Identical Particles

9 Key excerpts on "Identical Particles"

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  Quantum Mechanics

  Non-Relativistic Theory

  • L D Landau, E.M. Lifshitz(Authors)
  • 1981(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  CHAPTER IX

  IDENTITY OF PARTICLES

  §61 The principle of indistinguishability of similar particles

  IN classical mechanics, Identical Particles (electrons, say) do not lose their “individuality”, despite the identity of their physical properties. For we can imagine the particles at some instant to be “numbered”, and follow the subsequent motion of each of these in its path; then at any instant the particles can be identified.

  In quantum mechanics the situation is entirely different. We have already mentioned several times that, by virtue of the uncertainty principle, the concept of the path of an electron ceases to have any meaning. If the position of an electron is exactly known at a given instant, its coordinates have no definite values even at the next instant. Hence, by localizing and numbering the electrons at some instant, we make no progress towards identifying them at subsequent instants; if we localize one of the electrons, at some other instant, at some point in space, we cannot say which of the electrons has arrived at this point.

  Thus, in quantum mechanics, there is in principle no possibility of separately following each of a number of similar particles and thereby distinguishing them. We may say that, in quantum mechanics, Identical Particles entirely lose their “individuality”. The identity of the particles with respect to their physical properties is here very far-reaching: it results in the complete indistinguishability of the particles.

  This principle of the indistinguishability of similar particles , as it is called, plays a fundamental part in the q u a n t um theory of systems composed of Identical Particles. Let us start by considering a system of only two particles. Because of the identity of the particles, the states of the system obtained from each other by merely interchanging the two particles must be completely equivalent physically. This means that, as a result of this interchange, the wave function of the system can change only by an unimportant phase factor. Let ψ(ξ1 , ξ2 ) be the wave function of the system, ξ1 and ξ2

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  Quantum Mechanics

  A Paradigms Approach

  • David H. McIntyre(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  C H A P T E R 13 Identical Particles To study systems like multielectron atoms, we need to properly account for the fact that all fundamental particles like electrons and protons are identical. In classical physics, particles are not identical—we can always find a way to uniquely identify a particular particle. Even if we make two classical particles “the same” to the utmost level of precision, we can still find a way to identify the two particles without affecting their classical motion. For example, billiard balls behave identically, but can be identified by their numbers. In quantum mechanics, there is no way to identify two different electrons—they are indistinguishable. Two hydrogen atoms are identical no matter where they are in the universe. Researchers rely on this fact when they compare their experimental results on the spectra of hydrogen atoms in different laboratories. To account for the indistinguishability of fundamental particles, we introduce a new postulate in quantum mechanics, which leads to the Pauli exclusion principle that is responsible for the periodic table and all of chemistry. We apply this new postulate to the helium atom to learn how the indistinguishability of the two electrons in the atom affects the energies and the allowed states. 13.1  TWO SPIN-1/2 PARTICLES To start our discussion of Identical Particles, let’s return to the system of two spin-1/2 particles that we studied in Chapter 11. We found that we could describe the system using either of two bases: 0 + + 9 , 0 + - 9 , 0 - + 9 , 0 - - 9 uncoupled basis 0 s 1 s 2 m 1 m 2 9 0 11 9 , 0 109 , 0 1, - 1 9 , 0 009 coupled basis 0 SM S 9 . (13.1) The coupled basis is preferred when the two particles or systems interact, such as in the hyperfine interaction or the spin-orbit interaction, because the Hamiltonian is diagonal in that basis.

