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The postulates of quantum statistical mechanics are postulates concerning the coefficients (c n, cm)' when (8.3) refers to a macroscopic observable of a macroscopic system in thermodynamic equilibrium. For definiteness, we consider a macroscopic system which, although not truly isolated, interacts so weakly with the external world that its energy is approximately constant. Let the number of particles in the system be N and the volume of the system be V, and let its energy lie between E and E + il(il E). Let ,)If' be the Hamiltonian of the system. For such a system it is convenient (but not necessary) to choose a standard set of complete orthonormal wave functions {<I>n} such that every <l>n is a wave function for N particles contained in the volume V and is an eigenfunction of ,)If' with the eigenvalue En:
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(8.4)
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The postulates of quantum statistical mechanics are the following:
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Postulate of Equal a Priori Probability
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(E<En<E+il) (otherwise)
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QUANTUM STATISTICAL MECHANICS
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Postulate of Random Phases
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As a consequence of these postulates we may effectively regard the wave function of the system as given by
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(8.7)
where (E < En < E + il) (otherwise)
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and where the phases of the complex numbers {bn } are random numbers. In this manner the effect of the external world is taken into account in an average way. The observed value of an observable associated with the operator (!) is then given by
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L Ibn 12 ( cI>n' (!}cI>n)
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(8.9)
It should be emphasized that for (8.7) and (8.8) to be valid the system must interact with the external world. Otherwise the postulate of random phases is false. By the randomness of the phases we mean no more and no less than the absence of interference of probability amplitudes, as expressed by (8.9). For a truly isolated system such a circumstance may be true at an instant, but it cannot be true for all times. The postulate of random phases implies that the state of a system in equilibrium may be regarded as an incoherent superposition of eigenstates. It is possible to think of the system as one member of an infinite collection of systems, each of which is in an eigenstate whose wave function is cI>n' Since these systems do not interfere with one another, it is possible to form a mental picture of each system one at a time. This mental picture is the quantum mechanical generalization of the Gibbsian ensemble. The ensemble defined by the previous postulates is the mierocanonical ensemble. The postulates of quantum statistical mechanics are to be regarded as working hypotheses whose justification lies in the fact that they lead to results in agreement with experiments. Such a point of view is not entirely satisfactory, because these postulates cannot be independent of, and should be derivable from, the quantum mechanics of molecular systems. A rigorous derivation is at present lacking. We return to this subject very briefly at the end of this chapter. We should recognize that the postulates of quantum statistical mechanics, even if regarded as phenomenological statements, are more fundamental than the laws of thermodynamics. The reason is twofold. First, the postulates of quantum statistical mechanics not only imply the laws of thermodynamics, they also lead to definite formulas for all the thermodynamic functions of a given substance.
STATISTICAL MECHANICS
Second, they are more directly related to molecular dynamics than are the laws of thermodynamics. The concept of an ensemble is a familiar one in quantum mechanics. A trivial example is the description of an incident beam of particles in the theory of scattering. The incident beam of particles in a scattering experiment is composed of many particles, but in the theory of scattering we consider the particles one at a time. That is, we calculate the scattering cross section for a single incident particle and then add the cross sections for all the particles to obtain the physical cross section. Inherent in this method is the assumption that the wave functions of the particles in the incident beam bear no definite phase with respect to one another. What we have described is in fact an ensemble of particles. A less trivial example is the description of a beam of incident electrons whose spin can be polarized. If an electron has the wave function
where A and B are definite complex numbers, the electron has a spin pointing in some definite direction. This corresponds to an incident beam of completely polarized electrons. In the cross section calculated with this wave function there will appear interference terms proportional to A*B + AB *. If we have an incident beam that is partially polarized, we first calculate the cross section with a wave function proportional to (~) and then do the same thing for (~), adding the two cross sections with appropriate weighting factors. This is eqUIvalent to and ( ~) occur with certain relative probabilities.
describing the incident beam by an ensemble of electrons in which the states ( ~ )