Indistinguishable particles

5132: 4388: 6502: 5127:{\displaystyle {\begin{aligned}\Psi _{n_{1}n_{2}\cdots n_{N}}^{(S)}(x_{1},x_{2},\ldots ,x_{N})&\equiv \langle x_{1}x_{2}\cdots x_{N};S|n_{1}n_{2}\cdots n_{N};S\rangle \\&={\sqrt {\frac {\prod _{j}n_{j}!}{N!}}}\sum _{p}\psi _{p(1)}(x_{1})\psi _{p(2)}(x_{2})\cdots \psi _{p(N)}(x_{N})\\\Psi _{n_{1}n_{2}\cdots n_{N}}^{(A)}(x_{1},x_{2},\ldots ,x_{N})&\equiv \langle x_{1}x_{2}\cdots x_{N};A|n_{1}n_{2}\cdots n_{N};A\rangle \\&={\frac {1}{\sqrt {N!}}}\sum _{p}\mathrm {sgn} (p)\psi _{p(1)}(x_{1})\psi _{p(2)}(x_{2})\cdots \psi _{p(N)}(x_{N})\end{aligned}}} 4373: 6026: 3979: 135: 5543: 27: 6497:{\displaystyle \Psi _{n_{1}\cdots n_{N}}^{(A)}(x_{1},\ldots ,x_{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\psi _{n_{1}}(x_{1})&\psi _{n_{1}}(x_{2})&\cdots &\psi _{n_{1}}(x_{N})\\\psi _{n_{2}}(x_{1})&\psi _{n_{2}}(x_{2})&\cdots &\psi _{n_{2}}(x_{N})\\\vdots &\vdots &\ddots &\vdots \\\psi _{n_{N}}(x_{1})&\psi _{n_{N}}(x_{2})&\cdots &\psi _{n_{N}}(x_{N})\\\end{matrix}}\right|} 628: 636: 4368:{\displaystyle {\begin{aligned}|x_{1}x_{2}\cdots x_{N};S\rangle &={\sqrt {\frac {\prod _{j}n_{j}!}{N!}}}\sum _{p}\left|x_{p(1)}\right\rangle \left|x_{p(2)}\right\rangle \cdots \left|x_{p(N)}\right\rangle \\|x_{1}x_{2}\cdots x_{N};A\rangle &={\frac {1}{\sqrt {N!}}}\sum _{p}\mathrm {sgn} (p)\left|x_{p(1)}\right\rangle \left|x_{p(2)}\right\rangle \cdots \left|x_{p(N)}\right\rangle \end{aligned}}} 5206: 2838: 2580: 3592: 5988: 5538:{\displaystyle {\begin{aligned}\Psi _{n_{1}\cdots n_{N}}^{(S)}(\cdots x_{i}\cdots x_{j}\cdots )=\Psi _{n_{1}\cdots n_{N}}^{(S)}(\cdots x_{j}\cdots x_{i}\cdots )\\\Psi _{n_{1}\cdots n_{N}}^{(A)}(\cdots x_{i}\cdots x_{j}\cdots )=-\Psi _{n_{1}\cdots n_{N}}^{(A)}(\cdots x_{j}\cdots x_{i}\cdots )\end{aligned}}} 8833:, then this homotopy class has countably many elements (i.e. a counterclockwise interchange by half a turn, a counterclockwise interchange by one and a half turns, two and a half turns, etc., a clockwise interchange by half a turn, etc.). In particular, a counterclockwise interchange by half a turn is 3178: 978:

Two states are physically equivalent only if they differ at most by a complex phase factor. For two indistinguishable particles, a state before the particle exchange must be physically equivalent to the state after the exchange, so these two states differ at most by a complex phase factor. This fact

613:

that give the probability of finding a particle at each position. As time passes, the wavefunctions tend to spread out and overlap. Once this happens, it becomes impossible to determine, in a subsequent measurement, which of the particle positions correspond to those measured earlier. The particles

601:

Even if the particles have equivalent physical properties, there remains a second method for distinguishing between particles, which is to track the trajectory of each particle. As long as the position of each particle can be measured with infinite precision (even when the particles collide), then

569:

arguments, which are sensitive to whether or not the objects being studied are identical. As a result, identical particles exhibit markedly different statistical behaviour from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibbs'

1979:

In other words, symmetric and antisymmetric states are essentially unchanged under the exchange of particle labels: they are only multiplied by a factor of +1 or −1, rather than being "rotated" somewhere else in the Hilbert space. This indicates that the particle labels have no physical

1423:

are the same, the antisymmetric expression gives zero, which cannot be a state vector since it cannot be normalized. In other words, more than one identical particle cannot occupy an antisymmetric state (one antisymmetric state can be occupied only by one particle). This is known as the

1404: 1242: 594:. If differences exist, it is possible to distinguish between the particles by measuring the relevant properties. However, as far as can be determined, microscopic particles of the same species have completely equivalent physical properties. For instance, every electron has the same 1605: 2649: 3755: 1987:

is Hermitian. As a result, it can be regarded as an observable of the system, which means that, in principle, a measurement can be performed to find out if a state is symmetric or antisymmetric. Furthermore, the equivalence of the particles indicates that the

2385: 3376: 5749: 1610:

There is actually an exception to this rule, which will be discussed later. On the other hand, it can be shown that the symmetric and antisymmetric states are in a sense special, by examining a particular symmetry of the multiple-particle states known as

5199:

The most important property of these wavefunctions is that exchanging any two of the coordinate variables changes the wavefunction by only a plus or minus sign. This is the manifestation of symmetry and antisymmetry in the wavefunction representation:

9042:, the universal covering space still contains infinitely many points which are physically indistinguishable from the original point, now generated by a counterclockwise rotation by one full turn. This generator, then, results in a multiplication by 7740:

The differences between the statistical behavior of fermions, bosons, and distinguishable particles can be illustrated using a system of two particles. The particles are designated A and B. Each particle can exist in two possible states, labelled

6997:

Two obvious irreducible subspaces are the one dimensional symmetric/bosonic subspace and anti-symmetric/fermionic subspace. There are however more types of irreducible subspaces. States associated with these other irreducible subspaces are called

5764: 2226:

is a constant of motion. If the quantum state is initially symmetric (antisymmetric), it will remain symmetric (antisymmetric) as the system evolves. Mathematically, this says that the state vector is confined to one of the two eigenspaces of

2162: 1704: 8993:

generated by making a counterclockwise half-turn interchange. Unlike the previous case, performing this interchange twice in a row does not recover the original state; so such an interchange can generically result in a multiplication by

7251: 2972: 8489: 8292: 6927: 482:

also behave in this way. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in

9092:

is not connected, so even if particle I and particle II are identical, they can still be distinguished via labels such as "the particle on the left" and "the particle on the right". There is no interchange symmetry here.

1072: 7474:

The discrepancy in the partition functions of distinguishable and indistinguishable particles was known as far back as the 19th century, before the advent of quantum mechanics. It leads to a difficulty known as the

4393: 1253: 1091: 6761: 1451: 2958:

term, because each single-particle state can appear only once in a fermionic state. Otherwise the sum would again be zero due to the antisymmetry, thus representing a physically impossible state. This is the

3968: 1974: 1873: 1444:

The importance of symmetric and antisymmetric states is ultimately based on empirical evidence. It appears to be a fact of nature that identical particles do not occupy states of a mixed symmetry, such as

5211: 7696: 8642:

To understand why particle statistics work the way that they do, note first that particles are point-localized excitations and that particles that are spacelike separated do not interact. In a flat

2833:{\displaystyle |n_{1}n_{2}\cdots n_{N};A\rangle ={\frac {1}{\sqrt {N!}}}\sum _{p}\operatorname {sgn} (p)\left|n_{p(1)}\right\rangle \left|n_{p(2)}\right\rangle \cdots \left|n_{p(N)}\right\rangle \ } 6005:! equivalent points in the integral space. Because it is usually more convenient to work with unrestricted integrals than restricted ones, the normalizing constant has been chosen to reflect this. 5194: 3882: 3984: 3361: 3265: 2231:, and is not allowed to range over the entire Hilbert space. Thus, that eigenspace might as well be treated as the actual Hilbert space of the system. This is the idea behind the definition of 3603: 3973:

Symmetric and antisymmetric multi-particle states can be constructed from continuous eigenstates in the same way as before. However, it is customary to use a different normalizing constant:

8406:

state is 0.33. Note that the probability of finding particles in the same state is relatively larger than in the distinguishable case. This demonstrates the tendency of bosons to "clump".

2575:{\displaystyle |n_{1}n_{2}\cdots n_{N};S\rangle ={\sqrt {\frac {\prod _{n}m_{n}!}{N!}}}\sum _{p}\left|n_{p(1)}\right\rangle \left|n_{p(2)}\right\rangle \cdots \left|n_{p(N)}\right\rangle } 7564: 3587:{\displaystyle P_{S/A}\left(n_{1},\ldots ,n_{N}\rightarrow m_{1},\ldots ,m_{N}\right)\equiv {\big |}\left\langle m_{1}\cdots m_{N};S/A\,|\,n_{1}\cdots n_{N};S/A\right\rangle {\big |}^{2}} 940: 887: 790: 734: 7466: 7397: 8932:, so its only effect is to multiply the phase by a square root of 1. If the root is +1, then the points have Bose statistics, and if the root is –1, the points have Fermi statistics. 5588: 2266:. The nature of symmetric states has important consequences for the statistical properties of systems composed of many identical bosons. These statistical properties are described as 6838: 8210: 8171: 8017: 7978: 7939: 7900: 8971: 7733:. This is known as the Pauli Exclusion Principle, and is responsible for much of chemistry, since the electrons in an atom (fermions) successively fill the many states within 836:

of the combined system from the individual spaces. This expression is valid for distinguishable particles, however, it is not appropriate for indistinguishable particles since

8829: 8785: 6797: 6686: 9086: 5983:{\displaystyle \int \!\int \!\cdots \!\int \;\left|\Psi _{n_{1}n_{2}\cdots n_{N}}^{(S/A)}(x_{1},x_{2},\ldots ,x_{N})\right|^{2}d^{3}\!x_{1}d^{3}\!x_{2}\cdots d^{3}\!x_{N}=1} 9128: 6965: 6560: 2956: 8866: 1998: 8545: 8517: 8404: 8376: 8348: 8320: 8129: 8101: 8073: 8045: 7854: 7826: 7795: 7767: 7310: 6653: 834: 1628: 2213: 1758: 666:

of the wavefunction.) For simplicity, consider a system composed of two particles that are not interacting with each other. Suppose that one particle is in the state

8616:

As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on

6992: 6587: 7413:

If the possibility of overlapping states is neglected, which is valid if the temperature is high, then the number of times each state is counted is approximately

7071: 582:

There are two methods for distinguishing between particles. The first method relies on differences in the intrinsic physical properties of the particles, such as

7410:

s occurs once in the sum, even though each of these permutations is describing the same multi-particle state. Thus, the number of states has been over-counted.

2919: 2876: 7860:.) After some time, the composite system will have an equal probability of occupying each of the states available to it. The particle states are then measured. 6862: 6627: 6607: 6530: 2896: 3173:{\displaystyle \langle n_{1}n_{2}\cdots n_{N};S|n_{1}n_{2}\cdots n_{N};S\rangle =1,\qquad \langle n_{1}n_{2}\cdots n_{N};A|n_{1}n_{2}\cdots n_{N};A\rangle =1.} 7856:

states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states. (This is discussed in the article on

7701:

which is perfectly extensive. However, the reason for this correction to the partition function remained obscure until the discovery of quantum mechanics

7402:

If the particles are identical, this equation is incorrect. Consider a state of the system, described by the single particle states . In the equation for

9051: 3288:

for another particle, and so forth. If the particles are bosons (fermions), the state after the measurement must remain symmetric (antisymmetric), i.e.

8412: 8215: 1724:. This symmetry may be described as the symmetry under the exchange of labels attached to the particles (i.e., to the single-particle Hilbert spaces). 6870: 609:. According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by 2243:

The choice of symmetry or antisymmetry is determined by the species of particle. For example, symmetric states must always be used when describing

985: 6864:

identical particles, and hence observables satisfying the equation above, we find that the following states (after normalization) are equivalent

421: 1399:{\displaystyle |n_{1},n_{2};A\rangle \equiv {\mbox{constant}}\times {\bigg (}|n_{1}\rangle |n_{2}\rangle -|n_{2}\rangle |n_{1}\rangle {\bigg )}} 1237:{\displaystyle |n_{1},n_{2};S\rangle \equiv {\mbox{constant}}\times {\bigg (}|n_{1}\rangle |n_{2}\rangle +|n_{2}\rangle |n_{1}\rangle {\bigg )}} 644: 1600:{\displaystyle |n_{1},n_{2};?\rangle ={\mbox{constant}}\times {\bigg (}|n_{1}\rangle |n_{2}\rangle +i|n_{2}\rangle |n_{1}\rangle {\bigg )}} 44: 91: 215: 6693: 63: 7062: 7019:

The indistinguishability of particles has a profound effect on their statistical properties. To illustrate this, consider a system of

8660:. If there is no overlap between the particles, so that they do not interact directly, then their locations must belong to the space 1879: 1778: 1622:, called the exchange operator. When it acts on a tensor product of two state vectors, it exchanges the values of the state vectors: 7717:

respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the

70: 454:

that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to,

7630: 8409:

If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state

9375: 77: 6844:

Two states are equivalent whenever their expectation values coincide for all observables. If we restrict to observables of

5143: 3901: 2281:, which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by 3822: 9361: 3801:

of values of the observable, not a single value as with discrete observables. For instance, if a particle is in a state |

3786:

So far, the discussion has included only discrete observables. It can be extended to continuous observables, such as the

3750:{\displaystyle \sum _{m_{1}\leq m_{2}\leq \dots \leq m_{N}}P_{S/A}(n_{1},\ldots ,n_{N}\rightarrow m_{1},\ldots ,m_{N})=1} 3294: 3198: 979:

suggests that a state for two indistinguishable (and non-interacting) particles is given by following two possibilities:

414: 59: 9395: 9331: 9218: 249: 5548:

The many-body wavefunction has the following significance: if the system is initially in a state with quantum numbers

643:

What follows is an example to make the above discussion concrete, using the formalism developed in the article on the

5758:! comes from our normalizing constant, which has been chosen so that, by analogy with single-particle wavefunctions, 110: 9400: 7061:. As the particles do not interact, the total energy of the system is the sum of the single-particle energies. The 7003: 257: 177: 7505: 892: 839: 742: 686: 6001:! times in the integral. In other words, the probability associated with each event is evenly distributed across 5744:{\displaystyle N!\;\left|\Psi _{n_{1}n_{2}\cdots n_{N}}^{(S/A)}(x_{1},x_{2},\ldots ,x_{N})\right|^{2}\;d^{3N}\!x} 1989: 7709:

There are important differences between the statistical behavior of bosons and fermions, which are described by

2329:. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin. 654:

denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the

7423: 7321: 407: 48: 5559:, and a position measurement is performed, the probability of finding particles in infinitesimal volumes near 7729:. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the 484: 9390: 8621: 7722: 7710: 6804: 2267: 182: 8176: 8137: 7983: 7944: 7905: 7866: 8938: 167: 84: 8617: 7714: 2282: 187: 8805: 8761: 6769: 6658: 9060: 2960: 2278: 1425: 511: 8650:, at any given time, the configuration of two identical particles can be specified as an element of 6589:

acts on this space by permuting the entries. By definition the expectation values for an observable

2157:{\displaystyle H={\frac {p_{1}^{2}}{2m}}+{\frac {p_{2}^{2}}{2m}}+U(|x_{1}-x_{2}|)+V(x_{1})+V(x_{2})} 9370:

in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013,

8929: 8881: 7492: 2924: 241: 8849: 8624:, these principles are extended to large number of particles, with qualitatively similar results. 7471:

Note that this "high temperature" approximation does not distinguish between fermions and bosons.

