Identical Particle

Full Title or Meme

Fermi–Dirac statistics applies to Identical Particles. In this context, “identical” means that the particles have the same mass, charge, and other intrinsic properties. Electrons fit this criterion because they are indistinguishable from one another at least in the context of the electrons bound to an atom.

Taxonomy

For the purposes of this page:

• Particle = any piece of matter or energy that is small enough for quantum effects to be observed.
• Wave or wavelet = any view of a particle that emphases its wave-like characteristics.
• Virtual particle = any particle that can be imagined, but not observed.
  • For example virtual particles were used in Feynman diagrams to get the full gamut of all possible paths.

In the Beginning

The working hypothesis, the postulates and consequences.

1. There was just energy, lots of it. All of it as photons, that is, as each boson has no independent identity. The founder of the big bang has suggested that there was just one truly energetic photon that begat all of the reset. Dirac shared that view with Lemaître.
2. There was a fairly simple plan about how aggregations of energy could be arraigned.
3. Particles (Fermions) condensed out of this massive amount of energy by some mechanism that result in all matter and essentially no antimatter.
4. Fermions can have individual identity as we know from the statistics that they follow (Fermi-Dirac statistics).
5. Energy does not like to be naked. When it is, as a photon, it tries to find a home where it is not naked.

Context

In Quantum Mechanics two particles (or two waves) that can be considered identical include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules,composite subatomic particles (such as atomic nuclei), as well as atoms and molecules. For example, all of the electrons bound to an atom are indistinguishable and so there is no set of state variables that can be identified with a particular electron.

Pauli Exclusion Principle

This principle states that no two identical fermions (particles with half-integer spin, like electrons) can occupy the same quantum state simultaneously. In other words, electrons cannot share the same energy level within an atom. In statistical mechanics

Energy States

Electrons in an atom occupy specific energy levels (orbitals). Each energy level can accommodate a certain number of electrons. The lowest energy level (ground state) can hold up to two electrons with opposite spins (due to the Pauli exclusion principle).

===Fermi–Dirac Distribution[[[ Fermi–Dirac statistics describe how these electrons distribute themselves among energy states. At absolute zero temperature, electrons fill the lowest energy levels first, following the Pauli exclusion principle. As temperature increases, more energy levels become accessible, and electrons occupy them accordingly.

Assume you have two particles A and B in states 1 and 2. If the two particles are distinguishable, then by exchanging the particles A and B, you will obtain a new state that will have the same properties as the old state i.e. you have degeneracy and you have to count both states when calculating the entropy for example. On the other hand, for indistinguishable particles, exchanging A and B is a transformation that does nothing and you have the same physical state. This means that for indistinguishable particles, particle labels are unphysical and they represent a redundancy in describing the physical state and that is why you would have to divide by some symmetry factor to get the proper counting of states.

https://www.feynmanlectures.caltech.edu/III_01.html

Statistical Physics

Solutions

For this page the distinction that is important in the statistical rules that apply. Maxwell-Boltzmann distribution is used for solving distinguishable particle and Fermi-Dirac, Bose-Einstein for indistinguishable particles.

Particles which exhibit symmetric states are called bosons. 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 Bose–Einstein statistics.

Particles which exhibit antisymmetric states are called fermions. Antisymmetry gives rise to the Pauli exclusion principle, which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by Fermi–Dirac statistics.

Looking Forward

Comments from Louis de Broglie about the primacy of a particle view as opposed to a systemic view:

[Reflecting on remarks from earlier in my book], one will see that the non-individuality of particles, the exclusion principle and exchange energy are three intimately connected mysteries: all three derive from the impossibility of representing exactly elementary physical entities in a three-dimensional space continuum (or more generally in the four-dimensional space-time continuum). Perhaps some day, by escaping from this framework, we shall succeed in penetrating better the meaning, still quite obscure today, of these great guiding principles of the new physics. From another point of view, it might be said that the physical notion of an individual is complementary, in the sense of Bohr, with the notion of system. The particle truly has a well- defined individuality only when it is isolated. As soon as it enters into an interaction with other particles, its individuality is diminished. Perhaps it has not been pointed out enough in classical theories that the notion of potential energy of a system implies a certain weakening of the individuality of the constituents of the system through the "pooling," in the form of the potential energy, of a part of the total energy. In the cases contemplated by the new mechanics, where particles of the same nature occupy, somehow simultaneously, the same region of space, the individuality of these particles is dissipated to the vanishing point. In going progressively from cases of isolated particles without interactions to the cases just cited, the notion of the individuality of the particles is seen to grow more and more dim as the individuality of the system more strongly asserts itself. It therefore seems that the individual and the system the somewhat complementary idealizations. This, perhaps, is an idea which merits a more thorough study.