Identical Particles in Quantum Mechanics

12 Key excerpts on "Identical Particles in Quantum Mechanics"

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  An Introduction to Quantum Physics

  A First Course for Physicists, Chemists, Materials Scientists, and Engineers

  • Stefanos Trachanas, Manolis Antonoyiannakis, Leonidas Tsetseris, Manolis Antonoyiannakis, Leonidas Tsetseris(Authors)
  • 2017(Publication Date)
  • Wiley-VCH

    (Publisher)

  Chapter 11 Identical Particles and the Pauli Principle

  11.1 Introduction

  We will now discuss the second foundational premise—the first one was the uncertainty principle—of quantum theory: the Pauli principle. This will allow us to pursue, in the following chapters, our ultimate goal, which is to understand the structure of matter from first principles. To understand the structure of atoms—and how their basic properties are mapped in a periodic table—and proceed from there to construct the quantum theory of the chemical bond and extend it to crystalline solids. Let us begin.

  11.2 The Principle of Indistinguishability of Identical Particles in Quantum Mechanics

  The concept of identical particles is surely the same in both classical and quantum mechanics. We call identical all those particles that share the exact same physical properties: mass, charge, spin, baryon or lepton number, and any other quantum number required for their complete identification. Put in a different way, all particles of the same species are identical: all electrons, all protons, all photons, and so on.

  But there is one fundamental difference between classical and quantum mechanics when it comes to distinguishing identical particles of a physical system. In classical mechanics, we can always tell one identical particle from another because of the uniqueness of their orbits that allows us to know at any moment which one is particle , which one is particle , and so on. In contrast, in quantum mechanics, it is impossible to distinguish between particles of the same physical system (e.g., electrons in an atom), since these are described by overlapping wavefunctions that allow the particles to be found at the same point in space, which renders their identification impossible. This fundamental difference between classical and quantum mechanics is demonstrated in Figure 11.1

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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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  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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  Lectures On Quantum Mechanics

  • Gordon Baym(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 18 IDENTICAL PARTICLES

  In following the motion of a collection of identical classical particles – say billiard balls – it is always possible for us to keep track of individual particles. For example, to find out where a given particle, “number 4,” is at any later time we merely have to paint a “4” on that particle (carefully enough so that the paint won’t affect its dynamics) and then notice at the later time which particle is labeled “4.” It is not, however, possible to do this when studying a system of identical quantum mechanical particles, such as electrons. The problem is that quantum mechanical particles are too small for one to attach physical labels; they don’t have enough degrees of freedom to enable one to mark each particle differently.

  Now one might say that it really isn’t necessary actually to label the particles in order to keep track of individuals; all that’s necessary is to take a movie of the particles and then trace out the trajectories of the individual particles. This works perfectly well with classical particles, but every time one observes a quantum mechanical system one disturbs the system in some uncontrollable fashion; if the wave functions of the individual particles overlap at all then it is not possible, in principle, to keep track of the individual trajectories of the particles without seriously affecting their motion.

  Identical quantum mechanical particles are completely indistinguishable from one another. If we have a state with one particle localized about the origin and a second identical particle localized about some other position r, then the state with the second particle localized about the origin and the first about r is not a different state from the first, since there is no way of distinguishing the two situations. We would now like to study the consequences of this fundamental indistinguishability of identical quantum mechanical particles.

  Suppose that |Ψ〉 is a state of n identical particles. Then the wave function of the state is

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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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  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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  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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  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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  Quantum Reality

