microstates is indifferent to the function).

What are called interference effects here were formulated in a more precisemathematical fashion by Birkhoff and von Neumann (1936), who identified thedifference between classical and quantum probability as the fact that classical setssatisfy the distribution law (i.e. form a Boolean lattice), whereas quantum sets donot. It can be readily shown that, with the microstates of the open quantum system

Quantum Statistical Mechanics

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being entropy microstates, ζnS, and the value of an operator in such a microstate

being OnnS , the macrostates (i.e. sets of microstates with given values of the same or

different operators) satisfy the distribution law, ∩ ∪ ∩ ∪ ∩=a b c a b a c( ) ( ) ( ),and hence form a Boolean lattice. It is of course the mutually exclusive, non-interfering nature of the microstates of the open quantum system that ensures thisproperty.

In the present account of the transition from quantum to classical behavior, it isinterference, or the lack thereof, that is the relevant quantity to focus on in quantumstatistical mechanics. The analysis shows that the statistical average of an observableoperator is the trace of it and the probability operator. The very existence of theprobability operator is of great significance. This is because it is diagonalizable (in theentropy representation, in fact). (The analysis also shows that the probability operatoris a real, symmetric, positive semi-definite operator (since℘ ≡ ℘ ⩾ 0n nn

S for all n), butthese are not needed here.) This is a qualitative difference from quantum mechanics,where, it will be recalled from section 1.1.2 that the expectation value corresponds tothe single wave function density operator, which cannot be diagonalized. There is thusa difference in kind between an expectation value and a statistical average, betweenthe single wave function density operator and the probability operator. It follows thatan open quantum system (i.e. a sub-system that can exchange conserved material witha reservoir or its environment) is qualitatively different from a closed quantum system.

The fact that the probability operator is diagonal in the entropy representationmeans that there are no interference effects. It does not matter that the off-diagonalparts of the operator, Omn

S , are non-zero because these have no weight, ℘ = 0mnS ,

≠m n. The pure entropy microstates are the microstates of the system. Any functionof the system (i.e. operator) has a value in each microstate, and the set of microstatesis indifferent to the operator. The average is the sum over entropy states of theproduct of the probability and the value of the observable in each state. This isprecisely what is required by classical probability theory. It therefore follows that anopen quantum system is a classical system.

There are additional arguments to show that the form of the statistical averagemarks the transition from a quantum to a classical system. As just mentioned,the existence of a diagonalizable probability operator is in marked contrast to theconventional single wave function density operator of a quantum system, which isnot diagonalizable. In addition to the conclusion drawn above (that this implies theabsence of interference), it also means that an open quantum system cannot becharacterized by a single wave function. It is characterized by a mixture of purequantum states, each element of the mixture being an independent eigenfunctionof the entropy operator. This is definitely not a superposition of states or ofwave functions.

In a mathematical sense one can formulate a density matrix averaged overmultiple decoherent wave functions such that it equals the probability operator,equation (1.6) or equation (1.9). But this is an artificial post facto rationalization ofthe quantum statistical average and it lacks a physical basis. An ensemble averageexists only in the minds of mathematicians, not in the physical universe (cf thedifference between subjective and objective probability (Attard 2002, 2012a)).

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A time average can perhaps be justified, but it does raise a number of practicalquestions. It is better to accept the probability operator as an operator in its ownright and to recognize that an open quantum system is characterized by a statisticalaverage that represents the classical limit of the underlying quantum sub-system.

1.5.4 Classical phase space and Hamilton’s equations

The above arguments show that an open quantum system is distinctly different froman isolated quantum system, and that qualitatively the statistical average for an openquantum system has the same hallmarks as a classical statistical average.Quantitatively, one can go beyond this to give the mathematical detail for howclassical behavior arises from the quantum statistical average.

The statistical average of any operator is given as the sum over pure entropyquantum microstates weighted by their probability. (Microstates signifies thatalthough the entropy operator may be incomplete, the sum is over both the principaland degenerate states.) This means that the set of pure entropy quantum microstatesmust be complete. That is, the only accessible states of the system are pure entropyquantum microstates, and any observed value of an operator must be its value in apure entropy quantum microstate. If the set was not complete, additional termswould have to be included in the expression for the statistical average to reflect theadditional states that are not included in the incomplete set. The derivation showsthat no additional terms beyond the pure entropy quantum microstates are required.That the pure entropy quantum microstates form a complete set of the states of theopen system proves that a superposition of entropy microstates is not an allowed (i.e.contributing) state of an open quantum system.

The pure entropy quantum microstates referred to above are those of the sub-system. The reservoir only enters by providing their weight and by collapsing thewave function. Because the set of pure entropy quantum microstates is completewhen the sub-system is an open quantum system (i.e. can exchange with a reservoir,a bath or the environment), it is also sufficiently complete when the sub-system isclosed or isolated. The phrase ‘sufficiently complete’ means that although the set ofpure entropy microstates is not complete for a closed quantum system (becausesuperpositions of entropy states are allowed in a closed system and by the definitionof ‘pure’ such superposition states do not belong to the set of pure entropymicrostates), the set of pure entropy microstates is sufficient to encompass all theallowed states when the quantum sub-system interacts with a reservoir, a bath or theenvironment. In practice, almost all systems in the universe are open quantumsystems, as is discussed further below. This means that one need not be concernedwith the superposition states of a closed quantum system, and that for all practicalpurposes one can treat the set of pure quantum microstates as sufficiently complete.