6940: 6535: 2294:

In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as

1699:{\displaystyle P{\bigg (}|\psi \rangle |\phi \rangle {\bigg )}\equiv |\phi \rangle |\psi \rangle } 631:

Antisymmetric wavefunction for a (fermionic) 2-particle state in an infinite square well potential

6629:

indistinguishable particles should be invariant under these permutation. This means that for all

2852: 2322: 303: 225: 149: 37: 8522: 8494: 8381: 8353: 8325: 8297: 8106: 8078: 8050: 8022: 7831: 7803: 7772: 7744: 7282: 6632: 813: 571: 2306:, a phenomenon observed in the two-dimensional electron gases that form the inversion layer of 9367: 7863:

If A and B are distinguishable particles, then the composite system has four distinct states:

2299: 2176: 1730: 562: 320: 126: 9235: 7246:{\displaystyle Z=\sum _{n_{1},n_{2},\ldots ,n_{N}}\exp \left\{-{\frac {1}{kT}}\left\right\}} 639:

Symmetric wavefunction for a (bosonic) 2-particle state in an infinite square well potential

9284: 9184: 9150: 9107: 8696:

describes the interchanged configuration. With identical particles, the state described by

8134:

If A and B are identical bosons, then the composite system has only three distinct states:

7857: 7800:

The composite system can evolve in time, interacting with a noisy environment. Because the

6970: 6565: 6013: 3892: 2597: 363: 8: 9275: 8928:. So, the only permissible interchange is to swap both particles. This interchange is an 2303: 455: 358: 348: 160: 9288: 9154: 2901: 2858: 9300: 9166: 9140: 7590: 7261: 6934: 6847: 6612: 6592: 6515: 6017: 2881: 2219: 1721: 1713: 566: 463: 233: 9346: 8294:. When the experiment is performed, the probability of obtaining two particles in the 9371: 9327: 9304: 9296: 9255: 9214: 3787: 655: 606: 488: 435: 383: 325: 9170: 9292: 9247: 9158: 9102: 8637: 2311: 1717: 378: 353: 202: 8484:{\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle -|1\rangle |0\rangle )} 8287:{\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle +|1\rangle |0\rangle )} 3797:

Recall that an eigenstate of a continuous observable represents an infinitesimal

3760:

which verifies that the total probability is 1. The sum has to be restricted to

595: 587: 368: 330: 310: 6922:{\displaystyle \Psi \sim \sum _{\sigma \in S_{n}}\lambda _{\sigma }\sigma \Psi } 8721: 7734: 7606: 6999: 2326: 2288: 807: 591: 373: 280: 192: 144: 9357: 605:

The problem with the second approach is that it contradicts the principles of

9384: 9259: 7480: 7476: 7273: 503: 479: 467: 1067:{\displaystyle |n_{1}\rangle |n_{2}\rangle \pm |n_{2}\rangle |n_{1}\rangle } 9351: 8990: 7605:

does not double accordingly. Such a system does not obey the postulates of

4379: 1980:

meaning, in agreement with the earlier discussion on indistinguishability.

610: 393: 9251: 3270:

and a measurement is performed on some other set of discrete observables,

7269: 6009: 2586: 1769: 663: 388: 290: 285: 275: 9273:

Bach, Alexaner (1993). "Classification of Indistinguishable Particles".

8892:

itself, only has two points which are physically indistinguishable from

3778:

to ensure that each multi-particle state is not counted more than once.

942:

as a result of exchanging the particles are generally different states.

7726: 2232: 2168: 1761: 561:

The fact that particles can be identical has important consequences in

492: 9187:. Diss. University of Cambridge, 1998. Section 2.3 Identical particles 9162: 9010:, the absolute value of the multiplication must be 1). This is called 8977:

has infinitely many points that are physically indistinguishable from

134: 9007: 8837: 8322:

state is now 0.33; the probability of obtaining two particles in the

7730: 7496: 1429: 8350:

state is 0.33; and the probability of obtaining one particle in the

8075:

state is 0.25; and the probability of obtaining one particle in the

26: 9347:

The Feynman Lectures on Physics Vol. III Ch. 4: Identical Particles

9185:

Linear-scaling methods in ab initio quantum-mechanical calculations

8633: 2252: 2248: 555: 539: 535: 527: 475: 459: 451: 315: 9145: 8491:. When the experiment is performed, one particle is always in the 6756:{\displaystyle (\sigma \Psi )^{t}a(\sigma \Psi )=\Psi ^{t}a\Psi ,} 5137:

where the single-particle wavefunctions are defined, as usual, by

2607:

stands for the number of times each of the single-particle states

2302:. Experimental evidence for the existence of anyons exists in the 627: 8047:

state is 0.25; the probability of obtaining two particles in the

2315: 2274: 635: 551: 507: 16:

Concept in quantum mechanics of perfectly substitutable particles

9234:

Baker, David John; Halvorson, Hans; Swanson, Noel (2015-12-01).

7704: 7574: 7054: 2307: 2256: 2244: 1969:{\displaystyle P|n_{1},n_{2};A\rangle =-|n_{1},n_{2};A\rangle } 1868:{\displaystyle P|n_{1},n_{2};S\rangle =+|n_{1},n_{2};S\rangle } 1433: 680:. The quantum state of the system is denoted by the expression 547: 523: 515: 9352:

Exchange of Identical and Possibly Indistinguishable Particles

7014: 7006:

provide a way to classify all of these irreducible subspaces.

2325:

relates the exchange symmetry of identical particles to their

2291:

are mathematically possible, but no examples exist in nature.

9011: 7718: 2295: 2263: 543: 531: 519: 499: 197: 7023:

distinguishable, non-interacting particles. Once again, let

806:). This is the canonical way of constructing a basis for a 602:

there would be no ambiguity about which particle is which.

583: 471: 8708:

ought to be indistinguishable from the state described by

7036:. If the particles have the same physical properties, the 6008:

Finally, antisymmetric wavefunction can be written as the

2167:

It is possible to show that such Hamiltonians satisfy the

8664:

the subspace with coincident points removed. The element

7691:{\displaystyle S=Nk\ln \left({\frac {V}{N}}\right)+Nf(T)} 3366:

The probability of obtaining a particular result for the

3192:

bosons (fermions) in the symmetric (antisymmetric) state

622: 2340:

The above discussion generalizes readily to the case of

2273:

Particles which exhibit antisymmetric states are called

9129:"P. A. M. Dirac and the discovery of quantum mechanics" 7737:

rather than all lying in the same lowest energy state.

6507: 5993:

Because each integral runs over all possible values of

3805:⟩, the probability of finding it in a region of volume 2585:

Here, the sum is taken over all different states under

8840:

to a clockwise interchange by half a turn. Lastly, if

6943: 6538: 6129: 5189:{\displaystyle \psi _{n}(x)\equiv \langle x|n\rangle } 3963:{\displaystyle \langle x|x'\rangle =\delta ^{3}(x-x')} 1496: 1298: 1136: 498:

There are two main categories of identical particles:

9233: 9063: 8941: 8852: 8808: 8764: 8525: 8497: 8415: 8384: 8356: 8328: 8300: 8218: 8179: 8140: 8109: 8081: 8053: 8025: 7986: 7947: 7908: 7869: 7834: 7806: 7775: 7747: 7633: 7508: 7426: 7324: 7285: 7074: 6973: 6873: 6850: 6807: 6772: 6696: 6661: 6635: 6615: 6595: 6568: 6518: 6029: 5767: 5591: 5209: 5146: 4391: 3982: 3904: 3877:{\displaystyle |\langle x|\psi \rangle |^{2}\;d^{3}x} 3825: 3606: 3379: 3297: 3201: 2975: 2927: 2904: 2898:

is composed of an even number of transpositions, and

2884: 2861: 2652: 2388: 2179: 2001: 1882: 1781: 1733: 1631: 1454: 1256: 1094: 988: 895: 842: 816: 745: 689: 8796:, then this homotopy class only has one element. If 8019:. The probability of obtaining two particles in the 7032:

denote the state (i.e. quantum numbers) of particle

2262:

