H.2 Euler-Maclaur in Sum Formula
26.5 Density Operators for Specific Ensembles
In this section, we present the statistical density operators for the three main ensembles, microcanonical, canonical, and grand canonical, employed in statistical thermodynamics.
These ensembles pertain to equilibrium states, so Eq. (26.29) applies and can be satisfied by choosingρˆSto be a function of a HamiltonianHˆ that is independent of time.ρˆS can therefore be expressed in terms of a set of probabilities and thestationaryeigenstates|En of Hˆ. It is for this reason that we only had to deal with the stationary eigenstates ofHˆ in our previous description of statistical mechanics, beginning with the microcanonical ensemble.
For brevity of notation we drop the superscriptSin the rest of this section, but bear in mind that we are dealing with astatistical operator for a system in equilibrium. The results can therefore be expressed easily in the energy representation where the matrix representations of Hˆ, and therefore also ρ(ˆ Hˆ), are diagonal. Specifically, we employ a complete set of orthonormal stationary eigenstates|Enthat satisfyHˆ|En =En|En. Note especially thatnlabels states, not energies, so there can be many values ofnfor a given energy in the case of degeneracy. For the case of the grand canonical ensemble, we will employ states that are also eigenstates of the number operatorNˆ. See Appendix I for more information about number operators.
4The operatorfˆis in the Schrửdinger representation so its only dependence on time is explicit; we therefore use a partial derivative for its time rate of change.
26.5.1 Microcanonical Ensemble
The microcanonical ensemble applies in principle to an isolated system having constant total energy E. We recognize, however, that a truly isolated system is an impossibil- ity because there will always be some interaction of a system with its environment, even if ever so slight. Because of the uncertainty relation E∼h¯/t, a constant en- ergy would require isolation for an infinite time. Therefore, we actually treat a quasi- isolated system (see [66, p. 14]) for which the energy lies in a very narrow rangeE−E to E. Within this range, the number of quantum states of the system is represented by , and each is assumed to be equally probable. Then the density operator has the form
ˆ
ρ=
n
|EnpnEn| = n=1
|En1
En|; pn=
1/ forE−E≤En≤E.
0 otherwise. (26.31)
The entropy is given byS=kBln. In terms ofρ, it can be calculated from the formulaˆ
S= −kBtr(ρˆlnρ)ˆ , (26.32)
where the function lnρˆ is to be understood as the operator whose eigenvalues, in a representation whereρˆis diagonal, are equal to the logarithm of the eigenvalues ofρˆ. The quantity−tr(ρˆlnρ)ˆ in Eq. (26.32) is just the expectation value of−lnρˆ in the statistical state represented byρˆ; in a representation whereρˆcan be represented by a diagonal matrix with diagonal elementsPn, Eq. (26.32) gives the familiar resultS = −kB
nPnlnPn. For the microcanonical ensemble we can evaluate the trace in an arbitrary, complete set of states|φmto obtain
−tr(ρˆlnρ)ˆ = −
m
φm| n=1
|Enln(1/)
En||φm
= n=1
m
En|φmφm|Enln
=
n=1
En|Enln
=ln. (26.33)
26.5.2 Canonical Ensemble
The canonical ensemble pertains to a system in contact with a heat reservoir that maintains the system at temperature T. The corresponding probabilities in the energy representation are just Pn = exp(−βEn)/Z, where β = 1/(kBT) and Z =
mexp(−βEm)is the canonical partition function. Thus we can write the density operator in the form
ˆ
ρ=
n
|Enexp(−βEn)
Z En| = exp(−βH)ˆ
Z = exp(−βH)ˆ tr
exp(−βH)ˆ . (26.34) In this case, the sum is over all energy states, a complete set. From the last form of Eq. (26.34), it is obvious that trρˆ=1. In this case, Eq. (26.32) leads to the familiar formula
458 THERMAL PHYSICS
S/kB= −tr(ρˆlnρ)ˆ = −
m
φm|
n
|EnPnlnPnEn||φm = −
n
PnlnPn, (26.35) wherePnare the probabilities of occupation of the states. Of course the expectation value of the energy itself is the internal energy
U= ˆH =tr(ρˆH)ˆ =tr
Hˆexp(−βH)ˆ tr
exp(−βH)ˆ . (26.36)
If the eigenvalues of Hˆ cannot be calculated, the last expression in Eq. (26.36) can be calculated, at least approximately, in any convenient representation. If the eigenvalues are known, then we retrieve the familiar result
U=
nEnexp(−βEn)
mexp(−βEm) . (26.37)
26.5.3 Grand Canonical Ensemble
Based on the considerations of Chapter 21, the density operator in the grand canonical ensemble will be diagonal in a set of states that are simultaneous eigenfunctions of the number operatorNˆ and the Hamiltonian operatorHˆfor a system havingNparticles. Such states|NsErssatisfy
H|Nˆ sErs =Ers|NsErs; Nˆ|NsErs =Ns|NsErs. (26.38) Thus withPrsbeing the probability of the state|NsErs, we have
ˆ
ρ=
r,s
|NsErsPrsNsErs| =
r,s
|NsErsexp[−β(Ers−μNs)]
Z NsErs|
= exp[−β(Hˆ −μNˆ)]
Z = exp[−β(Hˆ −μNˆ)]
tr
exp[−β(Hˆ −μNˆ)], (26.39) where the grand partition function (see Eq. (21.21))
Z=
s
exp(βμNs)
r
exp(−βErs)=
s
λNs
r
exp(−βErs). (26.40) Here, λ = exp(βμ) is the absolute activity. The expectation value of some observable having operatorfˆis therefore
f =(1/Z)tr
fˆexp[−β(Hˆ −μNˆ)]
=
NλNfNZN
NλNZN , (26.41)
where ZN is the canonical partition function for a system of N particles and fN is the canonical average of fˆ for that system. From Eq. (26.32), the entropy is just S= −kB
r,sPrslnPrsas expected.