The significance for the quantum to classical transition of this result can best beappreciated in terms of classical phase space. A classical phase space point for anisolated system is Γ = p q{ , }, where p and q are the 3N-dimensional vectors ofmomenta and positions, respectively. For any such point there exists a purequantum entropy microstate labeled n with expectation value ζΓ Γ=( )n

S . Thisfollows because the set is sufficiently complete.

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For the canonical equilibrium system, ˆ = − ˆ T/r , which means that the entropyeigenfunctions are also energy eigenfunctions, ζ ζˆ | ⟩ = | ⟩n n n

S S . This last conditioncan be ensured if ζn

S is simultaneously an eigenfunction of q and p. Since these do notcommute, this can only hold with an uncertainty of Δ ˆΔ ˆ ⩾ ℏq p /2. As an example,consider the minimum uncertainty wave packet (Messiah 1961; Merzbacher 1970),

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ζ

ξ=

− − · −−

ℏ· −− ( ) ( ) ( )Cq

q q q qp q q( ) exp

4

1i

. (1.62)n

n n

n n1

2

q q

q q

The expectation values are

ζ ζ ζ ζ ζ ζ= ˆ = = ˆ =q q q p p p( ) , and ( ) , (1.63)n n n n n n n nq p

and the uncertainties (root mean square fluctuation) for each component are

ξ ξΔ ˆ = Δˆ = ℏq p, and /2 . (1.64)

These mean that the probability amplitude ζ ζ*n n has width ξ in position space, andwidth ξℏ/2 in momentum space. If a suitable width exists (e.g. Δ ˆ ≪p mk TB , andthe potential energy varies negligibly over length scales of order ξ), then to within theuncertainty, this wave packet is a simultaneous eigenfunction of the position andmomentum operators. One has

ζ ζ ζˆ = ≈q q q q q q( ) ( ) ( ) , (1.65)n n n nq

and

⎡⎣⎢

⎤⎦⎥∣ ⟩ ∣ ⟩ ∣ ⟩ζ

ξζ ζˆ = ℏ − + ≈( )i

p q q q p q p q( )2

( ) ( ) . (1.66)n n n n n n2 q p p

These follow from the sharply peaked nature of the wave packet, which means thatany prefactor that is a slowly-varying function of q can be evaluated at qn. Becausethe wave packet is a simultaneous position-momentum eigenfunction, it is also anenergy and hence an entropy eigenfunction that can be labeled ζn

S. This approx-imation is elaborated upon and systematically corrected in appendix C.

Ehrenfest theoremIn the position representation, the position and momentum operators are ˆ =q qand ˆ = − ℏ∂ ∂ = − ℏ∇qp i / i , where q is the 3N-dimensional vector of positions. Thesimplest form of the Hamiltonian operator is ˆ = −ℏ ∇ +m U q( /2 ) ( )2 2 , whereU q( )is the potential energy.

For a wave state ψ, the expected position is

ψ ψ ψψ ψ

=ˆ

qq

( ) (1.67)

Quantum Statistical Mechanics

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and its rate of change is

⎡⎣ ⎤⎦

∫

∫

ψψ ψ

ψ ψ ψ ψ

ψ ψψ ψ

ψ ψψ ψ

ψ ψψ ψ

ψ ψψ ψ

ψ ψψ ψ

˙ = ∣ˆ ∣ ˙ + ˙ ∣ˆ ∣

=ℏ

ˆ ˆ − ˆ ˆ

= −ℏℏ

∇ − ∇

= − ℏ ∇

= ˆ

= ∂ ˆ

∂ˆ

*

*

{ }

{ }m

m

m

q q q

q q

q q q q q

q q q

p

p

( )1

1i

2i

d ( ) ( )

id ( ) ( )

1

1. (1.68)

0 0 0

22 2

These are of course adiabatic time derivatives and the norm of the wave function is aconstant of the motion. The Hermitian nature of the Hamiltonian operator gives thesecond equality and integration by parts gives the fourth equality after somecancelation. The rate of change of momentum is

⎡⎣ ⎤⎦

∫

∫

ψψ ψ

ψ ψ ψ ψ

ψ ψψ ψ

ψ ψψ ψ

ψ ψψ ψ

ψ ψψ ψ

˙ = ∣ˆ ∣ ˙ + ˙ ∣ˆ ∣

=ℏ

ˆ ˆ − ˆ ˆ

= − ∇ − ∇

= − ∇

= − ∂ ˆ

∂ˆ

*

*

{ }

U U

U

p p p

p p

q q q q q

q q q q

q

( )1

1i

1d ( ) { ( ) ( ) } ( )

1d ( ) ( ) ( )

1. (1.69)

0 0 0

The fourth equality follows from the product rule for differentiation, since thegradient operator acts on everything to its right. These are called Ehrenfest’stheorem (Messiah 1961, Merzbacher 1970).

