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paradox will result from (8.19) because the correct counting of states is automatically implied by the definition of f(E) in (8.17). The only new result following from (8.19) that is not obtainable in classical statistical mechanics is the third law of thermodynamics, which we discuss separately in Section 8.4.
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Canonical Ensemble
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The derivation of the canonical ensemble from the microcanonical ensemble given in 8 did not make essential use of classical mechanics. That derivation continues to be valid in quantum statistical mechanics, with the trivial change that the integration over f space is replaced by a sum over all the states of the system: 1 (8.20) N!h 3N dpdq --4 ~
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Thus the canonical ensemble is defined by the density matrix
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e-f3En (8.21) mn where 13 = l/kT. This result states that at the temperature T the relative probability for the system to have the energy eigenvalue En is e- f3En , which is called the Boltzmann factor. The partition function is given by
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Pmn -
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where it must be emphasized that the sum on the right side is a sum over states and not over energy eigenvalues. The connection with thermodynamics is the same as in classical statistical mechanics. The density operator p is
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P = LIcI>n) e-f3En (cI>nl = e- f3 LIcI>n)(cI>nl
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where ' is the Hamiltonian operator. Now the operator
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L IcI>n) (cI>nI is
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identity operator, by the completeness property of eigenstates. Therefore
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P = e- f3
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The partition function can be written in the form
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T) = Tre- f3
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where the trace is to be taken over all states of the system that has N particles in the volume V. This form, which is explicitly independent of the representation, is sometimes convenient for calculations. The ensemble average of (!) in the canonical ensemble is Tr (!) e-{3 } (!})=---(8.25)
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Grand Canonical Ensemble
For the grand canonical ensemble the density operator p operates on a Hilbert space with an indefinite number of particles. We do not display it because we do not need it. It is sufficient to state that the grand partition function is
2(z, V, T) =
where QN is the partition function for N particles. The connection between log 2 and thermodynamics is the same as in classical statistical mechanics. The ensemble average of (!) in the grand canonical ensemble is (!) =
"i L
ZN (!)N
where (!) N is the ensemble average (8.25) in the canonical ensemble for N particles. These equations can be written more generally in the forms
2(z, V, T) = Tre- P( -/1N)
(!) =
1 "i Tr[(!) e- P( -/1N)]
where N is an operator representing a conserved quantity (i.e., one that commutes with the Hamiltonian), and the trace is taken over all states without restriction on the eigenvalues of N. The only restrictions on the trace are boundary conditions, which specify the volume containing the system, and the symmetry property of the states under the interchange of identical particles.
The definition of entropy is given by (8.19). At the absolute zero of temperature a system is in its ground state, i.e., a state of lowest energy. For a system whose energy eigenvalues are discrete, (8.19) implies that at absolute zero S = k log G, where G is the degeneracy of the ground state. If the ground state is unique, then S = 0 at absolute zero. If the ground state is not unique, but G $ N, where N is the total number of molecules in the system, then at absolute zero S $ k log N. In both of these cases the third law of thermodynamics holds, because the entropy per molecule at absolute zero is of order (log N)jN. The energy eigenvalues for most macroscopic systems, however, essentially form a continuous spectrum. For these systems the previous argument only shows that the entropy per molecule approaches zero when the temperature T is so low that
kT !:iE
where !:iE is the energy difference between the first excited state and the ground
state. As an estimate let us put
!:i E
li 2 -----z:;t3
where m is the mass of a nucleon, V = 1 em3 . Then we find that T ::=: 5 X 10 -15 K. Clearly this phenomenon has nothing to do with the third law of thermodynamics, which is a phenomenological statement based on experiments performed above 1 K. To verify the third law of thermodynamics for systems having an almost continuous energy spectrum we must study the behavior of the density of states w(E) near E = O. Most of the substances known to us become crystalline solids near absolute zero. For these substances all thermodynamic functions near absolute zero may be obtained through Debye's theory, which is discussed in Section 12.2. It is shown there that the third law of thermodynamics is fulfilled. The only known substance that remains a liquid at absolute zero is helium, which is discussed in 13. There it is shown that near absolute zero the density of states for liquid helium is qualitatively the same as that for a crystalline solid. Therefore the third law of thermodynamics is also fulfilled for liquid helium. Apart from these specific examples, which include all known substances, we cannot give a more universal proof of the third law of thermodynamics. But this is perhaps sufficient; after all, the third law of thermodynamics is a summary of empirical data gathered from known substances.