Particles which exhibit symmetric states are called

3356:{\displaystyle |m_{1}m_{2}\cdots m_{N};S/A\rangle } 3260:{\displaystyle |n_{1}n_{2}\cdots n_{N};S/A\rangle } 1081:, while states involving the difference are called 739:where the order of the tensor product matters ( if 577: 51:. Unsourced material may be challenged and removed. 9080: 8965: 8860: 8823: 8779: 8539: 8511: 8483: 8398: 8370: 8342: 8314: 8286: 8204: 8165: 8123: 8095: 8067: 8039: 8011: 7972: 7933: 7894: 7848: 7820: 7789: 7761: 7690: 7558: 7460: 7391: 7304: 7245: 6986: 6959: 6921: 6856: 6832: 6791: 6755: 6680: 6647: 6621: 6601: 6581: 6554: 6524: 6496: 5982: 5743: 5537: 5188: 5126: 4367: 3962: 3876: 3749: 3586: 3355: 3259: 3172: 2950: 2913: 2890: 2870: 2832: 2574: 2207: 2156: 1968: 1867: 1752: 1698: 1599: 1398: 1236: 1085:. More completely, symmetric states have the form 1066: 934: 881: 828: 784: 728: 9240:The British Journal for the Philosophy of Science 5963: 5939: 5918: 5779: 5775: 5771: 5737: 1720:. Because it is unitary, it can be regarded as a 1666: 1637: 1592: 1507: 1391: 1309: 1229: 1147: 617: 9382: 2251:atoms, and antisymmetric states when describing 8676:describes the configuration with particle I at 2596:elements. The square root left to the sum is a 2314:, which are associated with particles known as 2310:. There is another type of statistic, known as 3781: 1992:can be written in a symmetrical form, such as 1428:, and it is the fundamental reason behind the 7705:Statistical properties of bosons and fermions 3573: 3474: 2371:. If the particles are bosons, they occupy a 645:mathematical formulation of quantum mechanics 415: 9358:Identity and Individuality in Quantum Theory 8534: 8506: 8475: 8464: 8450: 8439: 8393: 8365: 8337: 8309: 8278: 8267: 8253: 8242: 8199: 8188: 8160: 8149: 8118: 8090: 8062: 8034: 8006: 7995: 7967: 7956: 7928: 7917: 7889: 7878: 7843: 7815: 7784: 7756: 7585:alone. The problem with this result is that 7559:{\displaystyle S=Nk\ln \left(V\right)+Nf(T)} 5183: 5169: 4954: 4868: 4588: 4502: 4222: 4031: 3924: 3905: 3845: 3831: 3350: 3254: 3161: 3075: 3062: 2976: 2697: 2433: 1963: 1920: 1862: 1819: 1772:are the symmetric and antisymmetric states: 1693: 1682: 1661: 1650: 1587: 1569: 1545: 1527: 1489: 1386: 1368: 1347: 1329: 1291: 1224: 1206: 1185: 1167: 1129: 1061: 1043: 1022: 1004: 935:{\displaystyle |n_{2}\rangle |n_{1}\rangle } 929: 911: 882:{\displaystyle |n_{1}\rangle |n_{2}\rangle } 876: 858: 785:{\displaystyle |n_{2}\rangle |n_{1}\rangle } 779: 761: 729:{\displaystyle |n_{1}\rangle |n_{2}\rangle } 723: 705: 7015:Statistical effects of indistinguishability 2375:, which is symmetric under the exchange of 5783: 5723: 5598: 3860: 2966:These states have been normalized so that 422: 408: 133: 9321: 9196: 9144: 9126: 9071: 8950: 8854: 8811: 8767: 7461:{\displaystyle Z={\frac {\xi ^{N}}{N!}}.} 7392:{\displaystyle \xi =\sum _{n}\exp \left.} 7045:s run over the same range of values. Let 7009: 6532:particles is given by the tensor product 3887:As a result, the continuous eigenstates | 3527: 3521: 1432:properties of atoms and the stability of 1247:while antisymmetric states have the form 792:, then the particle 1 occupies the state 111:Learn how and when to remove this message 799:while the particle 2 occupies the state 634: 626: 9236:"The Conventionality of Parastatistics" 9050:. This phase factor here is called the 9030:is now physically distinguishable from 8550:The results are summarized in Table 1: 1768:are +1 and −1. The corresponding 614:are then said to be indistinguishable. 9383: 9208: 3274:. In general, this yields some result 2238: 1077:States where it is a sum are known as 967:state and the particle 2 occupies the 960:state" ≠ "the particle 1 occupies the 953:state and the particle 2 occupies the 623:Symmetrical and antisymmetrical states 8870:, then this homotopy class is empty. 8554:Table 1: Statistics of two particles 7417:!. The correct partition function is 6833:{\displaystyle \sigma ^{t}a\sigma =a} 9272: 8989:. This is described by the infinite 8205:{\displaystyle |1\rangle |1\rangle } 8166:{\displaystyle |0\rangle |0\rangle } 8012:{\displaystyle |1\rangle |0\rangle } 7973:{\displaystyle |0\rangle |1\rangle } 7934:{\displaystyle |1\rangle |1\rangle } 7895:{\displaystyle |0\rangle |0\rangle } 7406:, every possible permutation of the 6508:Operator approach and parastatistics 5997:, each multi-particle state appears 1439: 510:, which cannot (as described by the 331:Grand potential / Landau free energy 49:adding citations to reliable sources 20: 9362:Stanford Encyclopedia of Philosophy 9014:statistics. In fact, even with two 8966:{\displaystyle M=\mathbb {R} ^{2},} 13: 6916: 6874: 6747: 6735: 6725: 6703: 6031: 5791: 5606: 5456: 5374: 5294: 5215: 4999: 4996: 4993: 4763: 4397: 4264: 4261: 4258: 2929: 2639:In the same vein, fermions occupy 14: 9412: 9340: 8627: 2277:. Antisymmetry gives rise to the 8973:the universal covering space of 8824:{\displaystyle \mathbb {R} ^{2}} 8780:{\displaystyle \mathbb {R} ^{d}} 6792:{\displaystyle \sigma \in S_{n}} 6681:{\displaystyle \sigma \in S_{n}} 1760:(the identity operator), so the 673:, and the other is in the state 578:Distinguishing between particles 25: 9213:. Addison-Wesley. p. 597. 9081:{\displaystyle M=\mathbb {R} ,} 6933:The equivalence classes are in 3074: 2921:if odd). Note that there is no 2348:particles with quantum numbers 2222:, this means that the value of 487:. They were first discussed by 36:needs additional citations for 9266: 9227: 9211:Introductory Quantum Mechanics 9202: 9190: 9177: 9120: 8527: 8519:state and the other is in the 8499: 8478: 8468: 8457: 8443: 8432: 8428: 8386: 8358: 8330: 8302: 8281: 8271: 8260: 8246: 8235: 8231: 8192: 8181: 8153: 8142: 8111: 8083: 8055: 8027: 7999: 7988: 7960: 7949: 7921: 7910: 7882: 7871: 7836: 7808: 7797:, which have the same energy. 7777: 7749: 7685: 7679: 7553: 7547: 7367: 7361: 7230: 7217: 7202: 7189: 7180: 7167: 6937:with irreducible subspaces of 6728: 6719: 6707: 6697: 6483: 6470: 6443: 6430: 6408: 6395: 6349: 6336: 6309: 6296: 6274: 6261: 6237: 6224: 6197: 6184: 6162: 6149: 6103: 6071: 6066: 6060: 5894: 5849: 5844: 5830: 5709: 5664: 5659: 5645: 5528: 5496: 5491: 5485: 5446: 5414: 5409: 5403: 5366: 5334: 5329: 5323: 5287: 5255: 5250: 5244: 5176: 5163: 5157: 5117: 5104: 5099: 5093: 5079: 5066: 5061: 5055: 5044: 5031: 5026: 5020: 5009: 5003: 4911: 4858: 4813: 4808: 4802: 4755: 4742: 4737: 4731: 4717: 4704: 4699: 4693: 4682: 4669: 4664: 4658: 4545: 4492: 4447: 4442: 4436: 4352: 4346: 4322: 4316: 4295: 4289: 4274: 4268: 4179: 4165: 4159: 4135: 4129: 4108: 4102: 3988: 3957: 3940: 3912: 3850: 3838: 3827: 3738: 3706: 3674: 3523: 3432: 3299: 3203: 3183: 3118: 3019: 2818: 2812: 2788: 2782: 2761: 2755: 2740: 2734: 2654: 2563: 2557: 2533: 2527: 2506: 2500: 2390: 2332: 2304:fractional quantum Hall effect 2151: 2138: 2129: 2116: 2107: 2103: 2075: 2071: 1930: 1887: 1829: 1786: 1686: 1675: 1654: 1643: 1573: 1555: 1531: 1513: 1456: 1372: 1354: 1333: 1315: 1258: 1210: 1192: 1171: 1153: 1096: 1047: 1029: 1008: 990: 915: 897: 862: 844: 765: 747: 709: 691: 618:Quantum mechanical description 1: 9315: 7612:Gibbs also showed that using 6960:{\textstyle \bigotimes _{n}H} 6555:{\textstyle \bigotimes _{n}H} 3188:Suppose there is a system of 2951:{\displaystyle \Pi _{n}m_{n}} 2344:particles. Suppose there are 946:"the particle 1 occupies the 565:, where calculations rely on 60:"Indistinguishable particles" 9113: 8861:{\displaystyle \mathbb {R} } 8609: 8606: 8603: 8595: 8592: 8589: 8581: 8578: 8575: 7483:showed that in the equation 2641:totally antisymmetric states 7: 9133:American Journal of Physics 9096: 9090:∖ {coincident points} 8975:∖ {coincident points} 8890:∖ {coincident points} 8888:, which is none other than 8886:∖ {coincident points} 8378:state and the other in the 8103:state and the other in the 6562:. The permutation group of 3782:Wavefunction representation 2615:-particle state. Note that 534:. Examples of fermions are 440:indistinguishable particles 173:Indistinguishable particles 10: 9417: 9297:10.1209/0295-5075/21/5/002 8631: 8540:{\displaystyle |1\rangle } 8512:{\displaystyle |0\rangle } 8399:{\displaystyle |1\rangle } 8371:{\displaystyle |0\rangle } 8343:{\displaystyle |1\rangle } 8315:{\displaystyle |0\rangle } 8124:{\displaystyle |1\rangle } 8096:{\displaystyle |0\rangle } 8068:{\displaystyle |1\rangle } 8040:{\displaystyle |0\rangle } 7849:{\displaystyle |1\rangle } 7821:{\displaystyle |0\rangle } 7790:{\displaystyle |1\rangle } 7762:{\displaystyle |0\rangle } 7723:Bose–Einstein condensation 7305:{\displaystyle Z=\xi ^{N}} 7272:. This expression can be 6648:{\displaystyle \psi \in H} 3812:surrounding some position 2855:of each permutation (i.e. 829:{\displaystyle H\otimes H} 514:). Examples of bosons are 9396:Pauli exclusion principle 8724:of continuous paths from 6766:or equivalently for each 2961:Pauli exclusion principle 2279:Pauli exclusion principle 1983:It will be recalled that 1618:Define a linear operator 1426:Pauli exclusion principle 512:Pauli exclusion principle 9322:Tuckerman, Mark (2010), 9209:Liboff, Richard (2003). 9127:Gottfried, Kurt (2011). 8882:universal covering space 8622:Bose–Einstein statistics 7711:Bose–Einstein statistics 3891:⟩ are normalized to the 2268:Bose–Einstein statistics 9401:Probabilistic arguments 9018:particles, even though 7624:! alters the result to 7057:of a particle in state 2373:totally symmetric state 2323:spin-statistics theorem 2208:{\displaystyle \left=0} 1753:{\displaystyle P^{2}=1} 448:indiscernible particles 216:Thermodynamic ensembles 168:Spin–statistics theorem 9082: 8967: 8862: 8825: 8781: 8750:\ {coincident points} 8662:\ {coincident points}, 8618:Fermi–Dirac statistics 8541: 8513: 8485: 8400: 8372: 8344: 8316: 8288: 8206: 8167: 8125: 8097: 8069: 8041: 8013: 7974: 7935: 7896: 7850: 7822: 7791: 7763: 7715:Fermi–Dirac statistics 7692: 7560: 7462: 7393: 7306: 7247: 7010:Statistical properties 6988: 6961: 6923: 6858: 6834: 6793: 6757: 6682: 6649: 6623: 6603: 6583: 6556: 6526: 6512:The Hilbert space for 6498: 5984: 5745: 5539: 5190: 5128: 4369: 3964: 3878: 3751: 3588: 3357: 3261: 3174: 2952: 2915: 2892: 2872: 2834: 2576: 2283:Fermi–Dirac statistics 2209: 2158: 1970: 1869: 1754: 1700: 1601: 1400: 1238: 1068: 936: 883: 830: 786: 730: 640: 632: 9324:Statistical Mechanics 9083: 9057:Finally, in the case 8968: 8863: 8826: 8782: 8542: 8514: 8486: 8401: 8373: 8345: 8317: 8289: 8207: 8168: 8126: 8098: 8070: 8042: 8014: 7975: 7936: 7897: 7851: 7823: 7792: 7764: 7693: 7561: 7463: 7394: 7307: 7248: 6989: 6987:{\displaystyle S_{n}} 6962: 6924: 6859: 6835: 6794: 6758: 6683: 6650: 6624: 6604: 6584: 6582:{\displaystyle S_{n}} 6557: 6527: 6499: 5985: 5746: 5540: 5191: 5129: 4370: 3965: 3879: 3752: 3597:It can be shown that 3589: 3358: 3262: 3175: 2953: 2916: 2893: 2873: 2835: 2577: 2300:fractional statistics 2210: 2159: 1971: 1870: 1755: 1701: 1602: 1401: 1239: 1069: 937: 884: 831: 787: 731: 638: 630: 563:statistical mechanics 321:Helmholtz free energy 250:Isoenthalpic–isobaric 127:Statistical mechanics 9368:Many-Electron States 9108:DeBroglie hypothesis 9061: 8939: 8850: 8806: 8762: 8523: 8495: 8413: 8382: 8354: 8326: 8298: 8216: 8177: 8138: 8107: 8079: 8051: 8023: 7984: 7945: 7906: 7867: 7858:quantum entanglement 7832: 7804: 7773: 7745: 7631: 7581:is some function of 7506: 7424: 7322: 7283: 7072: 7000:parastatistic states 6971: 6941: 6871: 6848: 6805: 6770: 6694: 6659: 6633: 6613: 6593: 6566: 6536: 6516: 6027: 5765: 5589: 5207: 5144: 4389: 3980: 3902: 3823: 3604: 3377: 3295: 3199: 2973: 2963:for many particles. 