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  The Odd Quantum

  • Sam Treiman(Author)
  • 2002(Publication Date)
  • Princeton University Press

    (Publisher)

  C H A P T E R S I X Identical Particles Symmetry, Antisymmetry Rules Although some of the principles of quantum mechanics were laid out earlier in general terms, for the most part we have concentrated so far on the case of a single particle. As the num- ber of particles in a quantum system increases, the computa- tional complications inevitably increase—often beyond reach if one is hoping for exact answers. Models based on physical in- sight and reasonable mathematical approximations have to in- tervene. However, as long as the particles in a system are all different one from another no new principles peculiar to mul- tiparticle systems come into play. But, remarkably, the various elementary particles of nature do in fact come in strictly iden- tical copies. Why this is so is something we’ll turn to later on. For the present, let us just see how quantum mechanics deals with particle identity. Both classically and quantum mechanically, two particles are said to be identical if they respond in exactly the same way to all conceivable probes. If the probe is a mass scale, they dis- play the same mass; if it is an electric or magnetic field, they reveal the same charge; they scatter light waves in the same way; and so on. Classically, of course, if the objects are macro- scopic you can mark and thereby distinguish them with identi- fying tags. But that’s cheating: the marked objects are no longer identical. We are concerned here with strictly identical entities that cannot be tagged. Anyhow, classically there is no need to 150 C H A P T E R S I X physically mark the particles. Although they are intrinsically identical, you can in principle keep an eye on them and simply declare at some initial instant that particle 1 is the one that’s here, particle 2 is the one over there, and so on. Thereafter you can (in principle) follow their movements and thereby maintain a consistent identification.

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  Quantum Mechanics for Scientists and Engineers

  • David A. B. Miller(Author)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  Chapter 13 Identical Particles Prerequisites: Chapters 2–5, and Chapters 9, 10, and 12. One aspect of quantum mechanics that is very different from the classical world is that particles can be absolutely identical – so identical that it is meaningless to say which is which. This “identicality” has substantial consequences for what states are allowed, quantum mechanically, and in the counting of possible states. Here, we examine this identicality, introducing the concepts of fermions and bosons and the Pauli exclusion principle that lies behind so much of the physics of materials. 13.1 Scattering of Identical Particles Suppose we have two electrons in the same spin state, 1 electrons that, for the moment, we imagine we can label as electron 1 and electron 2. We write the spatial coordinates of electron 1 as r 1 and those of electron 2 as r 2 . As far as we know, there is absolutely no difference between one electron and another. They are absolutely interchangeable. We might think, because of something we know about the history of these electrons, that it is more likely that we are looking at electron 1 rather than electron 2, but there is no way by making a measurement so that we can actually know for sure at which one we are looking. We could imagine that the two electrons were traveling through space, each in some kind of wavepacket. The wavepackets might each be quite localized in space at any given time. These wavepackets, however, each extend out arbitrarily far, even though the amplitude becomes small and, hence, the wavefunctions always overlap to some degree. We may find the following argument more convincing if we imagine that the wavepackets are initially directed toward one another and that these wavepackets substantially overlap for some time as they “bounce” off one another as shown in Fig. 13.1, repelled by the electron Coulomb repulsion or even some other force as yet undetermined.

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  Interpreting Bodies

  Classical and Quantum Objects in Modern Physics

  • Elena Castellani(Author)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  10. Ibid., 105. 11. Reichenbach, The Direction of Time , 211–224. [Note of the editor.] 5 The Problem of Indistinguishable Particles Bas C. van Fraassen In the quantum-mechanical description of nature, an elementary parti-cle is first of all characterized by some constant features, such as mass and charge. These features serve to classify them into basic kinds or types; physicists sometimes refer to particles characterized by the same constants as “Identical Particles.” (In deference to philosophical usage I shall use “identical” only in the strict sense in which no two distinct en-tities are identical.) In addition to these constant features, each particle is capable of various states of motion (represented by a Hilbert space). And that is all. So if two particles are of the same kind, and have the same state of mo-tion, nothing in the quantum-mechanical description distinguishes them. Yet this is possible. We have a dilemma: either this possibility violates the principle of identity of indiscernibles, or the quantum-mechanical description of nature is not complete. The dilemma could also be un-dercut: perhaps to conceive of such a particle as an individual, to which such a principle even could apply, is one of those many conceptual mis-takes fostered by an upbringing in classical physics. A closer look very quickly reveals a whole cluster of problems, of which this dilemma is the center. Some sorts of particles obey the exclusion principle, and cannot have two in the same state of motion—but they too have been cited as a violation of identity of indiscernibles. Both sorts of particles exhibit (for reasons that appear to be related) statistical correlations in their behavior which seem to defy causal explanation. And so forth. In this paper I shall only try to identify, relate, and clarify the prob-lems in this problem cluster. Though I will describe attempts at solution, including my own, I advocate none at this point.

