Indistinguishable Particles

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Indistinguishable (or identical ) particles in physics are characterized by the fact that they cannot be distinguished from one another in any way on the basis of certain properties that are unaffected by their respective state . In this sense, all fundamental particles of the same kind are indistinguishable (e.g. electrons , photons , quarks ). The indistinguishability also applies to all systems composed of them (e.g. protons , neutrons , atomic nuclei , atoms , molecules ), provided they are in the same state.

The impossibility of differentiating between several identical particles means that the assignment of serial numbers has no effect on experimental results. In scattering experiments it would lead to incorrect predictions. The indistinguishability of identical particles thus contradicts the principle formulated by Gottfried Wilhelm Leibniz in 1663 , according to which there cannot be two things in the world that do not differ in anything.

The indistinguishability of the fundamental particles has an impact on the possibilities of building systems composed of them. It thus contributes to the understanding of the behavior of matter .

Illustration in the thought experiment

The indistinguishability of the same particles causes effects that are incomprehensible to classical physics (and common sense). A thought experiment is intended to illustrate this: two particles with the same amount of momentum fly towards each other 10,000 times in succession , one from precisely north and one from precisely south. The flight directions of these "north" and "south" particles are fixed, but the distance between their flight paths is not. So all the gaps occur and everything is called a "push". If the distance is small enough, the two particles exert forces on each other and thereby change their flight direction (in opposite directions). Since all distances occur with a certain probability distribution, all deflection angles also occur with a certain (different) probability distribution. The number of times one of the particles (regardless of which one) is randomly deflected by exactly 90 ° and then flies eastwards is counted. Then the other particle always flies away in the opposite direction, i.e. to the west. This is only about these final states.

Different particles

For each pair of north and south particles there are two distinguishable final states, the frequencies of which are added together: (1) the north particle flies to the east after the collision and the south particle to the west, or (2) vice versa.

If north and south particles differ (e.g. by their color), one can count how many of the particles originally coming from the north fly eastwards, e.g. B. 16. For reasons of symmetry (because at 90 ° the deflection angle for north and south particles is the same), the same number of south particles will certainly be deflected there. This means that a total of 32 particles arrive on the east side, just like (for reasons of symmetry) on the west side.

Indistinguishable Particles

But if the particles are indistinguishable (in the sense of the complete indistinguishability of which we are talking here), does it then remain with the total of 32 observed particles?

The statistical effect : In the case of indistinguishable particles, the two final states just mentioned no longer have a physically detectable distinguishing feature. Then, in quantum physical counting of the states, there are not two different states at all, but only one. The probability that this one state will be hit with the random distribution of the deflection angles is therefore (with the same form of the forces) only half as great as the probability for the two states of the distinguishable particles together. Accordingly, instead of 32, only 16 particles arrive in the east. (The other 16, despite the same type of shooting at each other and the same form of forces, arrived at different deflection angles and increased the count rate there!) Because of the indistinguishability of the particles, the question of "which" of the particles it is, i.e. how many, is forbidden of them come from the north or south. This way of counting the possible states has proven to be the only correct one in the collision experiments with particles and in statistical physics .

The dynamic effect : In the scattering experiment shown here, there is another special feature of the identical particles. Then (with the same form of the forces) - depending on the particle class boson or fermion of the two collision partners - actually either 64 (for bosons) or none at all (for fermions) fly away to the east, instead of the just calculated number of 16 particles. This has been verified in appropriate experiments. It corresponds exactly to the prediction of quantum mechanics that the wave function (or the state vector ) must have a special form for indistinguishable particles . There are always exactly two particles with opposite flight directions. In the initial state they fly towards each other in a north-south direction and in the end state in an east-west direction away from each other. But in the initial state each of the two particles comes from north and south with the same probability amplitude , in the final state each of the two particles flies with the same amplitude to east and west. Thus it is already conceptually excluded to want to ascribe a certain origin or a certain path to that of the two indistinguishable particles that was observed. If the particles or their coordinates are numbered consecutively, as is usual in the representation by a wave function, this wave function must therefore assume a form in which each number occurs together with each of the single-particle states. This results in interference between the two probability amplitudes with which each of the two individual final states (north particles to the east or south particles to the east) would occur. With 90 ° deflection, both amplitudes are equal and must be added for bosons ( constructive interference, therefore doubling the number of observed particles from 32 to 64), for fermions subtracted ( destructive interference, therefore result 0). If the intensity is also recorded at other scattering angles, minima and maxima alternate as a function of the angle and show a pronounced interference pattern.