2925: 2902: 2882: 2859: 2650: 2598:normalizing constant 2386: 2177: 2169:commutation relation 1999: 1880: 1779: 1731: 1629: 1452: 1254: 1092: 986: 893: 840: 814: 743: 687: 662:to be the quantized 456:elementary particles 45:improve this article 9391:Particle statistics 9289:1993EL.....21..515B 9276:Europhysics Letters 9252:10.1093/bjps/axu018 9155:2011AmJPh..79..261G 8873:Suppose first that 8748:, within the space 8720:. Now consider the 8680:and particle II at 8646:-dimensional space 8555: 6070: 5848: 5663: 5495: 5413: 5333: 5254: 4812: 4446: 2239:Fermions and bosons 2220:Heisenberg equation 2054: 2024: 464:subatomic particles 258:Isothermal–isobaric 161:Particle statistics 9078: 8963: 8858: 8821: 8777: 8553: 8537: 8509: 8481: 8396: 8368: 8340: 8312: 8284: 8202: 8163: 8121: 8093: 8065: 8037: 8009: 7970: 7931: 7892: 7846: 7818: 7787: 7759: 7688: 7556: 7458: 7389: 7340: 7302: 7262:Boltzmann constant 7243: 7129: 7063:partition function 6984: 6957: 6953: 6935:bijective relation 6919: 6902: 6854: 6830: 6789: 6753: 6678: 6645: 6619: 6599: 6579: 6552: 6548: 6522: 6494: 6488: 6030: 6018:Slater determinant 5980: 5790: 5741: 5605: 5535: 5533: 5455: 5373: 5293: 5214: 5186: 5124: 5122: 4991: 4762: 4649: 4614: 4396: 4365: 4363: 4256: 4089: 4054: 3960: 3895:instead of unity: 3874: 3747: 3655: 3584: 3353: 3281:for one particle, 3257: 3170: 2948: 2914:{\displaystyle -1} 2911: 2888: 2871:{\displaystyle +1} 2868: 2830: 2727: 2572: 2487: 2452: 2205: 2154: 2040: 2010: 1966: 1865: 1750: 1696: 1597: 1500: 1396: 1302: 1234: 1140: 1064: 932: 879: 826: 782: 726: 641: 633: 502:, which can share 485:quantum statistics 198:Anyonic statistics 9376:978-3-89336-884-6 9354:by John S. Denker 9163:10.1119/1.3536639 9052:mutual statistics 8614: 8613: 8426: 8425: 8229: 8228: 7664: 7453: 7379: 7331: 7157: 7081: 7065:of the system is 6944: 6880: 6857:{\displaystyle n} 6622:{\displaystyle n} 6602:{\displaystyle a} 6539: 6525:{\displaystyle n} 6122: 6121: 4982: 4980: 4979: 4640: 4638: 4637: 4605: 4247: 4245: 4244: 4080: 4078: 4077: 4045: 3607: 2891:{\displaystyle p} 2829: 2718: 2716: 2715: 2478: 2476: 2475: 2443: 2379:particle labels: 2218:According to the 2063: 2033: 1722:symmetry operator 1613:exchange symmetry 1499: 1440:Exchange symmetry 1301: 1139: 656:particle in a box 607:quantum mechanics 489:Werner Heisenberg 436:quantum mechanics 432: 431: 326:Gibbs free energy 178:Maxwell–Boltzmann 121: 120: 113: 95: 9408: 9336: 9309: 9308: 9270: 9264: 9263: 9231: 9225: 9224: 9206: 9200: 9194: 9188: 9181: 9175: 9174: 9148: 9124: 9103:Quasi-set theory 9091: 9087: 9085: 9084: 9079: 9074: 9049: 9041: 9029: 9005: 9001: 8988: 8976: 8972: 8970: 8969: 8964: 8959: 8958: 8953: 8927: 8915: 8903: 8891: 8887: 8879: 8869: 8867: 8865: 8864: 8859: 8857: 8843: 8832: 8830: 8828: 8827: 8822: 8820: 8819: 8814: 8799: 8795: 8788: 8786: 8784: 8783: 8778: 8776: 8775: 8770: 8755: 8751: 8747: 8735: 8719: 8707: 8695: 8683: 8679: 8675: 8663: 8659: 8649: 8645: 8638:Braid statistics 8568:One 0 and one 1 8556: 8552: 8546: 8544: 8543: 8538: 8530: 8518: 8516: 8515: 8510: 8502: 8490: 8488: 8487: 8482: 8471: 8460: 8446: 8435: 8427: 8421: 8417: 8405: 8403: 8402: 8397: 8389: 8377: 8375: 8374: 8369: 8361: 8349: 8347: 8346: 8341: 8333: 8321: 8319: 8318: 8313: 8305: 8293: 8291: 8290: 8285: 8274: 8263: 8249: 8238: 8230: 8224: 8220: 8211: 8209: 8208: 8203: 8195: 8184: 8172: 8170: 8169: 8164: 8156: 8145: 8130: 8128: 8127: 8122: 8114: 8102: 8100: 8099: 8094: 8086: 8074: 8072: 8071: 8066: 8058: 8046: 8044: 8043: 8038: 8030: 8018: 8016: 8015: 8010: 8002: 7991: 7979: 7977: 7976: 7971: 7963: 7952: 7940: 7938: 7937: 7932: 7924: 7913: 7901: 7899: 7898: 7893: 7885: 7874: 7855: 7853: 7852: 7847: 7839: 7827: 7825: 7824: 7819: 7811: 7796: 7794: 7793: 7788: 7780: 7768: 7766: 7765: 7760: 7752: 7697: 7695: 7694: 7689: 7669: 7665: 7657: 7565: 7563: 7562: 7557: 7537: 7467: 7465: 7464: 7459: 7454: 7452: 7444: 7443: 7434: 7398: 7396: 7395: 7390: 7385: 7381: 7380: 7378: 7370: 7356: 7339: 7311: 7309: 7308: 7303: 7301: 7300: 7252: 7250: 7249: 7244: 7242: 7238: 7237: 7233: 7229: 7228: 7201: 7200: 7179: 7178: 7158: 7156: 7145: 7128: 7127: 7126: 7108: 7107: 7095: 7094: 6993: 6991: 6990: 6985: 6983: 6982: 6966: 6964: 6963: 6958: 6952: 6928: 6926: 6925: 6920: 6912: 6911: 6901: 6900: 6899: 6863: 6861: 6860: 6855: 6839: 6837: 6836: 6831: 6817: 6816: 6798: 6796: 6795: 6790: 6788: 6787: 6762: 6760: 6759: 6754: 6743: 6742: 6715: 6714: 6687: 6685: 6684: 6679: 6677: 6676: 6654: 6652: 6651: 6646: 6628: 6626: 6625: 6620: 6608: 6606: 6605: 6600: 6588: 6586: 6585: 6580: 6578: 6577: 6561: 6559: 6558: 6553: 6547: 6531: 6529: 6528: 6523: 6503: 6501: 6500: 6495: 6493: 6489: 6482: 6481: 6469: 6468: 6467: 6466: 6442: 6441: 6429: 6428: 6427: 6426: 6407: 6406: 6394: 6393: 6392: 6391: 6348: 6347: 6335: 6334: 6333: 6332: 6308: 6307: 6295: 6294: 6293: 6292: 6273: 6272: 6260: 6259: 6258: 6257: 6236: 6235: 6223: 6222: 6221: 6220: 6196: 6195: 6183: 6182: 6181: 6180: 6161: 6160: 6148: 6147: 6146: 6145: 6123: 6114: 6110: 6102: 6101: 6083: 6082: 6069: 6058: 6057: 6056: 6044: 6043: 5989: 5987: 5986: 5981: 5973: 5972: 5962: 5961: 5949: 5948: 5938: 5937: 5928: 5927: 5917: 5916: 5907: 5906: 5901: 5897: 5893: 5892: 5874: 5873: 5861: 5860: 5847: 5840: 5828: 5827: 5826: 5814: 5813: 5804: 5803: 5750: 5748: 5747: 5742: 5736: 5735: 5722: 5721: 5716: 5712: 5708: 5707: 5689: 5688: 5676: 5675: 5662: 5655: 5643: 5642: 5641: 5629: 5628: 5619: 5618: 5544: 5542: 5541: 5536: 5534: 5524: 5523: 5511: 5510: 5494: 5483: 5482: 5481: 5469: 5468: 5442: 5441: 5429: 5428: 5412: 5401: 5400: 5399: 5387: 5386: 5362: 5361: 5349: 5348: 5332: 5321: 5320: 5319: 5307: 5306: 5283: 5282: 5270: 5269: 5253: 5242: 5241: 5240: 5228: 5227: 5195: 5193: 5192: 5187: 5179: 5156: 5155: 5133: 5131: 5130: 5125: 5123: 5116: 5115: 5103: 5102: 5078: 5077: 5065: 5064: 5043: 5042: 5030: 5029: 5002: 4990: 4981: 4972: 4968: 4960: 4947: 4946: 4934: 4933: 4924: 4923: 4914: 4903: 4902: 4890: 4889: 4880: 4879: 4857: 4856: 4838: 4837: 4825: 4824: 4811: 4800: 4799: 4798: 4786: 4785: 4776: 4775: 4754: 4753: 4741: 4740: 4716: 4715: 4703: 4702: 4681: 4680: 4668: 4667: 4648: 4639: 4636: 4628: 4624: 4623: 4613: 4603: 4602: 4594: 4581: 4580: 4568: 4567: 4558: 4557: 4548: 4537: 4536: 4524: 4523: 4514: 4513: 4491: 4490: 4472: 4471: 4459: 4458: 4445: 4434: 4433: 4432: 4420: 4419: 4410: 4409: 4382:can be written, 4374: 4372: 4371: 4366: 4364: 4360: 4356: 4355: 4330: 4326: 4325: 4303: 4299: 4298: 4267: 4255: 4246: 4237: 4233: 4215: 4214: 4202: 4201: 4192: 4191: 4182: 4173: 4169: 4168: 4143: 4139: 4138: 4116: 4112: 4111: 4088: 4079: 4076: 4068: 4064: 4063: 4053: 4043: 4042: 4024: 4023: 4011: 4010: 4001: 4000: 3991: 3969: 3967: 3966: 3961: 3956: 3939: 3938: 3923: 3915: 3883: 3881: 3880: 3875: 3870: 3869: 3859: 3858: 3853: 3841: 3830: 3756: 3754: 3753: 3748: 3737: 3736: 3718: 3717: 3705: 3704: 3686: 3685: 3673: 3672: 3668: 3654: 3653: 3652: 3634: 3633: 3621: 3620: 3593: 3591: 3590: 3585: 3583: 3582: 3577: 3576: 3569: 3565: 3561: 3550: 3549: 3537: 3536: 3526: 3517: 3506: 3505: 3493: 3492: 3478: 3477: 3468: 3464: 3463: 3462: 3444: 3443: 3431: 3430: 3412: 3411: 3397: 3396: 3392: 3362: 3360: 3359: 3354: 3346: 3335: 3334: 3322: 3321: 3312: 3311: 3302: 3266: 3264: 3263: 3258: 3250: 3239: 3238: 3226: 3225: 3216: 3215: 3206: 3179: 3177: 3176: 3171: 3154: 3153: 3141: 3140: 3131: 3130: 3121: 3110: 3109: 3097: 3096: 3087: 3086: 3055: 3054: 3042: 3041: 3032: 3031: 3022: 3011: 3010: 2998: 2997: 2988: 2987: 2957: 2955: 2954: 2949: 2947: 2946: 2937: 2936: 2920: 2918: 2917: 2912: 2897: 2895: 2894: 2889: 2877: 2875: 2874: 2869: 2850: 2839: 2837: 2836: 2831: 2827: 2826: 2822: 2821: 2796: 2792: 2791: 2769: 2765: 2764: 2726: 2717: 2708: 2704: 2690: 2689: 2677: 2676: 2667: 2666: 2657: 2635: 2581: 2579: 2578: 2573: 2571: 2567: 2566: 2541: 2537: 2536: 2514: 2510: 2509: 2486: 2477: 2474: 2466: 2462: 2461: 2451: 2441: 2440: 2426: 2425: 2413: 2412: 2403: 2402: 2393: 2312:braid statistics 2298:, and they obey 2214: 2212: 2211: 2206: 2198: 2194: 2163: 2161: 2160: 2155: 2150: 2149: 2128: 2127: 2106: 2101: 2100: 2088: 2087: 2078: 2064: 2062: 2053: 2048: 2039: 2034: 2032: 2023: 2018: 2009: 1975: 1973: 1972: 1967: 1956: 1955: 1943: 1942: 1933: 1913: 1912: 1900: 1899: 1890: 1874: 1872: 1871: 1866: 1855: 1854: 1842: 1841: 1832: 1812: 1811: 1799: 1798: 1789: 1759: 1757: 1756: 1751: 1743: 1742: 1705: 1703: 1702: 1697: 1689: 1678: 1670: 1669: 1657: 1646: 1641: 1640: 1606: 1604: 1603: 1598: 1596: 1595: 1586: 1585: 1576: 1568: 1567: 1558: 1544: 1543: 1534: 1526: 1525: 1516: 1511: 1510: 1501: 1497: 1482: 1481: 1469: 1468: 1459: 1405: 1403: 1402: 1397: 1395: 1394: 1385: 1384: 1375: 1367: 1366: 1357: 1346: 1345: 1336: 1328: 1327: 1318: 1313: 1312: 1303: 1299: 1284: 1283: 1271: 1270: 1261: 1243: 1241: 1240: 1235: 1233: 1232: 1223: 1222: 1213: 1205: 1204: 1195: 1184: 1183: 1174: 1166: 1165: 1156: 1151: 1150: 1141: 1137: 1122: 1121: 1109: 1108: 1099: 1073: 1071: 1070: 1065: 1060: 1059: 1050: 1042: 1041: 1032: 1021: 1020: 1011: 1003: 1002: 993: 941: 939: 938: 933: 928: 927: 918: 910: 909: 900: 888: 886: 885: 880: 875: 874: 865: 857: 856: 847: 835: 833: 832: 827: 791: 789: 788: 783: 778: 777: 768: 760: 759: 750: 735: 733: 732: 727: 722: 721: 712: 704: 703: 694: 424: 417: 410: 203:Braid statistics 137: 123: 122: 116: 109: 105: 102: 96: 94: 53: 29: 21: 9416: 9415: 9411: 9410: 9409: 9407: 9406: 9405: 9381: 