Identical particles

Indistinguishable ( or identical ) particles in physics are characterized in that they can be distinguished from each other based on specific, prospective on the condition of properties in any way. All fundamental particles of the same type are indistinguishable in this sense (eg, electrons, photons, quarks ). The indistinguishability is also true for any resulting composite systems ( eg protons, neutrons, atomic nuclei, atoms, molecules, etc. ), provided they are located, relative to their center of gravity, in the same state. The impossibility of any distinction of several identical particles with the result that even the assignment of serial numbers (or other symbolic identifications ) is not permitted; it would lead, for example, in scattering experiments to incorrect predictions. Thus, the indistinguishability of identical particles refuted the 1663 formulated by Gottfried Wilhelm Leibniz logical principle according to which there can be no two things in the world, which differ in nothing. The indistinguishability of fundamental particles has to make a strong impact on the possibilities of composite systems from them. For example, it affects the structure and properties not only of atoms, but also of all macroscopic matter formed therefrom.

• 4.1 Formulation of wave mechanics
• 4.2 Formulation in the second quantization

Illustrate the thought experiment

The indistinguishability of identical particles causes effects that are for classical physics ( and common sense ) incomprehensible. A thought experiment it is intended to illustrate: 1000 times in succession two particles flying towards each other, one from the north and one from the south. Collide, forces thus exert on each other, and thereby change their direction of flight. Counting how often a particle is randomly deflected by exactly 90 °, and then flies to the east. Then the other particle always flies off in the opposite direction, ie to the west.

Different particles

For each pair of particles, there are two final states: (1) the northern flying particles after the collision to the east and the south- west particles or (2) vice versa. Are there differences between North and Südteilchen (eg color ), so you can count how much of the originally coming from the north particles flying to the East, for example, 16 [Note 1]. For reasons of symmetry (because at 90 ° angle of deflection for North and South particles is the same ) are also distracted sure the same number of South particles there. So come on the east side to a total of 32 particles, such as ( for reasons of symmetry ) on the west side also.

Indistinguishable particles

If the particles are indistinguishable but (in the sense of utter indistinguishability, from here the speech ), then it remains in the 32 observed particles on each side?

The statistical effect: For indistinguishable particles, the two just mentioned final states now have no physically detectable differentiator more. Then there are quantum physical counting no longer two different states, but only one. The probability that, in the collision of a random-controlled state is made is, therefore, only half as large as the probability of two distinct states of the particles together (with the same type of forces). Accordingly, come instead of 32 so only 16 particles to the east. It is inappropriate due to their indistinguishability the question of how many of them come from the north or south. This counting of possible states has proved effective in the collision experiments with particles and in statistical physics as the only true.

The dynamic effect: In the illustrated scattering experiment still occurs, add a further characteristic of the identical particles. Then fly (with the same type of forces) - depending on the fraction of particles boson or fermion of the two collision partners - in fact either 64 (for bosons ) or none (for fermions ) to the east away, instead of just calculated number of 16 particles [NB. . 2] This has been verified in corresponding experiments. It corresponds exactly to the prediction of quantum mechanics that for indistinguishable particles, the wave function ( or state vector ) must have a special form. Although it always come exactly two particles having opposite directions of flight. In the initial state, they fly in a north-south direction to each other and in the final state in east-west direction away from one another. But in the initial state is either of the two particles with equal probability amplitude from the north and from the south, in the final state of each of the two particles fly with the same amplitude to east and to west. Thus, it is already excluded conceptual, whichever of the two indistinguishable particles, which was observed to try to ascribe a specific origin or a specific way. If, as usual in the representation by a wave function, the particles and their coordinates are numbered, so this wave function [note must take a form in which each number with each of the single-particle states occur together. 3] This results in interference with the probability amplitudes, with each of the two individual final states (North- particles to the east or south - east particles ) would occur. [Note 4] At 90 ° deflection both amplitudes are equal and must be added for bosons ( constructive interference, thus doubling the number of observed particles from 32 to 64 ), with fermions subtract ( destructive interference, therefore result is 0 ). Taking the intensity in other scattering angles, switch off in response to the minimum and maximum angle, and show a strong interference pattern.