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  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  476 Ch. 17: Identical Particles Another common argument claims that the special properties of the states of Identical Particles, such as their restriction to be either symmetric or antisymmetric, are related to the indeterminacy principle. According to this argument, Identical Particles could be distinguished in classical mechanics by continuously following them along their trajectories. But in quantum mechanics the indeterminacy relation (8.33) does not allow position and momentum to both be sharp in any state. Therefore we can-not identify separate trajectories, and so the argument concludes that we cannot distinguish the particles. However, the pragmatic indistinguishabil-ity that is deduced from this argument implies nothing about the symmetry of the state vector. The derivation of the indeterminacy relations in Sec. 8.4 uses no property of the state vector or state operator except its existence. Even if we used an absurd state vector, having the wrong symmetry and violating the Schrodinger equation, we would still not violate the indetermi-nacy relations. Therefore the indeterminacy relations tell us nothing about the properties of the state vector. ]] We now examine the empirical consquences of the symmetrization postulate. Consider the three-particle antisymmetric function given in Sec. 17.1. ty a pi = {|a)|/3>| 7 > - |/3>|a)| 7 > - |a>| 7 >|/3) -l)P)a) + | 7 >|a>|/3> + |/3>| 7 >|a)}/>/S. If we put a = P we obtain Wpp-y = 0. A similar result clearly holds for an antisymmetrized product state vector for any number of particles. This is the basis of the Pauli exclusion principle, which asserts that in a system of identical fermions no more than one particle can have exactly the same single particle quantum numbers. The exclusion principle forms the basis of the theory of atomic structure and atomic spectra, and so is very well established. Thus we have strong empirical evidence that electrons are fermions.

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  Sneaking a Look at God's Cards

  Unraveling the Mysteries of Quantum Mechanics - Revised Edition

  • Giancarlo Ghirardi, Gerald Malsbary(Authors)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  We can sum up the position of this great thinker in his own words as follows: “Nowhere in nature can be found two entities so exactly alike that some inner difference cannot be found.” I will now illustrate, in the context of quantum formalism, the specific implications of the fact that, as far as we know (and, I would add, as far as is shown by all the significant consequences of the hypothesis), every electron is identical to any other electron, every proton to every other proton, and so on. 14.1. Identification of Indiscernibles in Classical and Quantum Physics For simplicity’s sake, we can consider the case of a physical system that comprises two elementary identical constituents, such as, for example, two particles of the same type, that is, two electrons or two protons, mesons, etc. First of all, we need to be precise about what we mean when we say that two objects of this kind must be considered identical. The phenomenology of elementary particles, as well as the experiments we can imagine carrying out on them, leads us to conclude that all the in-trinsic physical properties that characterize them are exactly identical: all the electrons, protons, mesons, etc., of the entire universe have exactly the same charge, mass, spin, and every other quality. If I were to present you with a specific electron to investigate (admitting, as we will have to ask ourselves soon, that this assertion has some meaning) you would not be able to identify any physical process or carry out any measurement that would permit you to distinguish this electron from any other one, in-dependently from its past “history,” or from the processes in which it has been involved, or from the production mechanism that has generated it (this electron could have its origin in a process whereby a photon created an electron-positron coupling, or in the decay of a neutron into a proton, an electron, and a neutrino).

