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Cambridge University Press eBooks

Systems of identical particles

Donald D. Fitts

Published Aug 26, 1999• 0 citations
Publisher

Abstract

The postulates 1 to 6 of quantum mechanics as stated in Sections 3.7 and 7.2 apply to multi-particle systems provided that each of the particles is distinguishable from the others. For example, the nucleus and the electron in a hydrogen-like atom are readily distinguishable by their differing masses and charges. When a system contains two or more identical particles, however, postulates 1 to 6 are not sufficient to predict the properties of the system. These postulates must be augmented by an additional postulate. This chapter introduces this new postulate and discusses its consequences. Permutations of identical particles Particles are identical if they cannot be distinguished one from another by any intrinsic property, such as mass, charge, or spin. There does not exist, in fact and in principle, any experimental procedure which can identify any one of the particles. In classical mechanics, even though all particles in the system may have the same intrinsic properties, each may be identified, at least in principle, by its precise trajectory as governed by Newton's laws of motion. This identification is not possible in quantum theory because each particle does not possess a trajectory; instead, the wave function gives the probability density for finding the particle at each point in space. When a particle is found to be in some small region, there is no way of determining either theoretically or experimentally which particle it is.

DOI
10.1017/cbo9780511813542.009

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This chapter introduces the partition function that involves a sum over independent quantum states of the N-particle system. It outlines the problem of distinguishability, that is, how to distinguish one electron from another when they are replicas of each other and are all the same. It also emphasizes that electrons have no feature or label based on which they can be told apart, and they repeat the same attributes endlessly. The chapter explains that identical particles lead to complications in the quantum mechanical description of the microscopic state of the system as there are no observable changes in the system when any two particles are exchanged. It analyses the uncertainty principle, wherein the position of a particle cannot be accurately specified at each stage, nor can the precise trajectory of a particle be followed.

Oxford University Press eBooks•Oct 21, 1999•0 citations
The postulates 1 to 6 of quantum mechanics as stated in Sections 3.7 and 7.2 apply to multi-particle systems provided that each of the particles is distinguishable from the others. For example, the nucleus and the electron in a hydrogen-like atom are readily distinguishable by their differing masses and charges. When a system contains two or more identical particles, however, postulates 1 to 6 are not sufficient to predict the properties of the system. These postulates must be augmented by an additional postulate. This chapter introduces this new postulate and discusses its consequences. Permutations of identical particles Particles are identical if they cannot be distinguished one from another by any intrinsic property, such as mass, charge, or spin. There does not exist, in fact and in principle, any experimental procedure which can identify any one of the particles. In classical mechanics, even though all particles in the system may have the same intrinsic properties, each may be identified, at least in principle, by its precise trajectory as governed by Newton's laws of motion. This identification is not possible in quantum theory because each particle does not possess a trajectory; instead, the wave function gives the probability density for finding the particle at each point in space. When a particle is found to be in some small region, there is no way of determining either theoretically or experimentally which particle it is.
Details of the many-particle system can also be read from my book titled “Quantum Field Theory: Feynman Path Integrals and Diagrammatic Techniques in Condensed Matter” [ 23 ]. The fundamental difference between classical and quantum mechanics relates the concept of indistinguishability of identical particles where each particle can be equipped with an identifying marker without influencing its behavior in classical mechanics. Besides in classical mechanics, the particle follows its own continuous path in phase space and so it is possible to determine the particular process taking place. So, principally, each particle in a group of identical particles can be identified which is not so in quantum mechanics. It is not possible to mark a particle without influencing its physical state, and worse, if a number of identical particles are brought to the same region in space, their wave functions will rapidly spread out and overlap with one another, thereby soon rendering it impossible to say which particle is where. One of the fundamental assumptions for many-particle systems is therefore that identical particles (i.e., particles characterized by the same quantum numbers such as mass, charge and spin) are, in principle, indistinguishable.
The treatment of identical particles in quantum mechanics rests on two (related) principles: the spin‐statistics connection and the Symmetrization Postulate. In light of recent theories (such as q‐deformed commutators) that allow for ‘small’ violations of the spin‐statistics connection and the Symmetrization Postulate, we revisit the issue of how quantum mechanics deals with identical particles and how it supports or fails to support various philosophical stances concerning individuality. As a consequence of the expanded possibilities for quantum statistics, we argue that permutation symmetry is best formulated as a formal property of the state function describing the system of particles rather than as a property of the individual particles.1 Introduction2 Philosophical background2.1 Important terminology2.1.1 Identity2.1.2 Indistinguishability2.1.3 Indiscernibility2.2 When are particles indistinguishable?2.3 The Principle of the Identity of Indiscernibles and quantum mechanics2.4 The Principle of the Identity of Indiscernibles and logic2.5 Particle history2.6 Transcendental individuality3 Some quantum formalism3.1 The Principle of Permutation Invariance and the Symmetrization Postulate3.2 The configuration‐space approach3.3 Commutators and anticommutators, and identical particle statistics3.4 Q‐mutators4 Identical particle statistics: a holistic point of view5 Conclusions
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
In classical physics it was assumed that a system of two bodies, both having the same mechanical and electrical properties, would show the same behavior regardless of their complete identity. This assumption, that gave rise to the paradoxes of Gibbs and of Einstein which are here discussed, must be abandoned in atomic physics. We must distinguish sharply between identical particles and those that are only quasiidentical, in the sense that they are the same in mass, charge and the potential fields to which they give rise. All of the results which have hitherto been obtained in quantum mechanics are valid even if we assume that two particles such as two electrons or two helium nuclei are only quasi-identical. The postulate of complete identity is a new mathematical condition which, operating simultaneously with the wave equation and its boundary conditions, restricts the number of possible solutions of the equation and thus gives rise mathematically to a number of exclusion laws such as those which have been empirically obtained.The assumption of identity is equivalent to the mathematical assumption that the ordinary configuration space is degenerate, in the sense that one portion is a mere reflection of another. By this method it is shown that if the wave function is a scalar, the laws of exclusion are obtained without ambiguity. Thus a perfectly symmetrical diatomic gas in its simplest electronic state can have only even rotational levels. It is pointed out that the experimental results with O2{\mathrm{O}}_{2}O2​(O16{\mathrm{O}}_{16}O16​O16{\mathrm{O}}_{16}O16​) and He2{\mathrm{He}}_{2}He2​ show that these atoms exist in a single nuclear state. It is predicted that with heavier atoms both even and odd states will be found owing to the existence of isobars. The monatomic gas is then discussed and it is shown that, if the Schr\"odinger function is a scalar, only the Bose-Einstein statistics are possible.The phenomenon of spin introduces a vector character into the wave function. If we assume that the spin factor changes sign upon reflection when the number of units of spin is odd but not when the number of units of spin is even, we have all the cases of exclusion now known, including the case of the nitrogen molecule. As an example of the method, the problem of the helium atom is discussed.
This chapter introduces the partition function that involves a sum over independent quantum states of the N-particle system. It outlines the problem of distinguishability, that is, how to distinguish one electron from another when they are replicas of each other and are all the same. It also emphasizes that electrons have no feature or label based on which they can be told apart, and they repeat the same attributes endlessly. The chapter explains that identical particles lead to complications in the quantum mechanical description of the microscopic state of the system as there are no observable changes in the system when any two particles are exchanged. It analyses the uncertainty principle, wherein the position of a particle cannot be accurately specified at each stage, nor can the precise trajectory of a particle be followed.