Meaning and History

The special role played by the indistinguishability of identical particles, was discovered in 1926 by Paul Dirac and Werner Heisenberg, as they the atoms with several electrons studied with the help of the then new quantum mechanics, what the older quantum theory had failed. Dirac and Heisenberg presented to the rule that it leaves the state of the atom unchanged when two electrons mutually interchanged in their orbitals. The quantum mechanical formalism ( wave function or state vector ) according to it is thus impossible under several electrons to identify and pursue his way a particular. This applies as described above, not only for the electrons in a given atom, but in general, for example, for free-flying electrons in scattering experiments. In a system from a plurality of electrons " that " of the electrons can be the total number of electrons and which identify states of them are filled, but not a certain condition holds. In the first textbook on quantum mechanics from 1928 Hermann Weyl put it, "From electrons can in principle not require proof of her alibi ." At the same time has been detected in two molecules of the same atoms, that this type of indistinguishability also applies to all the atoms, thus plays the same two nuclei, and thus applies to all building blocks of the material.

In everyday life, we find an equally perfect indistinguishability not real things, but only in the abstract, as in the case of equality of both sides of a mathematical equation as: At the conclusion of a "one" can not be determined whether it by halving is a two originated or by the addition of the two fractions. Such a principle indistinguishability does not occur with material things in everyday life. On the other hand, it is also attributed to all composite systems based on the formalism of quantum mechanics: atoms, molecules, etc., up to macroscopic bodies, only if they are in the exact same condition overall (based on their focus and their orientation in space). The generally accepted sense of individuality of an object of everyday life is therefore based exclusively on quantum mechanical state in which the goods are located exactly. In contrast, it is not a quality that can be permanently the matter of which is the object attribute, itself. The practical point of absolute certainty with which one can identify an object ( eg on the Fund Office) based solely on the practically negligible probability that an " other " everyday object is constructed not only from the same ingredients, but also still in same quantum mechanical state.

In philosophy, it was considered from time immemorial, and especially since Leibniz impossible that there could be a thing additional copies, resulting in literally leave nothing different from the thing ( Principle identitatis indiscernibilium - pii ). For this set there was also a formal logical proof. But after exactly this phenomenon was observed on the electron, this sentence and his evidence is heavily disputed. [Note 5] Weyl, for example, led the quoted sentence so on ". electrons from one can not demand proof of their alibi So basically continues in the modern quantum theory of Leibniz's principle of the coincidentia indiscernibilium through. " For an overview of the ongoing discussion and see.

Indistinguishability in statistical physics

In statistical physics, the indistinguishability is an important point in the counting of states of a system. A system of indistinguishable particles in comparison to a system consisting of an equal number of distinguishable particles a restricted state space (see above thought experiment ). Seemingly different states in which only particles were interchanged, are in reality always one and the same state. Since there are ways to exchange particles against each other, the indistinguishability leads to a reduction of the partition function by a factor. This Zählvorschrift brings the theoretical formula of Sackur and Tetrode for the entropy of an ideal gas in accordance with the measured values ​​, replacing for example, the Gibbs paradox.

Indistinguishability in quantum mechanics

Formulation of wave mechanics

In the mechanical shaft of quantum mechanics, each well-defined state of the entire N- particle system will be described by a wave function, which is dependent on sets of coordinates as much as there are particles in the system. The set of coordinates for the -th particle contains all its coordinates ( for space and, where appropriate, spin, charge, etc). The interchanging two particles, expressed by the operator corresponding to the interchange of the two sets of coordinates.