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  Fundamentals and New Frontiers of Bose-Einstein Condensation

  • Masahito Ueda(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  Chapter 1 Fundamentals of Bose–Einstein Condensation 1.1 Indistinguishability of Identical Particles Quantum statistics is governed by the principle of indistinguishability of Identical Particles. Particles with integer (half-integer) spin (in multiples of , where is the Planck constant divided by 2 π ) are called bosons (fermions). Bosons obey Bose–Einstein statistics in which there is no re-striction on the occupation number of any single-particle state. Fermions obey Fermi–Dirac statistics in which not more than one particle can occupy any single-particle state. The many-body wave function of identical bosons (fermions) must be symmetric (antisymmetric) under the exchange of any two particles. This symmetry requirement drastically reduces the number of available quantum states of the system, resulting in highly nonclassical phenomena at low temperature. To understand this, let us suppose that we obtain a wave function Φ( ξ 1 , ξ 2 ) of a two-particle system by solving the Schr¨odinger equation, where ξ 1 and ξ 2 represent the space and possibly spin coordinates of the two particles. For identical bosons (fermions), the symmetrized (antisym-metrized) wave function is given by Ψ( ξ 1 , ξ 2 ) = 1 √ 2 Φ( ξ 1 , ξ 2 ) ± Φ( ξ 2 , ξ 1 ) , (1.1) where the plus (minus) sign indicates bosons (fermions). The joint proba-bility of finding the two particles at ξ 1 and ξ 2 is given by | Ψ( ξ 1 , ξ 2 ) | 2 = 1 2 {| Φ( ξ 1 , ξ 2 ) | 2 + | Φ( ξ 2 , ξ 1 ) | 2 ± 2Re[Φ ∗ ( ξ 1 , ξ 2 )Φ( ξ 2 , ξ 1 )] } , (1.2) where Re denotes the real part. Because of the last interference term in Eq. (1.2), the probability of finding the two identical bosons at the same 1 2 Fundamentals and New Frontiers of Bose–Einstein Condensation coordinate, | Ψ( ξ, ξ ) | 2 , is twice as high as | Φ( ξ, ξ ) | 2 , which gives the corre-sponding probability for distinguishable particles. In contrast, for fermions, | Ψ( ξ, ξ ) | 2 vanishes in accordance with Pauli’s exclusion principle.

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  Fundamentals of Quantum Mechanics

  • Ajit Kumar(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  In quantum mechanics, colouring the particles means putting separate tags on them which we cannot do. This is because putting a tag on them means specifying some distinct physical characteristic for each of the particles of the system and this cannot be achieved in view of the fact that all of them have the same maximal set of commuting observables. Secondly, due to the uncertainty principle, even if the position of a particle is known at a given instant of time, its momentum is completely indeterminate. Therefore, the very concept of trajectory of a quantum particle loses its meaning and we cannot follow trajectories of the individual particles, the way we proposed to do in classical mechanics. Therefore, there is no way to distinguish between Identical Particles in quantum mechanics. Clearly, Identical Particles are inevitably indistinguishable in quantum mechanics. This indistinguishability of identical quantum particles has some interesting consequences, which we are going to discuss here. It turns out that, due to indistinguishability, it is possible to deduce some important properties of the wave functions of a system of N Identical Particles without solving (10.1.8). For this purpose, let us define the so-called permutation operator ˆ P jk , which interchanges the particles that are at the positions ~ r j and ~ r k . Its action on the wave function of the system will then read ˆ P jk φ ( ~ r 1 , ~ r 2 , ..., ~ r j , ..., ~ r k | {z } , ..., ~ r N ) = φ ( ~ r 1 , ~ r 2 , ..., ~ r k , ..., ~ r j | {z } , ..., ~ r N ). (10.3.1) Since the particles are indistinguishable, no experiment can determine which of the particles of the system is at ~ r j and which one is at ~ r k . The probability density, therefore, should remain unchanged, that is, 326 Fundamentals of Quantum Mechanics |φ ( ~ r 1 , ~ r 2 , ..., ~ r j , ..., ~ r k , ..., ~ r N )| 2 = |φ ( ~ r 1 , ~ r 2 , ..., ~ r k , ..., ~ r j , ..., ~ r N )| 2 .

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