جسيمات متماثلة

الجسيمات المتماثلة Identical particles أو الجسيمات غير المتمايزة 'indistinguishable particles هي جسيمات لا يمكن تفريقها عن بعضها البعض حتى من حيث المبدأ و هي تتضمن جسيمات أولية مثل الالكترونات أو جسيمات مركبة مجهرية مثل الذرات .

هناك مجموعتين رئيسيتين ضمن الجسيمات المتماثلة : البوزونات التي يمكن أن تتشارك بنفس الحالة الكمومية ، و الفرميونات التي يحظر عليها التشارك بالحالة الكمومية ( أو ما يدعى بمبدأ انتفاء باولي ) .

البوزونات : مثل الفوتونات ، الگلوونات ، الفونونات ، ذرات الهليوم-4 . الفرميونات : مثل الالكترونات ، النيوترونات ، الكواركات ، الپروتونات ، النيوترينوات ، ذرات الهليوم-3 .

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التمييز بين الجسيمات

There are two methods for distinguishing between particles. The first method relies on differences in the intrinsic physical properties of the particles, such as mass, electric charge, and spin. If differences exist, it is possible to distinguish between the particles by measuring the relevant properties. However, it is an empirical fact that microscopic particles of the same species have completely equivalent physical properties. For instance, every electron in the universe has exactly the same electric charge; this is why it is possible to speak of such a thing as "the charge of the electron".

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 there would be no ambiguity about which particle is which.

The problem with the second approach is that it contradicts the principles of quantum mechanics. According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by wavefunctions 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 are then said to be indistinguishable.

وصف حسب ميكانيكا الكم للجسيمات المتماثلة

الحالات التماثلية وعكس التماثلية

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

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

وجود هذه الجسيمات المتماثلة له نتائج مهمة على صعيد الميكانيك الاحصائي حيث تعتمد حسابات الميكانيك الاحصائي على صيغ رياضية احتمالية حساسة لخاصة تمايز الجسيمات أو عدم تمايزها .

|x_{1}x_{2}\cdots x_{N};S\rangle ={\frac {\prod _{j}N_{j}!}{N!}}\sum _{p}|x_{p(1)}\rangle |x_{p(2)}\rangle \cdots |x_{p(N)}\rangle

|x_{1}x_{2}\cdots x_{N};A\rangle ={\frac {1}{N!}}\sum _{p}\mathrm {sgn} (p)|x_{p(1)}\rangle |x_{p(2)}\rangle \cdots |x_{p(N)}\rangle

وبذلك يمكن كتابة دالة موجية عديدة الأجسام،

$\Psi _{n_{1}n_{2}\cdots n_{N}}^{(S)}(x_{1},x_{2},\cdots x_{N})$	$\equiv \langle x_{1}x_{2}\cdots x_{N};S\|n_{1}n_{2}\cdots n_{N};S\rangle$
	$={\sqrt {\frac {\prod _{j}N_{j}!}{N!}}}\sum _{p}\psi _{p(1)}(x_{1})\psi _{p(2)}(x_{2})\cdots \psi _{p(N)}(x_{N})$
$\Psi _{n_{1}n_{2}\cdots n_{N}}^{(A)}(x_{1},x_{2},\cdots x_{N})$	$\equiv \langle x_{1}x_{2}\cdots x_{N};A\|n_{1}n_{2}\cdots n_{N};A\rangle$
	$={\frac {1}{\sqrt {N!}}}\sum _{p}\mathrm {sgn} (p)\psi _{p(1)}(x_{1})\psi _{p(2)}(x_{2})\cdots \psi _{p(N)}(x_{N})$

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

\psi _{n}(x)\equiv \langle x|n\rangle

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:

\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 )

وأخيراً يجدر الإشارة إلى أن دالة الموجة عكس التماثلية يمكن كتابتها كمحددة مصفوفة، تـُعرف باسم محددة سليتر:

\Psi _{n_{1}\cdots n_{N}}^{(A)}(x_{1},\cdots 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})\\\cdots &\cdots &\cdots &\cdots \\\psi _{n_{N}}(x_{1})&\psi _{n_{N}}(x_{2})&\cdots &\psi _{n_{N}}(x_{N})\\\end{matrix}}\right|

مقاربة المُعامِل وشبه الإحصائيات

The Hilbert space for $n$ particles is given by the tensor product ${\textstyle \bigotimes _{n}H}$ . The permutation group of $S_{n}$ acts on this space by permuting the entries. By definition the expectation values for an observable $a$ of $n$ indistinguishable particles should be invariant under these permutation. This means that for all $\psi \in H$ and $\sigma \in S_{n}$

(\sigma \Psi )^{t}a(\sigma \Psi )=\Psi ^{t}a\Psi ,

or equivalently for each $\sigma \in S_{n}$

\sigma ^{t}a\sigma =a

.

Two states are equivalent whenever their expectation values coincide for all observables. If we restrict to observables of $n$ identical particles, and hence observables satisfying the equation above, we find that the following states (after normalization) are equivalent

\Psi \sim \sum _{\sigma \in S_{n}}\lambda _{\sigma }\sigma \Psi

.

The equivalence classes are in bijective relation with irreducible subspaces of ${\textstyle \bigotimes _{n}H}$ under $S_{n}$ .

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 parastatistic states.^[1] Young tableaux provide a way to classify all of these irreducible subspaces.