9380: 9343: 9334: 9318: 9313: 9312: 9271: 9267: 9232: 9228: 9221: 9207: 9203: 9197:Tuckerman (2010 9195: 9191: 9182: 9178: 9125: 9121: 9116: 9099: 9089: 9070: 9062: 9059: 9058: 9043: 9031: 9019: 9016:distinguishable 9003: 8995: 8978: 8974: 8954: 8949: 8948: 8940: 8937: 8936: 8917: 8905: 8893: 8889: 8885: 8874: 8853: 8851: 8848: 8847: 8845: 8841: 8815: 8810: 8809: 8807: 8804: 8803: 8801: 8797: 8790: 8771: 8766: 8765: 8763: 8760: 8759: 8757: 8753: 8749: 8737: 8725: 8709: 8697: 8685: 8681: 8677: 8665: 8661: 8651: 8647: 8643: 8640: 8630: 8573:Distinguishable 8526: 8524: 8521: 8520: 8498: 8496: 8493: 8492: 8467: 8456: 8442: 8431: 8416: 8414: 8411: 8410: 8385: 8383: 8380: 8379: 8357: 8355: 8352: 8351: 8329: 8327: 8324: 8323: 8301: 8299: 8296: 8295: 8270: 8259: 8245: 8234: 8219: 8217: 8214: 8213: 8191: 8180: 8178: 8175: 8174: 8152: 8141: 8139: 8136: 8135: 8110: 8108: 8105: 8104: 8082: 8080: 8077: 8076: 8054: 8052: 8049: 8048: 8026: 8024: 8021: 8020: 7998: 7987: 7985: 7982: 7981: 7959: 7948: 7946: 7943: 7942: 7920: 7909: 7907: 7904: 7903: 7881: 7870: 7868: 7865: 7864: 7835: 7833: 7830: 7829: 7807: 7805: 7802: 7801: 7776: 7774: 7771: 7770: 7748: 7746: 7743: 7742: 7707: 7656: 7652: 7632: 7629: 7628: 7577:of the gas and 7527: 7507: 7504: 7503: 7495:of a classical 7445: 7439: 7435: 7433: 7425: 7422: 7421: 7371: 7357: 7355: 7351: 7347: 7335: 7323: 7320: 7319: 7296: 7292: 7284: 7281: 7280: 7224: 7220: 7196: 7192: 7174: 7170: 7163: 7159: 7149: 7144: 7140: 7136: 7122: 7118: 7103: 7099: 7090: 7086: 7085: 7073: 7070: 7069: 7044: 7031: 7017: 7012: 6978: 6974: 6972: 6969: 6968: 6948: 6942: 6939: 6938: 6907: 6903: 6895: 6891: 6884: 6872: 6869: 6868: 6849: 6846: 6845: 6812: 6808: 6806: 6803: 6802: 6783: 6779: 6771: 6768: 6767: 6738: 6734: 6710: 6706: 6695: 6692: 6691: 6672: 6668: 6660: 6657: 6656: 6634: 6631: 6630: 6614: 6611: 6610: 6594: 6591: 6590: 6573: 6569: 6567: 6564: 6563: 6543: 6537: 6534: 6533: 6517: 6514: 6513: 6510: 6487: 6486: 6477: 6473: 6462: 6458: 6457: 6453: 6451: 6446: 6437: 6433: 6422: 6418: 6417: 6413: 6411: 6402: 6398: 6387: 6383: 6382: 6378: 6375: 6374: 6369: 6364: 6359: 6353: 6352: 6343: 6339: 6328: 6324: 6323: 6319: 6317: 6312: 6303: 6299: 6288: 6284: 6283: 6279: 6277: 6268: 6264: 6253: 6249: 6248: 6244: 6241: 6240: 6231: 6227: 6216: 6212: 6211: 6207: 6205: 6200: 6191: 6187: 6176: 6172: 6171: 6167: 6165: 6156: 6152: 6141: 6137: 6136: 6132: 6128: 6124: 6109: 6097: 6093: 6078: 6074: 6059: 6052: 6048: 6039: 6035: 6034: 6028: 6025: 6024: 5968: 5964: 5957: 5953: 5944: 5940: 5933: 5929: 5923: 5919: 5912: 5908: 5902: 5888: 5884: 5869: 5865: 5856: 5852: 5836: 5829: 5822: 5818: 5809: 5805: 5799: 5795: 5794: 5789: 5785: 5784: 5766: 5763: 5762: 5728: 5724: 5717: 5703: 5699: 5684: 5680: 5671: 5667: 5651: 5644: 5637: 5633: 5624: 5620: 5614: 5610: 5609: 5604: 5600: 5599: 5590: 5587: 5586: 5581: 5572: 5565: 5558: 5554: 5532: 5531: 5519: 5515: 5506: 5502: 5484: 5477: 5473: 5464: 5460: 5459: 5437: 5433: 5424: 5420: 5402: 5395: 5391: 5382: 5378: 5377: 5370: 5369: 5357: 5353: 5344: 5340: 5322: 5315: 5311: 5302: 5298: 5297: 5278: 5274: 5265: 5261: 5243: 5236: 5232: 5223: 5219: 5218: 5210: 5208: 5205: 5204: 5175: 5151: 5147: 5145: 5142: 5141: 5121: 5120: 5111: 5107: 5089: 5085: 5073: 5069: 5051: 5047: 5038: 5034: 5016: 5012: 4992: 4986: 4967: 4958: 4957: 4942: 4938: 4929: 4925: 4919: 4915: 4910: 4898: 4894: 4885: 4881: 4875: 4871: 4861: 4852: 4848: 4833: 4829: 4820: 4816: 4801: 4794: 4790: 4781: 4777: 4771: 4767: 4766: 4759: 4758: 4749: 4745: 4727: 4723: 4711: 4707: 4689: 4685: 4676: 4672: 4654: 4650: 4644: 4629: 4619: 4615: 4609: 4604: 4601: 4592: 4591: 4576: 4572: 4563: 4559: 4553: 4549: 4544: 4532: 4528: 4519: 4515: 4509: 4505: 4495: 4486: 4482: 4467: 4463: 4454: 4450: 4435: 4428: 4424: 4415: 4411: 4405: 4401: 4400: 4392: 4390: 4387: 4386: 4362: 4361: 4342: 4338: 4334: 4312: 4308: 4304: 4285: 4281: 4277: 4257: 4251: 4232: 4225: 4210: 4206: 4197: 4193: 4187: 4183: 4178: 4175: 4174: 4155: 4151: 4147: 4125: 4121: 4117: 4098: 4094: 4090: 4084: 4069: 4059: 4055: 4049: 4044: 4041: 4034: 4019: 4015: 4006: 4002: 3996: 3992: 3987: 3983: 3981: 3978: 3977: 3949: 3934: 3930: 3916: 3911: 3903: 3900: 3899: 3865: 3861: 3854: 3849: 3848: 3837: 3826: 3824: 3821: 3820: 3784: 3776: 3770: 3732: 3728: 3713: 3709: 3700: 3696: 3681: 3677: 3664: 3660: 3656: 3648: 3644: 3629: 3625: 3616: 3612: 3611: 3605: 3602: 3601: 3578: 3572: 3571: 3570: 3557: 3545: 3541: 3532: 3528: 3522: 3513: 3501: 3497: 3488: 3484: 3483: 3479: 3473: 3472: 3458: 3454: 3439: 3435: 3426: 3422: 3407: 3403: 3402: 3398: 3388: 3384: 3380: 3378: 3375: 3374: 3370:measurement is 3342: 3330: 3326: 3317: 3313: 3307: 3303: 3298: 3296: 3293: 3292: 3287: 3280: 3246: 3234: 3230: 3221: 3217: 3211: 3207: 3202: 3200: 3197: 3196: 3186: 3149: 3145: 3136: 3132: 3126: 3122: 3117: 3105: 3101: 3092: 3088: 3082: 3078: 3050: 3046: 3037: 3033: 3027: 3023: 3018: 3006: 3002: 2993: 2989: 2983: 2979: 2974: 2971: 2970: 2942: 2938: 2932: 2928: 2926: 2923: 2922: 2903: 2900: 2899: 2883: 2880: 2879: 2860: 2857: 2856: 2844: 2808: 2804: 2800: 2778: 2774: 2770: 2751: 2747: 2743: 2722: 2703: 2685: 2681: 2672: 2668: 2662: 2658: 2653: 2651: 2648: 2647: 2630: 2622: 2616: 2611:appears in the 2605: 2600:. The quantity 2553: 2549: 2545: 2523: 2519: 2515: 2496: 2492: 2488: 2482: 2467: 2457: 2453: 2447: 2442: 2439: 2421: 2417: 2408: 2404: 2398: 2394: 2389: 2387: 2384: 2383: 2370: 2361: 2354: 2338: 2241: 2184: 2180: 2178: 2175: 2174: 2145: 2141: 2123: 2119: 2102: 2096: 2092: 2083: 2079: 2074: 2055: 2049: 2044: 2038: 2025: 2019: 2014: 2008: 2000: 1997: 1996: 1951: 1947: 1938: 1934: 1929: 1908: 1904: 1895: 1891: 1886: 1881: 1878: 1877: 1850: 1846: 1837: 1833: 1828: 1807: 1803: 1794: 1790: 1785: 1780: 1777: 1776: 1738: 1734: 1732: 1729: 1728: 1685: 1674: 1665: 1664: 1653: 1642: 1636: 1635: 1630: 1627: 1626: 1591: 1590: 1581: 1577: 1572: 1563: 1559: 1554: 1539: 1535: 1530: 1521: 1517: 1512: 1506: 1505: 1495: 1477: 1473: 1464: 1460: 1455: 1453: 1450: 1449: 1442: 1422: 1415: 1390: 1389: 1380: 1376: 1371: 1362: 1358: 1353: 1341: 1337: 1332: 1323: 1319: 1314: 1308: 1307: 1297: 1279: 1275: 1266: 1262: 1257: 1255: 1252: 1251: 1228: 1227: 1218: 1214: 1209: 1200: 1196: 1191: 1179: 1175: 1170: 1161: 1157: 1152: 1146: 1145: 1135: 1117: 1113: 1104: 1100: 1095: 1093: 1090: 1089: 1055: 1051: 1046: 1037: 1033: 1028: 1016: 1012: 1007: 998: 994: 989: 987: 984: 983: 973: 966: 959: 952: 923: 919: 914: 905: 901: 896: 894: 891: 890: 870: 866: 861: 852: 848: 843: 841: 838: 837: 815: 812: 811: 805: 798: 773: 769: 764: 755: 751: 746: 744: 741: 740: 717: 713: 708: 699: 695: 690: 688: 685: 684: 679: 672: 625: 620: 596:electric charge 588:electric charge 580: 530:nuclei and all 428: 399: 398: 344: 336: 335: 311:Internal energy 306: 296: 295: 271: 263: 262: 242:Grand canonical 218: 208: 207: 163: 117: 106: 100: 97: 54: 52: 42: 30: 17: 12: 11: 5: 9414: 9404: 9403: 9398: 9393: 9379: 9378: 9365: 9355: 9349: 9342: 9341:External links 9339: 9338: 9337: 9333:978-0198525264 9332: 9326:, OUP Oxford, 9317: 9314: 9311: 9310: 9283:(5): 515–520. 9265: 9246:(4): 929–976. 9226: 9220:978-0805387148 9219: 9201: 9199:, p. 385) 9189: 9176: 9118: 9117: 9115: 9112: 9111: 9110: 9105: 9098: 9095: 9077: 9073: 9069: 9066: 8962: 8957: 8952: 8947: 8944: 8856: 8818: 8813: 8774: 8769: 8722:homotopy class 8629: 8628:Homotopy class 8626: 8612: 8611: 8608: 8605: 8602: 8598: 8597: 8594: 8591: 8588: 8584: 8583: 8580: 8577: 8574: 8570: 8569: 8566: 8563: 8560: 8536: 8533: 8529: 8508: 8505: 8501: 8480: 8477: 8474: 8470: 8466: 8463: 8459: 8455: 8452: 8449: 8445: 8441: 8438: 8434: 8430: 8424: 8420: 8395: 8392: 8388: 8367: 8364: 8360: 8339: 8336: 8332: 8311: 8308: 8304: 8283: 8280: 8277: 8273: 8269: 8266: 8262: 8258: 8255: 8252: 8248: 8244: 8241: 8237: 8233: 8227: 8223: 8201: 8198: 8194: 8190: 8187: 8183: 8162: 8159: 8155: 8151: 8148: 8144: 8131:state is 0.5. 8120: 8117: 8113: 8092: 8089: 8085: 8064: 8061: 8057: 8036: 8033: 8029: 8008: 8005: 8001: 7997: 7994: 7990: 7969: 7966: 7962: 7958: 7955: 7951: 7930: 7927: 7923: 7919: 7916: 7912: 7891: 7888: 7884: 7880: 7877: 7873: 7845: 7842: 7838: 7817: 7814: 7810: 7786: 7783: 7779: 7758: 7755: 7751: 7706: 7703: 7699: 7698: 7687: 7684: 7681: 7678: 7675: 7672: 7668: 7663: 7660: 7655: 7651: 7648: 7645: 7642: 7639: 7636: 7607:thermodynamics 7567: 7566: 7555: 7552: 7549: 7546: 7543: 7540: 7536: 7533: 7530: 7526: 7523: 7520: 7517: 7514: 7511: 7469: 7468: 7457: 7451: 7448: 7442: 7438: 7432: 7429: 7400: 7399: 7388: 7384: 7377: 7374: 7369: 7366: 7363: 7360: 7354: 7350: 7346: 7343: 7338: 7334: 7330: 7327: 7313: 7312: 7299: 7295: 7291: 7288: 7254: 7253: 7241: 7236: 7232: 7227: 7223: 7219: 7216: 7213: 7210: 7207: 7204: 7199: 7195: 7191: 7188: 7185: 7182: 7177: 7173: 7169: 7166: 7162: 7155: 7152: 7148: 7143: 7139: 7135: 7132: 7125: 7121: 7117: 7114: 7111: 7106: 7102: 7098: 7093: 7089: 7084: 7080: 7077: 7040: 7027: 7016: 7013: 7011: 7008: 7004:Young tableaux 6981: 6977: 6956: 6951: 6947: 6931: 6930: 6918: 6915: 6910: 6906: 6898: 6894: 6890: 6887: 6883: 6879: 6876: 6853: 6842: 6841: 6829: 6826: 6823: 6820: 6815: 6811: 6786: 6782: 6778: 6775: 6764: 6763: 6752: 6749: 6746: 6741: 6737: 6733: 6730: 6727: 6724: 6721: 6718: 6713: 6709: 6705: 6702: 6699: 6675: 6671: 6667: 6664: 6644: 6641: 6638: 6618: 6598: 6576: 6572: 6551: 6546: 6542: 6521: 6509: 6506: 6505: 6504: 6492: 6485: 6480: 6476: 6472: 6465: 6461: 6456: 6452: 6450: 6447: 6445: 6440: 6436: 6432: 6425: 6421: 6416: 6412: 6410: 6405: 6401: 6397: 6390: 6386: 6381: 6377: 6376: 6373: 6370: 6368: 6365: 6363: 6360: 6358: 6355: 6354: 6351: 6346: 6342: 6338: 6331: 6327: 6322: 6318: 6316: 6313: 6311: 6306: 6302: 6298: 6291: 6287: 6282: 6278: 6276: 6271: 6267: 6263: 6256: 6252: 6247: 6243: 6242: 6239: 6234: 6230: 6226: 6219: 6215: 6210: 6206: 6204: 6201: 6199: 6194: 6190: 6186: 6179: 6175: 6170: 6166: 6164: 6159: 6155: 6151: 6144: 6140: 6135: 6131: 6130: 6127: 6120: 6117: 6113: 6108: 6105: 6100: 6096: 6092: 6089: 6086: 6081: 6077: 6073: 6068: 6065: 6062: 6055: 6051: 6047: 6042: 6038: 6033: 5991: 5990: 5979: 5976: 5971: 5967: 5960: 5956: 5952: 5947: 5943: 5936: 5932: 5926: 5922: 5915: 5911: 5905: 5900: 5896: 5891: 5887: 5883: 5880: 5877: 5872: 5868: 5864: 5859: 5855: 5851: 5846: 5843: 5839: 5835: 5832: 5825: 5821: 5817: 5812: 5808: 5802: 5798: 5793: 5788: 5782: 5778: 5774: 5770: 5754:The factor of 5752: 5751: 5740: 5734: 5731: 5727: 5720: 5715: 5711: 5706: 5702: 5698: 5695: 5692: 5687: 5683: 5679: 5674: 5670: 5666: 5661: 5658: 5654: 5650: 5647: 5640: 5636: 