For identical particles we have, as in statistical physics, that of the interchange only the same physical state can emerge. For the wave function means that it is multiplied at most by a phase factor. If you for further requires that a repetition of the exchange not only the state but also the wave function can itself unchanged, the phase factor can only be:

All observations, the following applies: If one exchanges two particles in an arbitrarily composite many-body system, in the case of two identical bosons, the wave function remains unchanged, while the sign changes for identical fermions. A theoretical justification provides the spin- statistics theorem. The wave functions, which always change sign at the interchange of any two particles, hot totally antisymmetric, the ones that always remain the same, totally symmetrical.

The simplest basis states for the modeling of a total wave function of a system of N particles is constructed with the basis wave functions of the individual particles. In the case of distinguishable particles one simply the product of N single particle. Such a condition is referred to as base configuration. To account for the indistinguishability of the particles, this product will still need symmetrized or antisymmetrized in the case of fermions (and normalized to 1 ) in the case of bosons. For a system of identical particles is obtained, wherein the two single particle are constructed with.

The fully antisymmetrized product of N single particle is called Slater determinant. In the event that, contrary to the Pauli principle, a Einteilchenfunktion repeatedly appears in it or that one of the functions is a linear combination of the other, the Slater determinant is always zero, ie no possible total wave function. Hence the requirement of antisymmetry provides a deeper foundation for the Pauli principle with all its important consequences. Like any determinant reserves the Slater determinant their value when single particle instead of N is used N linear combinations of uses thereof which are linearly independent and orthonormal. Therefore, not even is not fixed in a Mehrteilchensystem with totally antisymmetric wave function for the simplest of basis states in the form of pure configurations, which are the N single particle states, which are occupied with one particle. The only certainty in which ( N-dimensional ) subspace of the one-particle state space are the occupied states. However, this information is lost in the case of linear combinations of several Slater determinants ( configuration mixing ), as they are necessary for a more precise description of the N- particle states of real physical systems.

In bosonic systems, the Pauli exclusion principle does not apply. Therefore sit bosons, if they do not repel, at low temperatures, preferably in the same lowest energy state, resulting in a particular system condition, the Bose -Einstein condensate.

Formulation in the second quantization

In the second quantization of the indistinguishability of identical particles is taken into account already in the basic concepts of the formalism in a perfect way. The state vector of a particle with wave function is formed by use of the corresponding generating operator on the state vector for the vacuum. The system should comprise a further particles wave function is applied to this state vector with a single particle of the corresponding production operator. Always be index indicates the selected operator for the letter to the type of particle, the exact respective single particle. If the last-mentioned particles of the same type as the first mentioned, his creation operator is instead to call with. Thus, the two-particle state produced is then the same, even if you filled in the formula above, both single-particle states in reversed order, is required for the creation operators commutativity:

Here, the plus sign applies if a boson produced, and the minus sign when a fermion generated. The Pauli principle for identical fermions then follows immediately, for example, by choosing in the commutation relation, because then yields the equation that is satisfied only for the zero operator. Others, in particular the annihilation operators, see Second quantization. For the wave-mechanical formulation passes back one by one - first for a particle in the state - asks for the amplitude with which the localized state occurs at the site in this state. This amplitude is the wave function and is given by the dot product

Accordingly, we calculated the Zweiteilchenwellenfunktion the state by the Skalarprodunkt with the two-particle state. The result is, and agrees exactly with the wave function, which is formed as described above by ( anti-) symmetrization of the product of the wavefunctions. ( To simplify the formulas referred to here again a complete set of coordinates, and there are normalization factors omitted. ) It should be noted that here are introduced and treated independently of the two particles, as two possible values ​​for the considered particle coordinate their movement. There is no indication of a more detailed assignment of a set of coordinates to one of the particles. In particular, it is not necessary, the particles than the " a " and the "other" or the "first" and "second" attribute a linguistic distinctness, they do not own physically. The calculated amplitude is expressed in words, the probability amplitude in state 1 occurs with the particles with the values ​​and at the same time 1 particles with the values ​​.