الخصائص الإحصائية

الآثار الإحصائية لعدم التمايز

The indistinguishability of particles has a profound effect on their statistical properties. To illustrate this, consider a system of N distinguishable, non-interacting particles. Once again, let n_j denote the state (i.e. quantum numbers) of particle j. If the particles have the same physical properties, the n_j's run over the same range of values. Let ε(n) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies. The partition function of the system is

Z=\sum _{n_{1},n_{2},\ldots ,n_{N}}\exp \left\{-{\frac {1}{kT}}\left[\varepsilon (n_{1})+\varepsilon (n_{2})+\cdots +\varepsilon (n_{N})\right]\right\}

where k is Boltzmann's constant and T is the temperature. This expression can be factored to obtain

Z=\xi ^{N}

where

\xi =\sum _{n}\exp \left[-{\frac {\varepsilon (n)}{kT}}\right].

If the particles are identical, this equation is incorrect. Consider a state of the system, described by the single particle states [n₁, ..., n_N]. In the equation for Z, every possible permutation of the n'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.

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 N!. The correct partition function is

Z={\frac {\xi ^{N}}{N!}}.

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

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 Gibbs paradox. Gibbs showed that in the equation Z = ξ^N, the entropy of a classical ideal gas is

S=Nk\ln \left(V\right)+Nf(T)

where V is the volume of the gas and f is some function of T alone. The problem with this result is that S is not extensive – if N and V are doubled, S does not double accordingly. Such a system does not obey the postulates of thermodynamics.

Gibbs also showed that using Z = ξ^N/N! alters the result to

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

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

الخصائص الإحصائية للبوزونات وفرميونات

There are important differences between the statistical behavior of bosons and fermions, which are described by Bose–Einstein statistics and Fermi–Dirac statistics respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the laser, Bose–Einstein condensation, and superfluidity. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the Fermi gas. 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 shells rather than all lying in the same lowest energy state.

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 $|0\rangle$ and $|1\rangle$ , which have the same energy.

The composite system can evolve in time, interacting with a noisy environment. Because the $|0\rangle$ and $|1\rangle$ 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 quantum entanglement.) 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.

If A and B are identical bosons, then the composite system has only three distinct states: $|0\rangle |0\rangle$ , $|1\rangle |1\rangle$ , and ${\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle +|1\rangle |0\rangle )$ . When the experiment is performed, the probability of obtaining two particles in the $|0\rangle$ state is now 0.33; the probability of obtaining two particles in the $|1\rangle$ state is 0.33; and the probability of obtaining one particle in the $|0\rangle$ state and the other in the $|1\rangle$ 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".

If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state ${\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle -|1\rangle |0\rangle )$ . When the experiment is performed, one particle is always in the $|0\rangle$ state and the other is in the $|1\rangle$ state.

The results are summarized in Table 1:

Table 1: Statistics of two particles
Particles	Both 0	Both 1	One 0 and one 1
Distinguishable	0.25	0.25	0.5
Bosons	0.33	0.33	0.33
Fermions	0	0	1

As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on Fermi–Dirac statistics and Bose–Einstein statistics, these principles are extended to large number of particles, with qualitatively similar results.

صف التوفيق

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 $d$ -dimensional space $M$ , at any given time, the configuration of two identical particles can be specified as an element of $M \times M$ . If there is no overlap between the particles, so that they do not interact directly, then their locations must belong to the space $[M × M] \ {coincident points},$ the subspace with coincident points removed. The element $(x, y)$ describes the configuration with particle I at $x$ and particle II at $y$ , while $(y, x)$ describes the interchanged configuration. With identical particles, the state described by $(x, y)$ ought to be indistinguishable from the state described by $(y, x)$ . Now consider the homotopy class of continuous paths from $(x, y)$ to $(y, x)$ , within the space $[M × M] \ {coincident points}$ . If $M$ is $\mathbb {R} ^{d}$ where $d \geq 3$ , then this homotopy class only has one element. If $M$ is $\mathbb {R} ^{2}$ , 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 not homotopic to a clockwise interchange by half a turn. Lastly, if $M$ is $\mathbb {R}$ , then this homotopy class is empty.

Suppose first that $d \geq 3$ . The universal covering space of $[M × M] \ {coincident points},$ which is none other than $[M × M] \ {coincident points}$ itself, only has two points which are physically indistinguishable from $(x, y)$ , namely $(x, y)$ itself and $(y, x)$ . So, the only permissible interchange is to swap both particles. This interchange is an involution, 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.

In the case $M=\mathbb {R} ^{2},$ the universal covering space of $[M × M] \ {coincident points}$ has infinitely many points that are physically indistinguishable from $(x, y)$ . This is described by the infinite cyclic group 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 $exp(iθ)$ for any real $θ$ (by unitarity, the absolute value of the multiplication must be 1). This is called anyonic statistics. In fact, even with two distinguishable particles, even though $(x, y)$ is now physically distinguishable from $(y, x)$ , 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 $exp(iφ)$ . This phase factor here is called the mutual statistics.

Finally, in the case $M=\mathbb {R} ,$ the space $[M × M] \ {coincident points}$ 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.

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انظر أيضاً

الهامش

^ Bach, Alexaner (1993). "Classification of Indistinguishable Particles". Europhysics Letters. 21 (5): 515–520. Bibcode:1993EL.....21..515B. doi:10.1209/0295-5075/21/5/002. S2CID 250835341.

المراجع

Tuckerman, Mark (2010), Statistical Mechanics, ISBN 978-0198525264

وصلات خارجية

The Feynman Lectures on Physics Vol. III Ch. 4: Identical Particles
Exchange of Identical and Possibly Indistinguishable Particles by John S. Denker
Identity and Individuality in Quantum Theory (Stanford Encyclopedia of Philosophy)
Many-Electron States in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013, ISBN 978-3-89336-884-6

[1] Bach, Alexaner (1993). "Classification of Indistinguishable Particles". Europhysics Letters. 21 (5): 515–520. Bibcode:1993EL.....21..515B. doi:10.1209/0295-5075/21/5/002. S2CID 250835341.

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