5632: 5627: 5623: 5617: 5613: 5608: 5603: 5597: 5594: 5577: 5570: 5563: 5556: 5552: 5546: 5545: 5530: 5527: 5522: 5518: 5514: 5509: 5505: 5501: 5498: 5493: 5490: 5487: 5480: 5476: 5472: 5467: 5463: 5458: 5454: 5451: 5448: 5445: 5440: 5436: 5432: 5427: 5423: 5419: 5416: 5411: 5408: 5405: 5398: 5394: 5390: 5385: 5381: 5376: 5372: 5371: 5368: 5365: 5360: 5356: 5352: 5347: 5343: 5339: 5336: 5331: 5328: 5325: 5318: 5314: 5310: 5305: 5301: 5296: 5292: 5289: 5286: 5281: 5277: 5273: 5268: 5264: 5260: 5257: 5252: 5249: 5246: 5239: 5235: 5231: 5226: 5222: 5217: 5213: 5212: 5197: 5196: 5185: 5182: 5178: 5174: 5171: 5168: 5165: 5162: 5159: 5154: 5150: 5135: 5134: 5119: 5114: 5110: 5106: 5101: 5098: 5095: 5092: 5088: 5084: 5081: 5076: 5072: 5068: 5063: 5060: 5057: 5054: 5050: 5046: 5041: 5037: 5033: 5028: 5025: 5022: 5019: 5015: 5011: 5008: 5005: 5001: 4998: 4995: 4989: 4985: 4978: 4975: 4971: 4966: 4963: 4961: 4959: 4956: 4953: 4950: 4945: 4941: 4937: 4932: 4928: 4922: 4918: 4913: 4909: 4906: 4901: 4897: 4893: 4888: 4884: 4878: 4874: 4870: 4867: 4864: 4862: 4860: 4855: 4851: 4847: 4844: 4841: 4836: 4832: 4828: 4823: 4819: 4815: 4810: 4807: 4804: 4797: 4793: 4789: 4784: 4780: 4774: 4770: 4765: 4761: 4760: 4757: 4752: 4748: 4744: 4739: 4736: 4733: 4730: 4726: 4722: 4719: 4714: 4710: 4706: 4701: 4698: 4695: 4692: 4688: 4684: 4679: 4675: 4671: 4666: 4663: 4660: 4657: 4653: 4647: 4643: 4635: 4632: 4627: 4622: 4618: 4612: 4608: 4600: 4597: 4595: 4593: 4590: 4587: 4584: 4579: 4575: 4571: 4566: 4562: 4556: 4552: 4547: 4543: 4540: 4535: 4531: 4527: 4522: 4518: 4512: 4508: 4504: 4501: 4498: 4496: 4494: 4489: 4485: 4481: 4478: 4475: 4470: 4466: 4462: 4457: 4453: 4449: 4444: 4441: 4438: 4431: 4427: 4423: 4418: 4414: 4408: 4404: 4399: 4395: 4394: 4376: 4375: 4359: 4354: 4351: 4348: 4345: 4341: 4337: 4333: 4329: 4324: 4321: 4318: 4315: 4311: 4307: 4302: 4297: 4294: 4291: 4288: 4284: 4280: 4276: 4273: 4270: 4266: 4263: 4260: 4254: 4250: 4243: 4240: 4236: 4231: 4228: 4226: 4224: 4221: 4218: 4213: 4209: 4205: 4200: 4196: 4190: 4186: 4181: 4177: 4176: 4172: 4167: 4164: 4161: 4158: 4154: 4150: 4146: 4142: 4137: 4134: 4131: 4128: 4124: 4120: 4115: 4110: 4107: 4104: 4101: 4097: 4093: 4087: 4083: 4075: 4072: 4067: 4062: 4058: 4052: 4048: 4040: 4037: 4035: 4033: 4030: 4027: 4022: 4018: 4014: 4009: 4005: 3999: 3995: 3990: 3986: 3985: 3971: 3970: 3959: 3955: 3952: 3948: 3945: 3942: 3937: 3933: 3929: 3926: 3922: 3919: 3914: 3910: 3907: 3893:delta function 3885: 3884: 3873: 3868: 3864: 3857: 3852: 3847: 3844: 3840: 3836: 3833: 3829: 3783: 3780: 3774: 3768: 3758: 3757: 3746: 3743: 3740: 3735: 3731: 3727: 3724: 3721: 3716: 3712: 3708: 3703: 3699: 3695: 3692: 3689: 3684: 3680: 3676: 3671: 3667: 3663: 3659: 3651: 3647: 3643: 3640: 3637: 3632: 3628: 3624: 3619: 3615: 3610: 3595: 3594: 3581: 3575: 3568: 3564: 3560: 3556: 3553: 3548: 3544: 3540: 3535: 3531: 3525: 3520: 3516: 3512: 3509: 3504: 3500: 3496: 3491: 3487: 3482: 3476: 3471: 3467: 3461: 3457: 3453: 3450: 3447: 3442: 3438: 3434: 3429: 3425: 3421: 3418: 3415: 3410: 3406: 3401: 3395: 3391: 3387: 3383: 3364: 3363: 3352: 3349: 3345: 3341: 3338: 3333: 3329: 3325: 3320: 3316: 3310: 3306: 3301: 3285: 3278: 3268: 3267: 3256: 3253: 3249: 3245: 3242: 3237: 3233: 3229: 3224: 3220: 3214: 3210: 3205: 3185: 3182: 3181: 3180: 3169: 3166: 3163: 3160: 3157: 3152: 3148: 3144: 3139: 3135: 3129: 3125: 3120: 3116: 3113: 3108: 3104: 3100: 3095: 3091: 3085: 3081: 3077: 3073: 3070: 3067: 3064: 3061: 3058: 3053: 3049: 3045: 3040: 3036: 3030: 3026: 3021: 3017: 3014: 3009: 3005: 3001: 2996: 2992: 2986: 2982: 2978: 2945: 2941: 2935: 2931: 2910: 2907: 2887: 2867: 2864: 2841: 2840: 2825: 2820: 2817: 2814: 2811: 2807: 2803: 2799: 2795: 2790: 2787: 2784: 2781: 2777: 2773: 2768: 2763: 2760: 2757: 2754: 2750: 2746: 2742: 2739: 2736: 2733: 2730: 2725: 2721: 2714: 2711: 2707: 2702: 2699: 2696: 2693: 2688: 2684: 2680: 2675: 2671: 2665: 2661: 2656: 2626: 2618: 2603: 2583: 2582: 2570: 2565: 2562: 2559: 2556: 2552: 2548: 2544: 2540: 2535: 2532: 2529: 2526: 2522: 2518: 2513: 2508: 2505: 2502: 2499: 2495: 2491: 2485: 2481: 2473: 2470: 2465: 2460: 2456: 2450: 2446: 2438: 2435: 2432: 2429: 2424: 2420: 2416: 2411: 2407: 2401: 2397: 2392: 2366: 2359: 2352: 2337: 2331: 2289:Parastatistics 2240: 2237: 2216: 2215: 2204: 2201: 2197: 2193: 2190: 2187: 2183: 2165: 2164: 2153: 2148: 2144: 2140: 2137: 2134: 2131: 2126: 2122: 2118: 2115: 2112: 2109: 2105: 2099: 2095: 2091: 2086: 2082: 2077: 2073: 2070: 2067: 2061: 2058: 2052: 2047: 2043: 2037: 2031: 2028: 2022: 2017: 2013: 2007: 2004: 1977: 1976: 1965: 1962: 1959: 1954: 1950: 1946: 1941: 1937: 1932: 1928: 1925: 1922: 1919: 1916: 1911: 1907: 1903: 1898: 1894: 1889: 1885: 1875: 1864: 1861: 1858: 1853: 1849: 1845: 1840: 1836: 1831: 1827: 1824: 1821: 1818: 1815: 1810: 1806: 1802: 1797: 1793: 1788: 1784: 1749: 1746: 1741: 1737: 1707: 1706: 1695: 1692: 1688: 1684: 1681: 1677: 1673: 1668: 1663: 1660: 1656: 1652: 1649: 1645: 1639: 1634: 1608: 1607: 1594: 1589: 1584: 1580: 1575: 1571: 1566: 1562: 1557: 1553: 1550: 1547: 1542: 1538: 1533: 1529: 1524: 1520: 1515: 1509: 1504: 1494: 1491: 1488: 1485: 1480: 1476: 1472: 1467: 1463: 1458: 1441: 1438: 1420: 1413: 1407: 1406: 1393: 1388: 1383: 1379: 1374: 1370: 1365: 1361: 1356: 1352: 1349: 1344: 1340: 1335: 1331: 1326: 1322: 1317: 1311: 1306: 1296: 1293: 1290: 1287: 1282: 1278: 1274: 1269: 1265: 1260: 1245: 1244: 1231: 1226: 1221: 1217: 1212: 1208: 1203: 1199: 1194: 1190: 1187: 1182: 1178: 1173: 1169: 1164: 1160: 1155: 1149: 1144: 1134: 1131: 1128: 1125: 1120: 1116: 1112: 1107: 1103: 1098: 1075: 1074: 1063: 1058: 1054: 1049: 1045: 1040: 1036: 1031: 1027: 1024: 1019: 1015: 1010: 1006: 1001: 997: 992: 976: 975: 971: 964: 957: 950: 931: 926: 922: 917: 913: 908: 904: 899: 878: 873: 869: 864: 860: 855: 851: 846: 825: 822: 819: 808:tensor product 803: 796: 781: 776: 772: 767: 763: 758: 754: 749: 737: 736: 725: 720: 716: 711: 707: 702: 698: 693: 677: 670: 658:problem, take 624: 621: 619: 616: 579: 576: 572:mixing paradox 504:quantum states 480:Quasiparticles 470:), as well as 430: 429: 427: 426: 419: 412: 404: 401: 400: 397: 396: 391: 386: 381: 376: 371: 366: 361: 356: 351: 345: 342: 341: 338: 337: 334: 333: 328: 323: 318: 313: 307: 302: 301: 298: 297: 294: 293: 288: 283: 278: 272: 269: 268: 265: 264: 261: 260: 252: 244: 236: 228: 226:Microcanonical 219: 214: 213: 210: 209: 206: 205: 200: 195: 193:Parastatistics 190: 185: 180: 175: 170: 164: 159: 158: 155: 154: 153: 152: 150:Kinetic theory 147: 145:Thermodynamics 139: 138: 130: 129: 119: 118: 101:September 2008 33: 31: 24: 15: 9: 6: 4: 3: 2: 9413: 9402: 9399: 9397: 9394: 9392: 9389: 9388: 9386: 9377: 9373: 9369: 9366: 9363: 9359: 9356: 9353: 9350: 9348: 9345: 9344: 9335: 9329: 9325: 9320: 9319: 9306: 9302: 9298: 9294: 9290: 9286: 9282: 9278: 9277: 9269: 9261: 9257: 9253: 9249: 9245: 9241: 9237: 9230: 9222: 9216: 9212: 9205: 9198: 9193: 9186: 9180: 9172: 9168: 9164: 9160: 9156: 9152: 9147: 9142: 9138: 9134: 9130: 9123: 9119: 9109: 9106: 9104: 9101: 9100: 9094: 9075: 9067: 9064: 9055: 9053: 9047: 9039: 9035: 9027: 9023: 9017: 9013: 9009: 9002:for any real 8999: 8992: 8986: 8982: 8960: 8955: 8945: 8942: 8933: 8931: 8925: 8921: 8913: 8909: 8901: 8897: 8883: 8877: 8871: 8839: 8836: 8816: 8793: 8772: 8745: 8741: 8733: 8729: 8723: 8717: 8713: 8705: 8701: 8693: 8689: 8673: 8669: 8658: 8654: 8639: 8635: 8625: 8623: 8619: 8600: 8599: 8586: 8585: 8572: 8571: 8567: 8564: 8561: 8558: 8557: 8551: 8548: 8531: 8503: 8472: 8461: 8453: 8447: 8436: 8422: 8418: 8407: 8390: 8362: 8334: 8306: 8275: 8264: 8256: 8250: 8239: 8225: 8221: 8196: 8185: 8157: 8146: 8132: 8115: 8087: 8059: 8031: 8003: 7992: 7964: 7953: 7925: 7914: 7886: 7875: 7861: 7859: 7840: 7812: 7798: 7781: 7753: 7738: 7736: 7732: 7728: 7727:superfluidity 7724: 7720: 7716: 7712: 7702: 7682: 7676: 7673: 7670: 7666: 7661: 7658: 7653: 7649: 7646: 7643: 7640: 7637: 7634: 7627: 7626: 7625: 7623: 7619: 7615: 7610: 7608: 7604: 7601:are doubled, 7600: 7596: 7592: 7588: 7584: 7580: 7576: 7572: 7550: 7544: 7541: 7538: 7534: 7531: 7528: 7524: 7521: 7518: 7515: 7512: 7509: 7502: 7501: 7500: 7498: 7494: 7490: 7486: 7482: 7478: 7477:Gibbs paradox 7472: 7455: 7449: 7446: 7440: 7436: 7430: 7427: 7420: 7419: 7418: 7416: 7411: 7409: 7405: 7386: 7382: 7375: 7372: 7364: 7358: 7352: 7348: 7344: 7341: 7336: 7332: 7328: 7325: 7318: 7317: 7316: 7297: 7293: 7289: 7286: 7279: 7278: 7277: 7275: 7271: 7267: 7263: 7259: 7239: 7234: 7225: 7221: 7214: 7211: 7208: 7205: 7197: 7193: 7186: 7183: 7175: 7171: 7164: 7160: 7153: 7150: 7146: 7141: 7137: 7133: 7130: 7123: 7119: 7115: 7112: 7109: 7104: 7100: 7096: 7091: 7087: 7082: 7078: 7075: 7068: 7067: 7066: 7064: 7060: 7056: 7053:) denote the 7052: 7048: 7043: 7039: 7035: 7030: 7026: 7022: 7007: 7005: 7001: 6995: 6979: 6975: 6954: 6949: 6945: 6936: 6913: 6908: 6904: 6896: 6892: 6888: 6885: 6881: 6877: 6867: 6866: 6865: 6851: 6827: 6824: 6821: 6818: 6813: 6809: 6801: 6800: 6799: 6784: 6780: 6776: 6773: 6750: 6744: 6739: 6731: 6722: 6716: 6711: 6700: 6690: 6689: 6688: 6673: 6669: 6665: 6662: 6642: 6639: 6636: 6616: 6596: 6574: 6570: 6549: 6544: 6540: 6519: 6490: 6478: 6474: 6463: 6459: 6454: 6448: 6438: 6434: 6423: 6419: 6414: 6403: 6399: 6388: 6384: 6379: 6371: 6366: 6361: 6356: 6344: 6340: 6329: 6325: 6320: 6314: 6304: 6300: 6289: 6285: 6280: 6269: 6265: 6254: 6250: 6245: 6232: 6228: 6217: 6213: 6208: 6202: 6192: 6188: 6177: 6173: 6168: 6157: 6153: 6142: 6138: 6133: 6125: 6118: 6115: 6111: 6106: 6098: 6094: 6090: 6087: 6084: 6079: 6075: 6063: 6053: 6049: 6045: 6040: 6036: 6023: 6022: 6021: 6019: 6016:, known as a 6015: 6011: 6006: 6004: 6000: 5996: 5977: 5974: 5969: 5965: 5958: 5954: 5950: 5945: 5941: 5934: 5930: 5924: 5920: 5913: 5909: 5903: 5898: 5889: 5885: 5881: 5878: 5875: 5870: 5866: 5862: 5857: 5853: 5841: 5837: 5833: 5823: 5819: 5815: 5810: 5806: 5800: 5796: 5786: 5780: 5776: 5772: 5768: 5761: 5760: 5759: 5757: 5738: 5732: 5729: 5725: 5718: 5713: 5704: 5700: 5696: 5693: 5690: 5685: 5681: 5677: 5672: 5668: 5656: 5652: 5648: 5638: 5634: 5630: 5625: 5621: 5615: 5611: 5601: 5595: 5592: 5585: 5584: 5583: 5580: 5576: 5569: 5562: 5551: 5525: 5520: 5516: 5512: 5507: 5503: 5499: 5488: 5478: 5474: 5470: 5465: 5461: 5452: 5449: 5443: 5438: 5434: 5430: 5425: 5421: 5417: 5406: 5396: 5392: 5388: 5383: 5379: 5363: 5358: 5354: 5350: 5345: 5341: 5337: 5326: 5316: 5312: 5308: 5303: 5299: 5290: 5284: 5279: 5275: 5271: 5266: 5262: 5258: 5247: 5237: 5233: 5229: 5224: 5220: 5203: 5202: 5201: 5180: 5172: 5166: 5160: 