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An ensemble is an incoherent superposition of states. Its relevance to physics has been postulated in the previous section. We note that only the square moduli Ibn 12 appear in (8.9). Hence it should be possible to describe an ensemble in such a way that the random phases of the states never need to be mentioned. Such a goal is achieved by introducing the density matrix. Before we define the density matrix let us note that an operator is defined when all its matrix elements with respect to a complete set of states are defined. Its matrix elements with respect to any other complete set of states can be found by the well-known rules of transformation theory in quantum mechanics. Therefore, if all the matrix elements of an operator are defined in one representation, the operator is thereby defined in any representation. We define the density matrix Pmn corresponding to a given ensemble by (8.10)
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where cI>n and bn have the same meaning as in (8.7). In this particular representation Pmn is a diagonal matrix, but in some other representation it need not be. Equation (8.10) also defines the density operator P whose matrix elements are Pmn The operator P operates on state vectors in the Hilbert space of the system under consideration. In terms of the density matrix, (8.9) can be rewritten in the form
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L(cI>n' (9pcI>n)
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Tr (9)
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The notation Tr A denotes the trace of the operator A and is the sum of all the diagonal matrix elements of A in any representation. An elementary property of the trace is that Tr(AB) = Tr(BA)
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It follows immediately that Tr A is independent of the representation; if Tr A is calculated in one representation, its value in another representation is
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Tr(SAS- 1 ) = Tr(S-lSA) = TrA
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The introduction of the density matrix merely introduces a notation. It does not introduce new physical content. The usefulness of the density matrix lies solely in the fact that with its help (8.11) is presented in a form that is manifestly independent of the choice of the basis {cI>n}' although this independence is a property that this expectation value always possesses. The density operator p defined by (8.10) contains all the information about an ensemble. It is independent of time if it commutes with the Hamiltonian of the system and if the Hamiltonian is independent of time. This statement is an immediate consequence of the equation of motion of p: Jp iha!
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which is the quantum mechanical version of Liouville's theorem. Formally we can represent the density operator p as
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where lcI>n) is the state vector whose wave function is cI>n' To prove (8.13), we verify that it has the matrix elements (8.10):
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Pmn == (cI>m' pcI>n) == (cI>mIPIcI>n)
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8mn lbnl
The time-averaging process through which we averaged out the effect of the external world on the system under consideration may be reformulated in terms of the density matrix.
Formula (8.2) is a general formula for the expectation value of any operator with respect to an arbitrary wave function 'Y. It may be trivially rewritten in the form Tr (R(!})
where R nm == (c m, cn) == (cI>n' RcI>m), the last identity being a definition of the operator R, and (!}nm == (cI>n, (!}cI>m)' Although R may depend on the time, Tr R is independent of time. The density operator is the time average of R:
The density matrix for the microcanonical ensemble in the representation in which the Hamiltonian is diagonal is
8mn lbnl2
Ibnl =
(E < En < E + il) (otherwise)
where {En} are the eigenvalues of the Hamiltonian. The density operator is
IcI>n) (cI>n I
The trace of P is equal to the number of states whose energy lies between E and E + il: (8.17) Trp = nn == r(E)
For macroscopic systems the spectrum {En} almost forms a continuum. For il E, we may take r(E) = w(E)il (8.18) where w(E) is the density of states at energy E. The connection between the microcanonical ensemble and thermodynamics is established by identifying the entropy as (8.19) S(E, V) = klogr(E) where k is Boltzmann's constant. This definition is the same as in classical statistical mechanics, except that r(E) must be calculated in quantum mechanics. From this point on all further developments become exactly the same as in classical statistical mechanics and so they need not be repeated. No Gibbs