Significance and history

The special role played by the indistinguishability of identical particles was discovered in 1926 by Paul Dirac and Werner Heisenberg when they studied atoms with several electrons with the help of quantum mechanics, which was new at the time, which is where the older quantum theories had failed. Dirac and Heisenberg established the rule that it leaves the state of the atom unchanged if two electrons in it exchange their orbitals . According to the quantum mechanical formalism ( wave function or state vector ) it becomes impossible to identify a certain electron among several electrons and to follow its path. This applies not only to the electrons in a particular atom, but in general, e.g. B. also for free-flying electrons in scattering experiments as described above. In a system of several electrons, the total number of electrons and which states are occupied by them can be identified, but not "which" of the electrons is in a certain state. In the first textbook on quantum mechanics from 1928, Hermann Weyl put it this way: “In principle, electrons cannot be required to provide evidence of their alibi”. At the same time, it was discovered in molecules made up of two identical atoms that this type of indistinguishability also applies to whole atoms, i.e. also applies to two identical atomic nuclei and thus applies to all building blocks of matter.

In everyday life, an equally perfect indistinguishability is not found in real things, but only in abstract ones, such as the equality of both sides of a mathematical equation such as  : The result, a “one”, can no longer be determined by halving it one two or by adding the two fractions. Such a fundamental indistinguishability does not occur in everyday life with material things. On the other hand, according to the formalism of quantum mechanics, it can also be assigned to all composite systems: the atoms, molecules, etc. up to the macroscopic bodies, if they are only in exactly the same overall state (based on their center of gravity and their orientation in space). The generally accepted unmistakable individuality of an object in everyday life is therefore based exclusively on the exact quantum mechanical state in which the object is. On the other hand, it is not a quality that one can permanently ascribe to the matter of which the object is made. From a practical point of view, the absolute certainty with which an object can be identified (e.g. on the lost property office) is based solely on the practically negligible probability that another everyday object is not only made up of the same components, but also in the same quantum mechanical part State.

In philosophy, it has long been considered impossible, especially since Leibniz , that there could be copies of a thing that literally cannot be distinguished from the thing ( Principium identitatis indiscernibilium - pii ). There was also a formal logical proof for this proposition. But after exactly this phenomenon was determined in the electrons, this theorem and its proof is heavily controversial. Weyl z. B. continued the quoted sentence as follows: “In principle, electrons cannot be required to prove their alibi. In modern quantum theory, Leibniz's principle of coincidentia indiscernibilium prevails . “For an overview of the ongoing discussion see and.

A (not perfect, but practical) indistinguishability plays a role in data modeling . In a database , all objects with a quantity attribute are considered indistinguishable. Example: 73 pieces (quantity attribute) identical mineral water bottles (of a certain type, size, etc.) in a store's inventory.

Indistinguishability in statistical physics

In statistical physics , indistinguishability is an important point when counting the states of a system . A system of indistinguishable particles has a limited state space compared to a system of the same number of distinguishable particles (see thought experiment above). Apparently different states in which only particles were exchanged for each other are in reality always one and the same state. Since there are possibilities to exchange particles for each other, the indistinguishability leads to a reduction of the partition function by a factor . This counting rule brings the theoretical formula of Sackur and Tetrode for the entropy of an ideal gas into harmony with the measured values ​​and thus also solves, for example, Gibbs' paradox .

Indistinguishability in quantum mechanics

Formulation in wave mechanics

In the wave mechanical formulation of quantum mechanics , every well-defined state of the entire N-particle system is described by a wave function that depends on as many sets of coordinates as there are particles in the system, namely N pieces. The set of coordinates for the -th particle contains all of its coordinates (for space and possibly spin, charge, etc.). The exchange of two sets of coordinates , expressed by the operator , corresponds to the exchange of the two sets of coordinates :

.

In the case of identical particles, as in statistical physics, only the same physical state can result from the exchange . For the wave function this means that it is at most multiplied by a phase factor. If one further demands that a repetition of the interchanging leaves not only the state but also the wave function itself unchanged, the phase factor can only be:

.

According to all observations applies: If in a arbitrarily assembled multi-particle system swaps two particles, in the case of two identical remains bosons the wave function unchanged while at identical fermions the sign changes. The spin statistics theorem provides a theoretical justification . The wave functions that always change their sign when exchanging any two particles are called totally antisymmetric , those that always remain the same are called totally symmetric .

The simplest base states for modeling an overall wave function of a system of particles are constructed with the base wave functions of the individual particles. In the case of distinguishable particles, one forms the product of single-particle functions. Such a basic state is called a configuration . In order to take into account the indistinguishability of the particles, this product must then be symmetrized in the case of bosons or antisymmetrized in the case of fermions (and normalized to 1 ). For a system of identical particles we get , where the two are single- particle functions with which we constructed.

