Statistics

In quantum statistics, the behavior of a macroscopic body cannot be determined solely by using Schrodinger's equation. The energy levels in a macroscopic body are extremely dense, making it impossible for the body to be in a strictly stationary state. The energy of the system is always broadened by interactions with surrounding bodies. It is also impractical to describe the state of a macroscopic body using a wave function, so a density matrix is used instead. The density matrix allows for calculations of mean values and probabilities of different quantities. The averaging process used to obtain the density matrix does not imply that the subsystem can be in multiple states with different probabilities. 5. The Statistical Matrix Turning now to the distinctive features of quantum statistics, we may note, first of all, that the purely mechanical approach to the problem of determining the behavior of a macroscopic body in quantum mechanics is, of course, just as hopeless as in classical mechanics. Such an approach would require the solution of Schrodinger's equation for a system consisting of all the particles in the body, a problem still more hopeless, one might even say, than the integration of the classical equations of motion. But even if it were possible in some particular case to find a general solution of Schrodinger's equation, it would be utterly impossible to select and write down the particular solution satisfying the precise conditions of the problem and specified by particular values of an enormous number of different quantum numbers. Moreover, we shall see below that for a macroscopic body, the concept of stationary states itself becomes to some extent arbitrary, a fact of fundamental significance. Let us first elucidate some purely quantum mechanical features of macroscopic bodies as compared with systems consisting of a relatively small number of particles. These features amount to an extremely high density of levels in the energy eigenvalue spectrum of a macroscopic body. The reason for this is easily seen if we note that, because of the very large number of particles in the body, a given quantity of energy can, roughly speaking, be distributed in innumerable ways among the various particles. The relation between this fact and the high density of levels becomes particularly clear if we take as an example a macroscopic body consisting of a gas of n particles, which do not interact at all, enclosed in some volume. The energy levels of such a system are just the sums of the energies of the individual particles, and the energy of each particle can range over an infinite series of discrete values. It is clear that, on choosing in all possible ways the values of the n terms in this sum, we shall obtain a very large number of possible values of the energy of the system in any appreciable finite part of the spectrum, and these values will therefore lie very close together. It may be shown, see 7.18, that the number of levels in a given finite range of the energy spectrum of a macroscopic body increases exponentially with the number of particles in the body, and the separations between levels are given by numbers of the form 10¹ⁿ, where n is a number of the order of the number of particles in the body, whatever the units, since a change in the unit of energy has no effect on such a fantastically small number. In consequence of the extremely high density of levels, a macroscopic body in practice can never be in a strictly stationary state. First of all, it is clear that the value of the energy of the system will always be broadened by an amount of the order of the energy of interaction between the system and the surrounding bodies. The latter is very large in comparison with the separations between levels, not only for quasi-closed subsystems but also for systems which from any other aspect could be regarded as strictly closed. In nature, of course, there are no completely closed systems, whose interaction with any other body is exactly zero, and whatever interaction does exist, even if it is so small that it does not affect other properties of the system, will still be very large in comparison with the infinitesimal intervals in the energy spectrum. In addition to this, there is another fundamental reason why a macroscopic body in practice cannot be in a stationary state. It is known from quantum mechanics that the state of a quantum mechanical system described by a wave function is the result of some process of interaction of the system with another system which obeys classical mechanics to a sufficient approximation. In this respect, the occurrence of a stationary state implies particular properties of the system. Here we must distinguish between the energy E of the system before the interaction and the energy E of the state which results from the interaction. The uncertainties dE and dE in the quantities E and E are related to the duration dt of the interaction process by the formula. The two errors dE and dE are in general of the same order of magnitude, and analysis shows that we cannot make dE dE. We can therefore say that dE dH dE. In order that the state may be regarded as stationary, the uncertainty dE must certainly be small in comparison with the separations between adjoining levels. Since the latter are extremely small, we see that, in order to bring the macroscopic body into a particular stationary state, an extremely long time dtp over H dEI would be necessary. In other words, we again conclude that strictly stationary states of a macroscopic body cannot exist. To describe the state of a macroscopic body by a wave function at all is impracticable, since the available data concerning the state of such a body are far short of the complete set of data necessary to establish its wave function. Here the position is somewhat similar to that which occurs in classical statistics, where the impossibility of taking account of the initial conditions for every particle in a body makes impossible an exact mechanical description of its behavior. The analogy is imperfect, however, since the impossibility of a complete quantum mechanical description and the lack of a wave function describing a macroscopic body may, as we have seen, possess a much more profound significance. The quantum mechanical description, based on an incomplete set of data concerning the system, is affected by means of what is called a density matrix. A knowledge of this matrix enables us to calculate the mean value of any quantity describing the system and also the probabilities of various values of such quantities. The incompleteness of the description lies in the fact that the results of various kinds of measurement, which can be predicted with a certain probability from a knowledge of the density matrix, might be predictable with greater or even complete certainty from a complete set of data for the system, from which its wave function could be derived. We shall not pause to write out here the formulae of quantum mechanics relating to the density matrix in the coordinate representation, since this representation is seldom used in statistical physics, but we shall show how the density matrix may be obtained directly in the energy representation, which is necessary for statistical applications. Let us consider some subsystem and define its stationary states as the states obtained when all interactions of the subsystem with the surrounding parts of a closed system are entirely neglected. Let PES and Q be the normalized wave functions of these states, without the time factor, Q conventionally denoting