5152: 5148: 5140: 5139: 5138: 5112: 5108: 5096: 5090: 5086: 5082: 5074: 5070: 5058: 5052: 5048: 5039: 5035: 5023: 5017: 5013: 5006: 4987: 4983: 4976: 4973: 4969: 4964: 4962: 4951: 4948: 4943: 4939: 4935: 4930: 4926: 4920: 4916: 4907: 4904: 4899: 4895: 4891: 4886: 4882: 4876: 4872: 4865: 4863: 4853: 4849: 4845: 4842: 4839: 4834: 4830: 4826: 4821: 4817: 4805: 4795: 4791: 4787: 4782: 4778: 4772: 4768: 4750: 4746: 4734: 4728: 4724: 4720: 4712: 4708: 4696: 4690: 4686: 4677: 4673: 4661: 4655: 4651: 4645: 4641: 4633: 4630: 4625: 4620: 4616: 4610: 4606: 4598: 4596: 4585: 4582: 4577: 4573: 4569: 4564: 4560: 4554: 4550: 4541: 4538: 4533: 4529: 4525: 4520: 4516: 4510: 4506: 4499: 4497: 4487: 4483: 4479: 4476: 4473: 4468: 4464: 4460: 4455: 4451: 4439: 4429: 4425: 4421: 4416: 4412: 4406: 4402: 4385: 4384: 4383: 4381: 4357: 4349: 4343: 4339: 4335: 4331: 4327: 4319: 4313: 4309: 4305: 4300: 4292: 4286: 4282: 4278: 4271: 4252: 4248: 4241: 4238: 4234: 4229: 4227: 4219: 4216: 4211: 4207: 4203: 4198: 4194: 4188: 4184: 4170: 4162: 4156: 4152: 4148: 4144: 4140: 4132: 4126: 4122: 4118: 4113: 4105: 4099: 4095: 4091: 4085: 4081: 4073: 4070: 4065: 4060: 4056: 4050: 4046: 4038: 4036: 4028: 4025: 4020: 4016: 4012: 4007: 4003: 3997: 3993: 3976: 3975: 3974: 3953: 3950: 3946: 3943: 3935: 3931: 3927: 3920: 3917: 3908: 3898: 3897: 3896: 3894: 3890: 3871: 3866: 3862: 3855: 3842: 3834: 3819: 3818: 3817: 3815: 3811: 3808: 3804: 3800: 3795: 3793: 3789: 3779: 3777: 3767: 3763: 3744: 3741: 3733: 3729: 3725: 3722: 3719: 3714: 3710: 3701: 3697: 3693: 3690: 3687: 3682: 3678: 3669: 3665: 3661: 3657: 3649: 3645: 3641: 3638: 3635: 3630: 3626: 3622: 3617: 3613: 3608: 3600: 3599: 3598: 3579: 3566: 3562: 3558: 3554: 3551: 3546: 3542: 3538: 3533: 3529: 3518: 3514: 3510: 3507: 3502: 3498: 3494: 3489: 3485: 3480: 3469: 3465: 3459: 3455: 3451: 3448: 3445: 3440: 3436: 3427: 3423: 3419: 3416: 3413: 3408: 3404: 3399: 3393: 3389: 3385: 3381: 3373: 3372: 3371: 3369: 3347: 3343: 3339: 3336: 3331: 3327: 3323: 3318: 3314: 3308: 3304: 3291: 3290: 3289: 3284: 3277: 3273: 3251: 3247: 3243: 3240: 3235: 3231: 3227: 3222: 3218: 3212: 3208: 3195: 3194: 3193: 3191: 3167: 3164: 3158: 3155: 3150: 3146: 3142: 3137: 3133: 3127: 3123: 3114: 3111: 3106: 3102: 3098: 3093: 3089: 3083: 3079: 3071: 3068: 3065: 3059: 3056: 3051: 3047: 3043: 3038: 3034: 3028: 3024: 3015: 3012: 3007: 3003: 2999: 2994: 2990: 2984: 2980: 2969: 2968: 2967: 2964: 2962: 2943: 2939: 2933: 2908: 2905: 2885: 2865: 2862: 2854: 2848: 2823: 2815: 2809: 2805: 2801: 2797: 2793: 2785: 2779: 2775: 2771: 2766: 2758: 2752: 2748: 2744: 2737: 2731: 2728: 2723: 2719: 2712: 2709: 2705: 2700: 2694: 2691: 2686: 2682: 2678: 2673: 2669: 2663: 2659: 2646: 2645: 2644: 2642: 2637: 2634: 2629: 2625: 2621: 2614: 2610: 2606: 2599: 2595: 2591: 2588: 2568: 2560: 2554: 2550: 2546: 2542: 2538: 2530: 2524: 2520: 2516: 2511: 2503: 2497: 2493: 2489: 2483: 2479: 2471: 2468: 2463: 2458: 2454: 2448: 2444: 2436: 2430: 2427: 2422: 2418: 2414: 2409: 2405: 2399: 2395: 2382: 2381: 2380: 2378: 2374: 2369: 2365: 2358: 2351: 2347: 2343: 2335: 2330: 2328: 2324: 2319: 2317: 2313: 2309: 2305: 2301: 2297: 2292: 2290: 2286: 2284: 2280: 2276: 2271: 2269: 2265: 2260: 2258: 2254: 2250: 2246: 2236: 2234: 2230: 2225: 2221: 2202: 2199: 2195: 2191: 2188: 2185: 2181: 2173: 2172: 2171: 2170: 2146: 2142: 2135: 2132: 2124: 2120: 2113: 2110: 2097: 2093: 2089: 2084: 2080: 2068: 2065: 2059: 2056: 2050: 2045: 2041: 2035: 2029: 2026: 2020: 2015: 2011: 2005: 2002: 1995: 1994: 1993: 1991: 1986: 1981: 1960: 1957: 1952: 1948: 1944: 1939: 1935: 1926: 1923: 1917: 1914: 1909: 1905: 1901: 1896: 1892: 1883: 1876: 1859: 1856: 1851: 1847: 1843: 1838: 1834: 1825: 1822: 1816: 1813: 1808: 1804: 1800: 1795: 1791: 1782: 1775: 1774: 1773: 1771: 1767: 1763: 1747: 1744: 1739: 1735: 1725: 1723: 1719: 1715: 1711: 1690: 1679: 1671: 1658: 1647: 1632: 1625: 1624: 1623: 1621: 1616: 1614: 1582: 1578: 1564: 1560: 1551: 1548: 1540: 1536: 1522: 1518: 1502: 1492: 1486: 1483: 1478: 1474: 1470: 1465: 1461: 1448: 1447: 1446: 1437: 1435: 1431: 1427: 1419: 1412: 1409:Note that if 1381: 1377: 1363: 1359: 1350: 1342: 1338: 1324: 1320: 1304: 1294: 1288: 1285: 1280: 1276: 1272: 1267: 1263: 1250: 1249: 1248: 1219: 1215: 1201: 1197: 1188: 1180: 1176: 1162: 1158: 1142: 1132: 1126: 1123: 1118: 1114: 1110: 1105: 1101: 1088: 1087: 1086: 1084: 1083:antisymmetric 1080: 1056: 1052: 1038: 1034: 1025: 1017: 1013: 999: 995: 982: 981: 980: 970: 963: 956: 949: 945: 944: 943: 924: 920: 906: 902: 871: 867: 853: 849: 823: 820: 817: 809: 802: 795: 774: 770: 756: 752: 718: 714: 700: 696: 683: 682: 681: 676: 669: 665: 661: 657: 653: 648: 646: 637: 629: 615: 612: 611:wavefunctions 608: 603: 599: 597: 593: 589: 585: 575: 573: 568: 567:probabilistic 564: 559: 557: 553: 549: 545: 541: 537: 533: 529: 525: 521: 517: 513: 509: 505: 501: 496: 494: 490: 486: 481: 477: 473: 469: 468:atomic nuclei 465: 462:), composite 461: 457: 453: 449: 445: 442:(also called 441: 437: 425: 420: 418: 413: 411: 406: 405: 403: 402: 395: 392: 390: 387: 385: 382: 380: 377: 375: 372: 370: 367: 365: 362: 360: 357: 355: 352: 350: 347: 346: 340: 339: 332: 329: 327: 324: 322: 319: 317: 314: 312: 309: 308: 305: 300: 299: 292: 289: 287: 284: 282: 279: 277: 274: 273: 267: 266: 259: 256: 253: 251: 248: 245: 243: 240: 237: 235: 232: 229: 227: 224: 221: 220: 217: 212: 211: 204: 201: 199: 196: 194: 191: 189: 186: 184: 183:Bose–Einstein 181: 179: 176: 174: 171: 169: 166: 165: 162: 157: 156: 151: 148: 146: 143: 142: 141: 140: 136: 132: 131: 128: 125: 124: 115: 112: 104: 93: 90: 86: 83: 79: 76: 72: 69: 65: 62: – 61: 57: 56:Find sources: 50: 46: 40: 39: 34:This article 32: 28: 23: 22: 19: 9323: 9280: 9274: 9268: 9243: 9239: 9229: 9210: 9204: 9192: 9179: 9139:(3): 2, 10. 9136: 9132: 9122: 9056: 9045: 9037: 9033: 9025: 9021: 9015: 8997: 8991:cyclic group 8984: 8980: 8935:In the case 8934: 8923: 8919: 8911: 8907: 8899: 8895: 8875: 8872: 8834: 8791: 8743: 8739: 8731: 8727: 8715: 8711: 8703: 8699: 8691: 8687: 8671: 8667: 8656: 8652: 8641: 8615: 8549: 8408: 8133: 7862: 7799: 7739: 7708: 7700: 7621: 7617: 7613: 7611: 7602: 7598: 7594: 7586: 7582: 7578: 7570: 7568: 7488: 7484: 7473: 7470: 7414: 7412: 7407: 7403: 7401: 7314: 7265: 7257: 7255: 7058: 7050: 7046: 7041: 7037: 7033: 7028: 7024: 7020: 7018: 6996: 6932: 6843: 6765: 6511: 6007: 6002: 5998: 5994: 5992: 5755: 5753: 5578: 5574: 5567: 5560: 5549: 5547: 5198: 5136: 4380:wavefunction 4378:A many-body 4377: 3972: 3888: 3886: 3813: 3809: 3806: 3802: 3798: 3796: 3791: 3785: 3772: 3765: 3761: 3759: 3596: 3367: 3365: 3282: 3275: 3271: 3269: 3189: 3187: 2965: 2846: 2842: 2640: 2638: 2632: 2627: 2623: 2619: 2612: 2608: 2601: 2593: 2589: 2587:permutations 2584: 2376: 2372: 2367: 2363: 2356: 2349: 2345: 2341: 2339: 2333: 2320: 2293: 2287: 2272: 2261: 2242: 2228: 2223: 2217: 2166: 1984: 1982: 1978: 1770:eigenvectors 1765: 1726: 1709: 1708: 1619: 1617: 1612: 1609: 1443: 1417: 1410: 1408: 1246: 1082: 1078: 1076: 977: 968: 961: 954: 947: 800: 793: 738: 674: 667: 659: 651: 649: 642: 604: 600: 581: 560: 497: 447: 443: 439: 433: 254: 246: 238: 230: 222: 172: 107: 98: 88: 81: 74: 67: 55: 43:Please help 38:verification 35: 18: 9183:Haynes, P. 8916:itself and 7270:temperature 6010:determinant 3184:Measurement 1990:Hamiltonian 1762:eigenvalues 664:wave vector 379:von Neumann 188:Fermi–Dirac 9385:Categories 9316:References 9088:the space 8930:involution 8632:See also: 7276:to obtain 3764:values of 2592:acting on 2233:Fock space 493:Paul Dirac 343:Scientists 304:Potentials 71:newspapers 9305:250835341 9260:0007-0882 9146:1006.4610 9114:Footnotes 9008:unitarity 8904:, namely 8838:homotopic 8559:Particles 8535:⟩ 8507:⟩ 8476:⟩ 8465:⟩ 8454:− 8451:⟩ 8440:⟩ 8394:⟩ 8366:⟩ 8338:⟩ 8310:⟩ 8279:⟩ 8268:⟩ 8254:⟩ 8243:⟩ 8200:⟩ 8189:⟩ 8161:⟩ 8150:⟩ 8119:⟩ 8091:⟩ 8063:⟩ 8035:⟩ 8007:⟩ 7996:⟩ 7968:⟩ 7957:⟩ 7929:⟩ 7918:⟩ 7890:⟩ 7879:⟩ 7844:⟩ 7816:⟩ 7785:⟩ 7757:⟩ 7731:Fermi gas 7650:⁡ 7591:extensive 7525:⁡ 7497:ideal gas 7437:ξ 7359:ε 7353:− 7345:⁡ 7333:∑ 7326:ξ 7294:ξ 7215:ε 7209:⋯ 7187:ε 7165:ε 7142:− 7134:⁡ 7113:… 7083:∑ 6946:⨂ 6917:Ψ 6914:σ 6909:σ 6905:λ 6889:∈ 6886:σ 6882:∑ 6878:∼ 6875:Ψ 6822:σ 6810:σ 6777:∈ 6774:σ 6748:Ψ 6736:Ψ 6726:Ψ 6723:σ 6704:Ψ 6701:σ 6666:∈ 6663:σ 6640:∈ 6637:ψ 6541:⨂ 6455:ψ 6449:⋯ 6415:ψ 6380:ψ 6372:⋮ 6367:⋱ 6362:⋮ 6357:⋮ 6321:ψ 6315:⋯ 6281:ψ 6246:ψ 6209:ψ 6203:⋯ 6169:ψ 6134:ψ 6088:… 6046:⋯ 6032:Ψ 5951:⋯ 5879:… 5816:⋯ 5792:Ψ 5781:∫ 5777:⋯ 5773:∫ 5769:∫ 5694:… 5631:⋯ 5607:Ψ 5526:⋯ 5513:⋯ 5500:⋯ 5471:⋯ 5457:Ψ 5453:− 5444:⋯ 5431:⋯ 5418:⋯ 5389:⋯ 5375:Ψ 5364:⋯ 5351:⋯ 5338:⋯ 5309:⋯ 5295:Ψ 5285:⋯ 5272:⋯ 5259:⋯ 5230:⋯ 5216:Ψ 5184:⟩ 5170:⟨ 5167:≡ 5149:ψ 5087:ψ 5083:⋯ 5049:ψ 5014:ψ 4984:∑ 4955:⟩ 4936:⋯ 4892:⋯ 4869:⟨ 4866:≡ 4843:… 4788:⋯ 4764:Ψ 4725:ψ 4721:⋯ 4687:ψ 4652:ψ 4642:∑ 4607:∏ 4589:⟩ 4570:⋯ 4526:⋯ 4503:⟨ 4500:≡ 4477:… 4422:⋯ 4398:Ψ 4332:⋯ 4249:∑ 4223:⟩ 4204:⋯ 4145:⋯ 4082:∑ 4047:∏ 4032:⟩ 4013:⋯ 3947:− 3932:δ 3925:⟩ 3906:⟨ 3846:⟩ 3843:ψ 3832:⟨ 3723:… 3707:→ 3691:… 3642:≤ 3639:⋯ 3636:≤ 3623:≤ 3609:∑ 3539:⋯ 3495:⋯ 3470:≡ 3449:… 3433:→ 3417:… 3351:⟩ 3324:⋯ 3255:⟩ 3228:⋯ 3162:⟩ 3143:⋯ 3099:⋯ 3076:⟨ 3063:⟩ 3044:⋯ 3000:⋯ 2977:⟨ 2930:Π 2906:− 2798:⋯ 2732:⁡ 2720:∑ 2698:⟩ 2679:⋯ 2543:⋯ 2480:∑ 2445:∏ 2434:⟩ 2415:⋯ 2336:particles 2253:electrons 2090:− 1964:⟩ 1927:− 1921:⟩ 1863:⟩ 1820:⟩ 1727:Clearly, 1714:Hermitian 1694:⟩ 1691:ψ 1683:⟩ 1680:ϕ 1672:≡ 1662:⟩ 1659:ϕ 1651:⟩ 1648:ψ 1588:⟩ 1570:⟩ 1546:⟩ 1528:⟩ 1503:× 1490:⟩ 1387:⟩ 1369:⟩ 1351:− 1348:⟩ 1330:⟩ 1305:× 1295:≡ 1292:⟩ 1225:⟩ 1207:⟩ 1186:⟩ 1168:⟩ 1143:× 1133:≡ 1130:⟩ 1079:symmetric 1062:⟩ 1044:⟩ 1026:± 1023:⟩ 1005:⟩ 930:⟩ 912:⟩ 877:⟩ 859:⟩ 821:⊗ 780:⟩ 762:⟩ 724:⟩ 706:⟩ 540:neutrinos 536:electrons 495:in 1926. 476:molecules 466:(such as 460:electrons 458:(such as 452:particles 444:identical 374:Ehrenfest 354:Boltzmann 234:Canonical 9171:18229595 9097:See also 8684:, while 8634:Homotopy 8601:Fermions 7274:factored 5555:, ..., n 4358:⟩ 4328:⟩ 4301:⟩ 4171:⟩ 4141:⟩ 4114:⟩ 3954:′ 3921:′ 3788:position 3567:⟩ 3481:⟨ 2824:⟩ 2794:⟩ 2767:⟩ 2569:⟩ 2539:⟩ 2512:⟩ 2316:plektons 2275:fermions 2249:helium-4 1712:is both 1498:constant 1430:chemical 1300:constant 1138:constant 558:nuclei. 556:helium-3 552:neutrons 528:helium-4 508:fermions 369:Einstein 316:Enthalpy 281:Einstein 9285:Bibcode 9151:Bibcode 9012:anyonic 8868:⁠ 8846:⁠ 8831:⁠ 8802:⁠ 8787:⁠ 8758:⁠ 8547:state. 7589:is not 7573:is the 7493:entropy 7268:is the 7260:is the 5573:, ..., 3771:, ..., 3762:ordered 2851:is the 2377:any two 2362:, ..., 2308:MOSFETs 2257:protons 2245:photons 1718:unitary 974:state". 