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Princeton University Press eBooks

The Quantum Statistics

John von Neumann

Published Feb 27, 2018• 3 citations
Publisher

Abstract

This chapter returns to the analysis of quantum mechanical theories. In the previous chapter, how quantum mechanics makes possible the determination of all possible values of one particular physical quantity—energy—was only discussed. These values are the eigenvalues of the energy operator (i.e., the numbers of its spectrum). On the other hand, no mention was made about the values of other quantities, as well as regarding the causal or statistical relations among the values of several quantities. The statements of the theory relative to this problem are thus considered in this chapter. It takes as a basis the wave mechanical method of description since the equivalence of the two theories has already been established.

DOI
10.23943/princeton/9780691178561.003.0004

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This chapter returns to the analysis of quantum mechanical theories. In the previous chapter, how quantum mechanics makes possible the determination of all possible values of one particular physical quantity—energy—was only discussed. These values are the eigenvalues of the energy operator (i.e., the numbers of its spectrum). On the other hand, no mention was made about the values of other quantities, as well as regarding the causal or statistical relations among the values of several quantities. The statements of the theory relative to this problem are thus considered in this chapter. It takes as a basis the wave mechanical method of description since the equivalence of the two theories has already been established.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
The quantum mechanics of fields recently developed by us leads to a modification of statistical mechanics of elementary particles which seems to overcome some of the difficulties (divergence of integrals) occurring in the usual quantum theory of fields. The main difference between the new theory and the usual one is as follows. In the usual theory the wave-vector k is introduced classically and, so to speak, kinematically by the Fourier analysis of the field. The Fourier coefficients of the field components are then treated according to quantum mechanics as non-commuting quantities; those belonging to the wave-vector k describe the corresponding “model” mechanical system, namely the kth radiation oscillator. But the statement that the Fourier coefficients belonging to a certain k all vanish, which statement classically is significant, is now meaningless because there is a lowest state with zero-point energy for each radiation oscillator. The field is thus made to be equivalent to the assembly of radiation oscillators of all possible wave-vectors which, being necessarily infinite in number, contribute an infinite zero-point energy for the pure field and lead to other divergent integrals for the interaction between different fields.
We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the ε j 's and g j 's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics. We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.
Classical and quantum physics both make considerable use of the expressions "physical system" and "state of a physical system." Indeed, any sufficiently isolated thing—a voltmeter, an electron, or a molecule—is a physical system. The expression physical system is taken in its familiar, intuitive sense. For instance, a partially but sufficiently isolated atom is a physical system or, for short, a system. This chapter summarizes the rules of quantum mechanics. Quantum mechanics is essentially a statistical theory. Except in special cases it makes no prediction that bears on individual systems. The observables which have a continuous eigenvalue spectrum have eigenvectors that are not normalizable. As regards the measurement of these observables, some technical changes should therefore be made in expression. The chapter introduces vector space which has the structure of a Hilbert space for describing ensembles of physical systems of the same type.
T he necessity for the treatment of the properties of quantum particles in terms of their most likely values is indicated in the previous chapter. Schrödinger’s equation, which provides the means to do this, is developed by examining the classical relationships describing the properties of a particle in motion and their transformation into quantum mechanical equivalents. The probable properties of quantum particles are obtained from the wavefunctions that are meaningful solutions to Schrödinger’s equation.