  Theory and Philosophy

  • Jonathan Allday(Author)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  In the classical case, we can always distinguish objects no matter how careful we are to make them identical. No manufacturing process can produce a sequence of objects that survive microscopic examination without finding some distinguishing feature. At the quantum level, two electrons, for example, are identical in every measurable aspect, so classical expectations do not apply. Incidentally, this profound identity of particles arises from their underlying nature as aspects of a quantum field, which we will explore in Chapter 32. Given what we have seen with experiments on identical atoms, we might expect that our multiparticle state | Φ 〉 will be different if the two particles that it represents are identical. Firstly, the basis can no longer be of the form { | n i 〉 | e j 〉 } as that implies the ability to tell one from the other. Instead, we have to use a basis { | i, j 〉 } which is not composed of products of single- particle basis states. Secondly, the identity of the particles places an extra constraint on the amplitudes. In the expansion: | Φ 〉 = C 11 | 1, 1 〉 + C 12 | 1, 2 〉 + C 13 | 1, 3 〉 ⋯ + C 21 | 2, 1 〉 + C 31 | 3, 1 〉 + ⋯ + C 55 | 5, 5 〉 C 13 must be linked with C 31, for example. After all, the particles’ identity means that we are unable to distinguish between {particle 1 in box 1 with particle 2 in box 3} and {particle 2 in box 1 with particle 1 in box 3}. The probability can only be {a particle in box 1 AND another in box 3}. As a result (and generalizing): | C i j | 2 = C i j C i j * = C j i C j i * = | C j i | 2 with the implication that either C i j = C j i ⇒ C i j * = C j i * or C i j = − C j i ⇒ C i j * = − C j i * giving us two possible. expansions: | Φ 〉 S = C 11 | 1, 1 〉 + C 12 | 1, 2 〉 + C 13 | 1, 3 〉 ⋯ + C 12 | 2, 1 〉 + C 13 | 3, 1 〉 + ⋯ + C 55 | 5, 5 〉 or | Φ 〉 A = C 11 | 1, 1 〉 + C 12 | 1, 2 〉 + C 13 | 1[--=PLGO-SEPARAT. OR=--], 3 〉 ⋯ − C 12 | 2, 1 〉 − C 13 | 3, 1 〉 + ⋯ + C 55 | 5, 5 〉 But this is slightly too hasty in the second case

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  Quantum Worlds

  Perspectives on the Ontology of Quantum Mechanics

  • Olimpia Lombardi, Sebastian Fortin, Cristian López, Federico Holik(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  The proper understanding of the relations between these concepts, and the kind of view that results from them in the context of quantum particles, is the topic of this chapter. These issues were also central to another contributor to the development of quantum theory. In a classical passage, in which the issues of identity and individu- ality were prominent, Hermann Weyl points out: . . . the possibility that one of the identical twins Mike and Ike is in the quantum state E 1 and the other in the quantum state E 2 does not include two differentiable cases which are permuted on permuting Mike and Ike; it is impossible for either of these individuals to retain his identity so that one of them will always be able to say ‘I’m Mike’ and the other ‘I’m Ike’. Even in principle one cannot demand an alibi of an electron! (Weyl 1950: 241) The questions of discernibility and of an “alibi” of a quantum particle are clearly posed. Once quantum particles, such as electrons, are in an entangled state, it cannot be determined which particle is in which state. In other words, it cannot be settled which particle is which. There is nothing – no property, no special ingredi- ent – that could act as an alibi to discern electrons. In this respect, it is their indiscernibility rather than their identity that should take center stage. Differently from what Schrödinger suggests, perhaps identity need not lose its meaning, provided that indiscernible things can still be numerically distinct (or identical). Making Sense of Nonindividuals in Quantum Mechanics 187 As will becomes clear, to articulate this proposal it is required that identity and indiscernibility be distinguished. In classical logic and standard mathematics, identity is formulated in terms of indiscernibility. So, in order to keep one and change the other, one needs to resist this identification and clearly separate the two notions.

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  Interpreting Bodies

  Classical and Quantum Objects in Modern Physics

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

    (Publisher)

  However, to get the correct results in quantum statistics possibilities 3 and 4 must be counted as one and the same. But this seems to run counter to the whole point of regarding the particles as individuals and labeling them. From the point of view of the statistics, the labels are otiose, which sug-gests that the particles should be regarded as nonindividuals, in some sense. More formally, the above result can be expressed in the form of the indistinguishability postulate: P Q P = Q ∀ ∀ Q ∀ where the represent physically realizable states, the P s are particle (label) permutation operators, and the Q s are operators representing physical observables. Now we have to be careful how we interpret this principle. The fore-going argument depends on understanding it as imposing restrictions on the set of possible observables, such that particle permutation operators cannot be so regarded. However, if we interpret it as a restriction on the set of states, then it says that nonsymmetric states, such as possibilities 3 and 4, are rendered inaccessible to the particles. This interpretation is consistent with the metaphysical view of particles as individuals; quan-tum statistics is recovered by regarding such states as possible but never actually realized. 12 Thus the formalism can be taken to support two very different meta-physical packages, one in which the particles are regarded as “nonin-dividuals” in some sense and another in which they are regarded as (philosophically) classical individuals for which certain sets of states are rendered inaccessible. (As we shall shortly see, this latter view needs to be supplemented with a particular view of the relations holding between the particles when quantum entanglement is considered.) 4.

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