The completely antisymmetrized product of single-particle functions is called the Slater determinant . In the case that, contrary to the Pauli principle, a single-particle function appears several times in it, or that one of the functions is a linear combination of the others, the Slater determinant is always zero, i.e. H. no possible total wave function. Therefore, the requirement of antisymmetry provides a deeper justification for the Pauli principle with all its significant consequences. Like every determinant, the Slater determinant retains its value if, instead of the one-particle functions used in it, one uses linear combinations thereof, which are linearly independent and orthonormal . Therefore, in a multi-particle system with a totally antisymmetric wave function, it is not clear, not even for the simplest basic states in the form of pure configurations, which individual single-particle states are each occupied by one particle. It is only certain in which ( -dimensional) subspace of the one-particle state space the occupied states are located. However, this information is also lost with linear combinations of several Slater determinants ( configuration mixture ), as is necessary for a more precise description of the particle states of real physical systems.

The Pauli principle does not apply in boson systems . Therefore, if they do not repel each other, bosons are preferably in the same, energetically lowest possible state at low temperatures, which leads to a special system state, the Bose-Einstein condensate .

Formulation in the 2nd quantization

In the second quantization , the indistinguishability of identical particles is already fully taken into account in the basic concepts of the formalism. The state vector of a particle with a wave function is formed by applying the corresponding creation operator to the state vector for the vacuum ,,: . If the system is a further particles with wave function contained is to this state vector with only one particle, the corresponding creation operator applied: . The letter chosen for the operator always shows the type of particle, its index the exact state of the single particle. If the last-mentioned particle is to be of the same type as the first-mentioned, its creation operator has to be denoted with instead of . In order that the generated two-particle state is the same, even if one occupies both single-particle states in the formula in exchanged order, exchangeability is required for the generation operators:

The plus sign applies when generating a boson and the minus sign when generating a fermion. The Pauli principle for identical fermions then follows e.g. B. immediately by choosing in the commutation relation, because then the equation results which is only satisfied for the zero operator . For more information, in particular on the annihilation operators , see under second quantization . One can get back to the wave-mechanical formulation by asking - first for a particle in the state - about the amplitude with which the state localized at the location occurs in this state. This amplitude is the wave function and is given by the scalar product

Correspondingly, one calculates the two-particle wave function of the state through the scalar product with the two-particle state . The result is and corresponds exactly to the wave function, which is formed as described above by (anti-) symmetrization of the product of the single-particle wave functions. (To simplify the formulas, here again denotes a complete set of coordinates, and normalization factors have been omitted.) It should be noted that here, independently of the two particles, are introduced and treated as two possible values ​​of their set of coordinates for the type of particle under consideration. There is no indication whatsoever for a more detailed assignment of a set of coordinates to one of the particles. In particular, it is not necessary to ascribe to the particles as the "one" and the "other" or the "first" and the "second" a linguistic distinctiveness which they do not physically possess. The calculated amplitude is, expressed in words, the probability amplitude with which 1 particle with the values and at the same time 1 particle with the values occur in the state.

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Remarks

  1. Details: North and South particles should have opposite momentum so that their center of gravity is at rest. To keep the presentation simple, the statistical fluctuations that would occur in the real experiment are not taken into account in the numerical examples. Likewise, in the real experiment, a small angular range around the assumed exactly 90 ° would have to be counted. The conclusions remain unaffected.
  2. If the particles have spin , they must have their spins aligned parallel to one another. Otherwise one could distinguish them by the spin position. If one takes into account the finite angular resolution of a real experiment, it is not quite 64 or not exactly 0.
  3. The formulation of quantum mechanics with creation and annihilation operators ( second quantization ) avoids this, there identical particles are not even assigned consecutive numbers.
  4. In the example the probability amplitude (for different particles) originally has the value 4 (because 4² = 16). For bosons it is 4 + 4 = 8, the intensity thus 8² = 64, for fermions 4-4 = 0, intensity zero.
  5. It took about 30 years, however, until the contradiction was recognized in the field of physics, and even longer until philosophy began to deal with it.

Individual evidence

  1. ^ GR Plattner, I. Sick: Coherence, interference and the Pauli principle: Coulomb scattering of carbon from carbon , European Journal of Physics, Vol. 2 (1981), pp. 109-113. Shown in detail in: Jörn Bleck-Neuhaus: Elementary Particles. Modern physics from the atoms to the standard model , Springer-Verlag (Heidelberg), 2010, chap. 5.7, ISBN = 978-3-540-85299-5
  2. ^ Hermann Weyl: group theory and quantum mechanics ; Leipzig 1928, p. 188
  3. See e.g. BFA Muller, S. Saunders: Discerning Fermions , in: Brit. Journ. Philos. Science, Vol. 59 (2008), pp. 499–54 ( online ; PDF; 359 kB)
  4. The identity of Indiscernibles entry in Edward N. Zalta (ed.): Stanford Encyclopedia of Philosophy .Template: SEP / Maintenance / Parameter 1 and neither parameter 2 nor parameter 3
  5. ^ Identity and Individuality in Quantum Theory Entry in Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .Template: SEP / Maintenance / Parameter 1 and neither parameter 2 nor parameter 3

See also