Abstract. Quantum physics against intuition. Since Galileo, deeper understanding of physical reality often contradicts everyday’s intuition. This trend has become dramatic a century ago, rendering modern physics abstruse to the laymen. The scientist can exert his intuition and his imagination within an elaborate mathematical language, but it is impossible to translate his ideas faithfully into ordinary language; this difficulty may be one of the causes of the present tarnish of the image of science. A first shock was brought in by relativity, which revolutionized our conceptions of space and time, but quantum physics presents even more counter-intuitive features. One has been led to represent physical quantities pertaining to a system, such as the position or the velocity of a particle, no longer by ordinary numbers but by “observables”, mathematical objects which belong to a non-commutative algebra. Namely, the product of two such observables is not a unique new observable but may depend on their order: the products AB and BA may differ.The values that a physical quantity can take constitute a set of ordinary numbers, the eigenvalues a( n) of the observable A. This set may be discrete as a consequence of the algebra. The discreteness of the eigenvalues of the energy of an atom explains the spectrum of its radiation. Any component of the intrinsic angular momentum of an electron can take only two opposite values, baffling our geometric intuition.If two observables commute (i.e., AB = BA), they are compatible: both may take well-defined values chosen among the eigenvalues. If they do not commute (i.e., AB ≠ BA), which occurs for the position and velocity of a particle, for the electric and magnetic field, or for the components of an angular momentum, they are incompatible: their values cannot be measured nor even specified simultaneously. Each of them presents statistical fluctuations which must be larger than some lower bound (Heisenberg’s inequality). The state of a system, prepared under some definite conditions, should be identified as the set of expectation values of all observables.Thus, quantum physics, our most fundamental description of nature, is a theory involving irreducible probabilities and unavoidable uncertainties. Its predictions do not refer to a single system, but to a statistical ensemble, and the observer cannot be completely disregarded. Moreover it may involve aspects which are apparently contradictory (Bohr’s complementarity). An electron, an atom or a molecule may behave in limiting cases either as a particle or as a wave; under intermediate conditions, it behaves as both, but each of the descriptions is blurred with probabilities. As a wave, it can interfere with itself. Likewise, an electromagnetic field has the double nature of a set of photons and of a wave. At the macroscopic scale, relative fluctuations are small, so that determinism emerges.Quantum probabilities differ from standard ones. In particular, quantum correlations involving four observables pair wise violate Bell’s inequalities, so that microscopic quantities are intrinsically indeterminate and depend on the observation process.The GHZ paradox, due to Greenberger, Horne and Zeilinger, exemplifies the most troublesome conceptual feature of quantum physics. Indeed, it baffles not only intuition, but also current logics. It deals with a physical system made of three correlated particles. Various observables, A1, A2, A3, B1, B2 and B3 ≡ B1 B2, which we do not specify here, can be measured on such a composite system. Each of them may take discrete values a1 , a2, a3, b1, b2 equal either to +1 or to –1. Since the three observables A1 B1, A2 B2 and A3 B3 are compatible, they can be simultaneously controlled. Consider then a statistical ensemble of systems, all prepared identically in such a way that A1 B1 = A2 B2 = A3 B3 = +1. If we measure A1 and B1, we shall find either a1 = b1 = +1 or a1 = b1 = –1, that is always a1 = b1; likewise for a2 = b2 = ±1 and a3 = b1 b2 = ±1. We are tempted to infer that a simultaneous determination of the values a1, a2 and a3 of A1, A2 and A3 (which are compatible) should verify a3 = a1 a2, that is, a1 a2 a3 = 1. However both theory and experiment provide in all cases a1 a2 a3 = –1. In fact the four statements a1 = b1, a2 = b2, a3 = b1 b2 and a1 a2 a3 = –1 which seem contradictory are separately “true”, in the sense that each one can be verified experimentally; but such experiments cannot be performed on a single system. We are not allowed to make a logical reasoning based on a simultaneous consideration of the values of B1 and A2 because these observables do not commute. No calculation, no experiment can simultaneously provide values for the numbers b1 and a2. Hence, we cannot regard as true together the statements a1 = b1 and a2 = b2, although they can be checked by separate experiments. Each one is true, but their union is meaningless.Thus, if theory does not allow us to imagine thought experiments in which several statements on a system might simultaneously be checked, we cannot draw conclusions by putting such statements together. We are led to regard the “truth” of a quantum statement as contextual. The values of physical quantities are not attached only to the system in itself, but also refer to the context in which their relations may be verified. Quantum mechanics does not deal with objects per se, but with what we know and what we can infer about them. It does not provide a direct image of the reality of the object since it involves its observation. Although it is a perfect tool for letting us make statistical predictions, and although it appears as our most basic theory owing to its precision and generality, it violates our daily intuition, our elementary mathematical intuition and our feelings about reality. Its probabilities and even its logics obey rather shocking new rules.