the set of all coordinates of the subsystem, and the suffix and the set of all quantum numbers which distinguish the various stationary states. The energies of these states will be denoted by N. Let us assume that at some instant the subsystem is in a completely described state, with wave function kicks. The latter may be expanded in terms of the functions sinQ, which form a complete set. We write the expansion as the mean value of any quantity f in this state can be calculated from the coefficients cn, cn, cn. By means of the formula are the matrix elements of the quantity f, at f, f being the corresponding operator. The change from the complete to the incomplete quantum mechanical description of the subsystem may be regarded as a kind of averaging over its various pecs-in-piece-ices states. In this averaging, the products cn, scm, cnn, c, e, m, cn, t, s, m give a double set two suffixes of quantities which we denote by wmn, wmn, wmn, and which cannot be expressed as products of any quantities forming a single set. The mean value of fOEF is now given by the set of quantities 1d, mn, wmn, which in general are functions of time, is the density matrix in the energy representation. In statistical physics, it is called the statistical matrix. If we regard the dm, dm, mn, wmn as the matrix elements of some statistical operator w, w, w, w, then the sum n, e, m, f, m, or sum n, u, min, f, m, n, e, n, f, m will be a diagonal matrix element of the operator product. A u, f, hat, e, n, hat, f, u, fa, and mean value f, n, bar, f, f, ba becomes the trace sum of diagonal elements of this operator. This formula has the advantage of enabling us to calculate with any complete set of orthonormal wave functions. The trace of an operator is independent of the particular set of functions with respect to which the matrix elements are defined. The other expressions of quantum mechanics which involve the quantities cn, cn, cn are similarly modified, the products cn, cm, cnn, c, m, cn, cm being everywhere replaced by the averaged values du, m, nu, nd, and du, dm. For example, the probability that the subsystem is in the nth state is equal to the corresponding diagonal element. When w, i, n, n, o, when n of the density matrix instead of the squared modulus cn, dn, cn, i, n, cn, cn it is evident that these elements which we shall denote by w, n, w, w, n, w, n are always positive and satisfy the normalization condition. It must be emphasized that the averaging over various cn, p, psi, cal states which we have used in order to illustrate the transition from a complete to an incomplete quantum mechanical description has only a very formal significance. In particular, it would be quite incorrect to suppose that the description by means of the density matrix signifies that the subsystem can be in various cn, p, psi, cal states with various probabilities and that the averaging is over these probabilities. Such a treatment would be in conflict with the basic principles of quantum mechanics. The states of a quantum mechanical system that are described by wave functions are sometimes called pure states, as distinct from mixed states which are described by a density matrix. Care should, however, be taken not to misunderstand the latter term in the way indicated above. The averaging by means of the statistical matrix according to 5.4 has a twofold nature. It comprises both the averaging due to the probabilistic nature of the quantum description, even when as complete as possible, and the statistical averaging necessitated by the incompleteness of our information concerning the object considered. For a pure state, only the first averaging remains, but in statistical cases, both types of averaging are always present. It must be borne in mind, however, that these constituents cannot be separated. The whole averaging procedure is carried out as a single operation and cannot be represented as the result of successive averagings, one purely quantum mechanical and the other purely statistical. The statistical matrix in quantum statistics takes the place of the distribution function in classical statistics. The whole of the discussion in the previous sections concerning classical statistics and the, in practice, deterministic nature of its predictions applies entirely to quantum statistics also. The proof given in Myrrh 2, that the relative fluctuations of additive physical quantities tend to zero as the number of particles increases, made no use of any specific properties of classical mechanics, and so remains entirely valid in the quantum case. We can therefore again assert that macroscopic quantities remain practically equal to their mean values. In classical statistics, the distribution function Q pqqpqpq gives directly the probability distribution of the various values of the coordinates and momenta of the particles of the body. In quantum statistics, this is no longer true. The quantities gnwnwn give only the probabilities of finding the body in a particular quantum state, with no direct indication of the values of the coordinates and momenta of the particles. From the very nature of quantum mechanics, the statistics based on it can deal only with the determination of the probability distribution for the coordinates and momenta separately, not together, since the coordinates and momenta of a particle cannot simultaneously have definite values. The required probability distributions must reflect both the statistical uncertainty and the uncertainty inherent in the quantum mechanical description. To find these distributions, we repeat the arguments given above. We first assume that the body is in a pure quantum state with the wave function 5.1. The probability distribution for the coordinates is given by the squared modulus, so that the probability that the coordinates have values in a given interval dqdq1dq2, dqsdqesdq1dq2 and l dots, dqes is dqdq1dq2, dqes is dwq2dqdwq, aot is crpc, ta2ddqdwq2dq. For a mixed state, the products cncmcn, cemcncm are replaced by the elements wnu and wmn of the statistical matrix, and anywangpzing2 thus becomes, by the definition of the matrix elements, and so thus we have the following formula for the coordinate probability distribution. In this expression, the functions poinir, psi, and poin may be any complete set of normalized wave functions. Let us next determine the momentum probability distribution. The quantum states in which all the momenta have definite values correspond to free motion of all the particles. We denote the wave functions of these states by PESPQRPSIP, CUPSPQ, the suffix PPP conventionally representing the set of values of all the momenta. As we know, the diagonal elements of the density matrix are the probabilities that the system is in the corresponding quantum states. Hence, having determined the density matrix with respect to the set of functions PESPPPSIP, we obtain the required momentum probability distribution from the formula. It is interesting that both distributions, coordinate and momentum, can be obtained by integrating the same function. Integration of this expression with respect to QQQ gives the momentum distribution, and with respect to PPP gives the coordinate distribution, the expression 5.8, with the functions PPPPSIP as the complete set of wave functions. It should be emphasized, however, that this does not mean that the function 5.10 may be regarded as a probability distribution for coordinates and momenta simultaneously. The expression 5.10 is complex, quite apart from the fact that such a view would conflict with the basic principles of quantum mechanics.