548:protons 524:phonons 516:photons 349:Maxwell 85:scholar 9374: 9330: 9303: 9258: 9217: 9169: 8880:. The 8789:where 8587:Bosons 8565:Both 1 8562:Both 0 8212:, and 7980:, and 7735:shells 7725:, and 7575:volume 7569:where 7491:, the 7315:where 7256:where 7055:energy 6967:under 6014:matrix 3790: 2843:Here, 2828: 2296:anyons 2264:bosons 1434:matter 810:space 590:, and 554:, and 544:quarks 532:mesons 520:gluons 506:, and 500:bosons 450:) are 384:Tolman 270:Models 87: 80: 73: 66: 58: 9301:S2CID 9167:S2CID 9141:arXiv 8752:. If 8596:0.33 7719:laser 7593:– if 7481:Gibbs 6012:of a 3799:range 472:atoms 394:Fermi 389:Debye 364:Gibbs 291:Potts 286:Ising 276:Debye 92:JSTOR 78:books 9372:ISBN 9328:ISBN 9256:ISSN 9215:ISBN 9044:exp( 9006:(by 8996:exp( 8636:and 8620:and 8593:0.33 8590:0.33 8582:0.5 8579:0.25 8576:0.25 7828:and 7769:and 7713:and 7597:and 7264:and 6655:and 2853:sign 2845:sgn( 2327:spin 2321:The 1716:and 1416:and 889:and 650:Let 592:spin 584:mass 491:and 474:and 359:Bose 64:news 9293:doi 9248:doi 9159:doi 8884:of 8878:≥ 3 8844:is 8835:not 8800:is 8794:≥ 3 8756:is 8736:to 7499:is 7342:exp 7131:exp 7002:. 6609:of 5582:is 3816:is 2878:if 2729:sgn 2255:or 2247:or 1764:of 446:or 434:In 255:NPT 247:NPH 239:µVT 231:NVT 223:NVE 47:by 9387:: 9299:. 9291:. 9281:21 9279:. 9254:. 9244:66 9242:. 9238:. 9165:. 9157:. 9149:. 9137:79 9135:. 9131:. 9054:. 9046:iφ 9036:, 9024:, 8998:iθ 8983:, 8922:, 8910:, 8898:, 8742:, 8730:, 8714:, 8702:, 8690:, 8670:, 8655:× 8610:1 8173:, 7941:, 7902:, 7721:, 7647:ln 7616:= 7609:. 7522:ln 7487:= 7479:. 6994:. 6020:: 5566:, 3794:. 3168:1. 2643:: 2636:. 2631:= 2355:, 2318:. 2285:. 2270:. 2259:. 2235:. 1615:. 1436:. 647:. 598:. 586:, 574:. 550:, 546:, 542:, 538:, 526:, 522:, 518:, 478:. 438:, 9364:) 9360:( 9307:. 9295:: 9287:: 9262:. 9250:: 9223:. 9173:. 9161:: 9153:: 9143:: 9076:, 9072:R 9068:= 9065:M 9048:) 9040:) 9038:x 9034:y 9032:( 9028:) 9026:y 9022:x 9020:( 9004:θ 9000:) 8987:) 8985:y 8981:x 8979:( 8961:, 8956:2 8951:R 8946:= 8943:M 8926:) 8924:x 8920:y 8918:( 8914:) 8912:y 8908:x 8906:( 8902:) 8900:y 8896:x 8894:( 8876:d 8855:R 8842:M 8817:2 8812:R 8798:M 8792:d 8773:d 8768:R 8754:M 8746:) 8744:x 8740:y 8738:( 8734:) 8732:y 8728:x 8726:( 8718:) 8716:x 8712:y 8710:( 8706:) 8704:y 8700:x 8698:( 8694:) 8692:x 8688:y 8686:( 8682:y 8678:x 8674:) 8672:y 8668:x 8666:( 8657:M 8653:M 8648:M 8644:d 8607:0 8604:0 8532:1 8528:| 8504:0 8500:| 8479:) 8473:0 8469:| 8462:1 8458:| 8448:1 8444:| 8437:0 8433:| 8429:( 8423:2 8419:1 8391:1 8387:| 8363:0 8359:| 8335:1 8331:| 8307:0 8303:| 8282:) 8276:0 8272:| 8265:1 8261:| 8257:+ 8251:1 8247:| 8240:0 8236:| 8232:( 8226:2 8222:1 8197:1 8193:| 8186:1 8182:| 8158:0 8154:| 8147:0 8143:| 8116:1 8112:| 8088:0 8084:| 8060:1 8056:| 8032:0 8028:| 8004:0 8000:| 7993:1 7989:| 7965:1 7961:| 7954:0 7950:| 7926:1 7922:| 7915:1 7911:| 7887:0 7883:| 7876:0 7872:| 7841:1 7837:| 7813:0 7809:| 7782:1 7778:| 7754:0 7750:| 7686:) 7683:T 7680:( 7677:f 7674:N 7671:+ 7667:) 7662:N 7659:V 7654:( 7644:k 7641:N 7638:= 7635:S 7622:N 7620:/ 7618:ξ 7614:Z 7603:S 7599:V 7595:N 7587:S 7583:T 7579:f 7571:V 7554:) 7551:T 7548:( 7545:f 7542:N 7539:+ 7535:) 7532:V 7529:( 7519:k 7516:N 7513:= 7510:S 7489:ξ 7485:Z 7456:. 7450:! 7447:N 7441:N 7431:= 7428:Z 7415:N 7408:n 7404:Z 7387:. 7383:] 7376:T 7373:k 7368:) 7365:n 7362:( 7349:[ 7337:n 7329:= 7298:N 7290:= 7287:Z 7266:T 7258:k 7240:} 7235:] 7231:) 7226:N 7222:n 7218:( 7212:+ 7206:+ 7203:) 7198:2 7194:n 7190:( 7184:+ 7181:) 7176:1 7172:n 7168:( 7161:[ 7154:T 7151:k 7147:1 7138:{ 7124:N 7120:n 7116:, 7110:, 7105:2 7101:n 7097:, 7092:1 7088:n 7079:= 7076:Z 7059:n 7051:n 7049:( 7047:ε 7042:j 7038:n 7034:j 7029:j 7025:n 7021:N 6980:n 6976:S 6955:H 6950:n 6929:. 6897:n 6893:S 6852:n 6840:. 6828:a 6825:= 6819:a 6814:t 6785:n 6781:S 6751:, 6745:a 6740:t 6732:= 6729:) 6720:( 6717:a 6712:t 6708:) 6698:( 6674:n 6670:S 6643:H 6617:n 6597:a 6575:n 6571:S 6550:H 6545:n 6520:n 6491:| 6484:) 6479:N 6475:x 6471:( 6464:N 6460:n 6444:) 6439:2 6435:x 6431:( 6424:N 6420:n 6409:) 6404:1 6400:x 6396:( 6389:N 6385:n 6350:) 6345:N 6341:x 6337:( 6330:2 6326:n 6310:) 6305:2 6301:x 6297:( 6290:2 6286:n 6275:) 6270:1 6266:x 6262:( 6255:2 6251:n 6238:) 6233:N 6229:x 6225:( 6218:1 6214:n 6198:) 6193:2 6189:x 6185:( 6178:1 6174:n 6163:) 6158:1 6154:x 6150:( 6143:1 6139:n 6126:| 6119:! 6116:N 6112:1 6107:= 6104:) 6099:N 6095:x 6091:, 6085:, 6080:1 6076:x 6072:( 6067:) 6064:A 6061:( 6054:N 6050:n 6041:1 6037:n 6003:N 5999:N 5995:x 5978:1 5975:= 5970:N 5966:x 5959:3 5955:d 5946:2 5942:x 5935:3 5931:d 5925:1 5921:x 5914:3 5910:d 5904:2 5899:| 5895:) 5890:N 5886:x 5882:, 5876:, 5871:2 5867:x 5863:, 5858:1 5854:x 5850:( 5845:) 5842:A 5838:/ 5834:S 5831:( 5824:N 5820:n 5811:2 5807:n 5801:1 5797:n 5787:| 5756:N 5739:x 5733:N 5730:3 5726:d 5719:2 5714:| 5710:) 5705:N 5701:x 5697:, 5691:, 5686:2 5682:x 5678:, 5673:1 5669:x 5665:( 5660:) 5657:A 5653:/ 5649:S 5646:( 5639:N 5635:n 5626:2 5622:n 5616:1 5612:n 5602:| 5596:! 5593:N 5579:N 5575:x 5571:2 5568:x 5564:1 5561:x 5557:N 5553:1 5550:n 5529:) 5521:i 5517:x 5508:j 5504:x 5497:( 5492:) 5489:A 5486:( 5479:N 5475:n 5466:1 5462:n 5450:= 5447:) 5439:j 5435:x 5426:i 5422:x 5415:( 5410:) 5407:A 5404:( 5397:N 5393:n 5384:1 5380:n 5367:) 5359:i 5355:x 5346:j 5342:x 5335:( 5330:) 5327:S 5324:( 5317:N 5313:n 5304:1 5300:n 5291:= 5288:) 5280:j 5276:x 5267:i 5263:x 5256:( 5251:) 5248:S 5245:( 5238:N 5234:n 5225:1 5221:n 5181:n 5177:| 5173:x 5164:) 5161:x 5158:( 5153:n 5118:) 5113:N 5109:x 5105:( 5100:) 5097:N 5094:( 5091:p 5080:) 5075:2 5071:x 5067:( 5062:) 5059:2 5056:( 5053:p 5045:) 5040:1 5036:x 5032:( 5027:) 5024:1 5021:( 5018:p 5010:) 5007:p 5004:( 5000:n 4997:g 4994:s 4988:p 4977:! 4974:N 4970:1 4965:= 4952:A 4949:; 4944:N 4940:n 4931:2 4927:n 4921:1 4917:n 4912:| 4908:A 4905:; 4900:N 4896:x 4887:2 4883:x 4877:1 4873:x 4859:) 4854:N 4850:x 4846:, 4840:, 4835:2 4831:x 4827:, 4822:1 4818:x 4814:( 4809:) 4806:A 4803:( 4796:N 4792:n 4783:2 4779:n 4773:1 4769:n 4756:) 4751:N 4747:x 4743:( 4738:) 4735:N 4732:( 4729:p 4718:) 4713:2 4709:x 4705:( 4700:) 4697:2 4694:( 4691:p 4683:) 4678:1 4674:x 4670:( 4665:) 4662:1 4659:( 4656:p 4646:p 4634:! 4631:N 4626:! 4621:j 4617:n 4611:j 4599:= 4586:S 4583:; 4578:N 4574:n 4565:2 4561:n 4555:1 4551:n 4546:| 4542:S 4539:; 4534:N 4530:x 4521:2 4517:x 4511:1 4507:x 4493:) 4488:N 4484:x 4480:, 4474:, 4469:2 4465:x 4461:, 4456:1 4452:x 4448:( 4443:) 4440:S 4437:( 4430:N 4426:n 4417:2 4413:n 4407:1 4403:n 4353:) 4350:N 4347:( 4344:p 4340:x 4336:| 4323:) 4320:2 4317:( 4314:p 4310:x 4306:| 4296:) 4293:1 4290:( 4287:p 4283:x 4279:| 4275:) 4272:p 4269:( 4265:n 4262:g 4259:s 4253:p 4242:! 4239:N 4235:1 4230:= 4220:A 4217:; 4212:N 4208:x 4199:2 4195:x 4189:1 4185:x 4180:| 4166:) 4163:N 4160:( 4157:p 4153:x 4149:| 4136:) 4133:2 4130:( 4127:p 4123:x 4119:| 4109:) 4106:1 4103:( 4100:p 4096:x 4092:| 4086:p 4074:! 4071:N 4066:! 4061:j 4057:n 4051:j 4039:= 4029:S 4026:; 4021:N 4017:x 4008:2 4004:x 3998:1 3994:x 3989:| 3958:) 3951:x 3944:x 3941:( 3936:3 3928:= 3918:x 3913:| 3909:x 3889:x 3872:x 3867:3 3863:d 3856:2 3851:| 3839:| 3835:x 3828:| 3814:x 3810:x 3807:d 3803:ψ 3792:x 3775:N 3773:m 3769:1 3766:m 3745:1 3742:= 3739:) 3734:N 3730:m 3726:, 3720:, 3715:1 3711:m 3702:N 3698:n 3694:, 3688:, 3683:1 3679:n 3675:( 3670:A 3666:/ 3662:S 3658:P 3650:N 3646:m 3631:2 3627:m 3618:1 3614:m 3580:2 3574:| 3563:A 3559:/ 3555:S 3552:; 3547:N 3543:n 3534:1 3530:n 3524:| 3519:A 3515:/ 3511:S 3508:; 3503:N 3499:m 3490:1 3486:m 3475:| 3466:) 3460:N 3456:m 3452:, 3446:, 3441:1 3437:m 3428:N 3424:n 3420:, 3414:, 3409:1 3405:n 3400:( 3394:A 3390:/ 3386:S 3382:P 3368:m 3348:A 3344:/ 3340:S 3337:; 3332:N 3328:m 3319:2 3315:m 3309:1 3305:m 3300:| 3286:2 3283:m 3279:1 3276:m 3272:m 3252:A 3248:/ 3244:S 3241:; 3236:N 3232:n 3223:2 3219:n 3213:1 3209:n 3204:| 3190:N 3165:= 3159:A 3156:; 3151:N 3147:n 3138:2 3134:n 3128:1 3124:n 3119:| 3115:A 3112:; 3107:N 3103:n 3094:2 3090:n 3084:1 3080:n 3072:, 3069:1 3066:= 3060:S 3057:; 3052:N 3048:n 3039:2 3035:n 3029:1 3025:n 3020:| 3016:S 3013:; 3008:N 3004:n 2995:2 2991:n 2985:1 2981:n 2944:n 2940:m 2934:n 2909:1 2886:p 2866:1 2863:+ 2849:) 2847:p 2819:) 2816:N 2813:( 2810:p 2806:n 2802:| 2789:) 2786:2 2783:( 2780:p 2776:n 2772:| 2762:) 2759:1 2756:( 2753:p 2749:n 2745:| 2741:) 2738:p 2735:( 2724:p 2713:! 2710:N 2706:1 2701:= 2695:A 2692:; 2687:N 2683:n 2674:2 2670:n 2664:1 2660:n 2655:| 2633:N 2628:n 2624:m 2620:n 2617:Σ 2613:N 2609:n 2604:n 2602:m 2594:N 2590:p 2564:) 2561:N 2558:( 2555:p 2551:n 2547:| 2534:) 2531:2 2528:( 2525:p 2521:n 2517:| 2507:) 2504:1 2501:( 2498:p 2494:n 2490:| 2484:p 2472:! 2469:N 2464:! 2459:n 2455:m 2449:n 2437:= 2431:S 2428:; 2423:N 2419:n 2410:2 2406:n 2400:1 2396:n 2391:| 2368:N 2364:n 2360:2 2357:n 2353:1 2350:n 2346:N 2342:N 2334:N 2229:P 2224:P 2203:0 2200:= 2196:] 2192:H 2189:, 2186:P 2182:[ 2152:) 2147:2 2143:x 2139:( 2136:V 2133:+ 2130:) 2125:1 2121:x 2117:( 2114:V 2111:+ 2108:) 2104:| 2098:2 2094:x 2085:1 2081:x 2076:| 2072:( 2069:U 2066:+ 2060:m 2057:2 2051:2 2046:2 2042:p 2036:+ 2030:m 2027:2 2021:2 2016:1 2012:p 2006:= 2003:H 1985:P 1961:A 1958:; 1953:2 1949:n 1945:, 1940:1 1936:n 1931:| 1924:= 1918:A 1915:; 1910:2 1906:n 1902:, 1897:1 1893:n 1888:| 1884:P 1860:S 1857:; 1852:2 1848:n 1844:, 1839:1 1835:n 1830:| 1826:+ 1823:= 1817:S 1814:; 1809:2 1805:n 1801:, 1796:1 1792:n 1787:| 1783:P 1766:P 1748:1 1745:= 1740:2 1736:P 1710:P 1687:| 1676:| 1667:) 1655:| 1644:| 1638:( 1633:P 1620:P 1593:) 1583:1 1579:n 1574:| 1565:2 1561:n 1556:| 1552:i 1549:+ 1541:2 1537:n 1532:| 1523:1 1519:n 1514:| 1508:( 1493:= 1487:? 1484:; 1479:2 1475:n 1471:, 1466:1 1462:n 1457:| 1421:2 1418:n 1414:1 1411:n 1392:) 1382:1 1378:n 1373:| 1364:2 1360:n 1355:| 1343:2 1339:n 1334:| 1325:1 1321:n 1316:| 1310:( 1289:A 1286:; 1281:2 1277:n 1273:, 1268:1 1264:n 1259:| 1230:) 1220:1 1216:n 1211:| 1202:2 1198:n 1193:| 1189:+ 1181:2 1177:n 1172:| 1163:1 1159:n 1154:| 1148:( 1127:S 1124:; 1119:2 1115:n 1111:, 1106:1 1102:n 1097:| 1057:1 1053:n 1048:| 1039:2 1035:n 1030:| 1018:2 1014:n 1009:| 1000:1 996:n 991:| 972:1 969:n 965:2 962:n 958:2 955:n 951:1 948:n 925:1 921:n 916:| 907:2 903:n 898:| 872:2 868:n 863:| 854:1 850:n 845:| 824:H 818:H 804:1 801:n 797:2 794:n 775:1 771:n 766:| 757:2 753:n 748:| 719:2 715:n 710:| 701:1 697:n 692:| 678:2 675:n 671:1 668:n 660:n 652:n 423:e 416:t 409:v 114:) 108:( 103:) 99:( 89:· 82:· 75:· 68:· 41:.

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Knowledge

Indistinguishable particles

Index