Hamilton’s equations of motionFor simultaneous eigenfunctions, the energy eigenvalue is = p q( , )n n n , where theright-hand side is the classical Hamiltonian function of phase space. It follows thatthe adiabatic time derivatives for the isolated system are

ζ ζ ζ˙ = ∂ ˆ

∂ˆ=∂

∂( )qp

p q

p

( , )(1.70)n n n

n n

n

0 S S S

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and

ζ ζ ζ˙ = −∂ ˆ

∂ˆ=−∂

∂( )pq

p q

q

( , ). (1.71)n n n

n n

n

0 S S S

The first equality in each case is Ehrenfest’s theorem. The second equality invokesthe simultaneous eigenfunctions: the expectation of a function of an operator in aneigenstate is equal to the function of the eigenvalue. Identifying the left-hand sides as˙ = ˙q q p q( , )n n n

0 0 and ˙ = ˙p p p q( , )n n n0 0 , one sees that these are the classical Hamilton

equations of motion.This shows that the set of pure entropy quantum microstates gives rise to classical

mechanics. Of course this assumes that the spacing between pure entropy states issufficiently small that the position and momentum states can be treated as acontinuum. Accepting this, one sees that Hamilton’s classical equations of motionhold on the set of pure entropy microstates, and that the set is sufficiently complete.This determines that the corresponding open quantum system behaves classically.

Classical universeIt remains to argue that almost all systems in the universe are in effect open quantumsystems and therefore behave classically. Most systems are part of a larger wholeand can exchange energy and other quantities with their broader environment.Similarly, a measurement is no more than interaction and exchange with an externalsystem. In both cases the systems will behave classically by the above analysis. Aremaining class of systems are those that are deliberately isolated from theirsurroundings and are to be treated as a whole. Apart from the practical issue ofthe efficacy of the necessary isolation procedures, any isolated macroscopic objectcan itself be divided mathematically into interacting sub-systems. The exchangesbetween them and the averaging over them collapse the wave function into pureentropy states, so that each such sub-system behaves classically. Whether one saysthat an object composed of classical sub-systems is itself classical, or whether onesays that a measurement of a macroscopic system is always made in practice onsome part of the system, the conclusion is the same: macroscopic objects behaveclassically.

This argument that all macroscopic systems are classical in nature needs to bequalified in two respects. In the above derivation of the quantum statistical average,the treatment of the entangled wave function of the total system assumed that energyexchange occurred between the sub-system and the reservoir, and also that theinteractions between the sub-system and the reservoir were relatively negligible, suchthat the two systems could be considered to be quasi-independent apart from energyconservation. (It was also assumed that the reservoir was very much larger than thesub-system so that the Taylor expansion of the entropy could be terminated at thelinear term.) The first of these assumptions can be violated at low temperatureswhere the thermal energy can be less than the spacing between energy states and noenergy exchange can occur. The second of these assumptions can be violated forsmall sub-systems, where the volume of the boundary region over which interactions

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between the sub-system and the reservoir occur becomes comparable to the volumeof the sub-system itself, which vitiates the assumption of quasi-independence. (Theobject also has to be large enough to be considered as a reservoir for any of itscomponent parts.) In these two cases the argument for classical behavior cannot beexpected to hold.

ReferencesAttard P 2002 Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximisation

(London: Academic)Attard P 2012a Non-Equilibrium Thermodynamics and Statistical Mechanics: Foundations and

Applications (Oxford: Oxford University Press)Birkhoff G and von Neumann J 1936 The logic of quantum mechanics Annals Math. 37 823Breuer H-P and Petruccione F 2002 The Theory of Open Quantum Systems (Oxford: Oxford

University Press)Davies E B 1976 Quantum Theory of Open Systems (London: Academic)Feynman R P and Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York:

McGraw-Hill)Feynman R P 1998 Statistical Mechanics 2nd edn (Boulder, CO: Westview)Jaynes E T 2003 Probability Theory: the Logic of Science (Cambridge: Cambridge University

Press)Kleinert H 2009 Path Integrals in Quantum Mechanics, Statistics Polymer Physics and Financial

Markets (Singapore: World Scientific)Messiah A 1961 Quantum Mechanics (vol 1 and 2) (Amsterdam: North-Holland)Merzbacher E 1970 Quantum Mechanics 2nd edn (New York: Wiley)Schulman L S 1981 Techniques and Applications of Path Integration (New York: Wiley)von Neumann J 1932 Mathematische Grundlagen der Quantenmechanik (Berlin: Springer)Weiss U 2008 Quantum Dissipative Systems 3rd edn (Singapore: World Scientific)Zeh H D 2001 The Physical Basis of the Direction of Time 4th edn (Berlin: Springer)Zurek W H 2003 Decoherence, einselection and the quantum origins of the classical arXiv:quant-

ph/0105127v3

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