Particle statistics
Encyclopedia
Particle statistics refers to the particular description of particles in statistical mechanics
.
and composite particles, atoms, molecules, electrons, etc.) in the system are considered distinguishable
. This means that one can label and track each individual particle in a system. As a consequence changing the position of any two particles in the system leads to a completely different configuration of the entire system. Furthermore there is no restriction on placing more than one particle in any given state accessible to the system. Classical statistics is called Maxwell-Boltzmann statistics (or M-B statistics).
that distinguishes it from classical mechanics is that particles of a particular type are indistinguishable
from one another. This means that in an assembly consisting of similar particles, interchanging any two particles does not lead to a new configuration of the system (in the language of quantum mechanics: the wavefunction
of the system is invariant with respect to the interchange of the constituent particles). In case of a system consisting of particles belonging to different nature (for example electrons and protons), the wavefunction of the system is invariant separately for the assembly of the two particles.
While this difference between classical and quantum description of systems is fundamental to all of quantum statistics, it is further divided into the following two classes on the basis of symmetry of the system.
It is important to emphasize that the wavefunction of the system has not changed itself. This has very important consequences on the state of the system: There is no restriction to the number of particles that can be placed in a single state (accessible to the system). It is found that the particles that obey Bose-Einstein statistics are the ones which have integer spins, which are therefore called bosons (named for Bose). Examples of bosons include photons and helium-4
atoms. One type of system obeying B-E statistics is the Bose-Einstein condensate where all particles of the assembly exist in the same state.
Again, the wavefunction of the system itself does not change. The consequence of the negative sign on the Fermi-Dirac statistics can be understood in the following way:
Suppose that the particles that are interchanged belong to the same state. Since the particles are considered indistinguishable from one another then changing the coordinates of the particles should not have any change on the system's wavefunction (because by our assumptions the particles are in the same state). Therefore, the wavefunction before interchanging similar states equals the wavefunction after interchanging similar states.
Combining (or adding, literally speaking) the above statement with the fundamental asymmetry of the Fermi-Dirac system leads us to conclude that the wavefunction of the system before interchanging equals zero.
This shows that in Fermi-Dirac statistics, more than one particle cannot occupy a single state accessible to the system. This is called Pauli's exclusion principle.
It is found that particles with half-integral spin (or fermions) obey the Fermi-Dirac statistics. This includes, electrons, protons, Helium-3 etc.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
.
Classical statistics
In classical mechanics all the particles (fundamentalElementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
and composite particles, atoms, molecules, electrons, etc.) in the system are considered distinguishable
Identical particles
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...
. This means that one can label and track each individual particle in a system. As a consequence changing the position of any two particles in the system leads to a completely different configuration of the entire system. Furthermore there is no restriction on placing more than one particle in any given state accessible to the system. Classical statistics is called Maxwell-Boltzmann statistics (or M-B statistics).
Quantum statistics
The fundamental feature of quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
that distinguishes it from classical mechanics is that particles of a particular type are indistinguishable
Identical particles
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...
from one another. This means that in an assembly consisting of similar particles, interchanging any two particles does not lead to a new configuration of the system (in the language of quantum mechanics: the wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
of the system is invariant with respect to the interchange of the constituent particles). In case of a system consisting of particles belonging to different nature (for example electrons and protons), the wavefunction of the system is invariant separately for the assembly of the two particles.
While this difference between classical and quantum description of systems is fundamental to all of quantum statistics, it is further divided into the following two classes on the basis of symmetry of the system.
Bose-Einstein statistics
In Bose-Einstein statistics (B-E Statistics) interchanging any two particles of the system leaves the resultant system in a symmetric state. That is, the wavefunction of the system before interchanging equals the wavefunction of the system after interchanging.It is important to emphasize that the wavefunction of the system has not changed itself. This has very important consequences on the state of the system: There is no restriction to the number of particles that can be placed in a single state (accessible to the system). It is found that the particles that obey Bose-Einstein statistics are the ones which have integer spins, which are therefore called bosons (named for Bose). Examples of bosons include photons and helium-4
Helium-4
Helium-4 is a non-radioactive isotope of helium. It is by far the most abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on earth. Its nucleus is the same as an alpha particle, consisting of two protons and two neutrons. Alpha decay of heavy...
atoms. One type of system obeying B-E statistics is the Bose-Einstein condensate where all particles of the assembly exist in the same state.
Fermi-Dirac Statistics
In Fermi-Dirac statistics (F-D statistics) interchanging any two particles of the system leaves the resultant system in an antisymmetric state. That is, the wavefunction of the system before interchanging is the wavefunction of the system after interchanging, with an overall minus sign.Again, the wavefunction of the system itself does not change. The consequence of the negative sign on the Fermi-Dirac statistics can be understood in the following way:
Suppose that the particles that are interchanged belong to the same state. Since the particles are considered indistinguishable from one another then changing the coordinates of the particles should not have any change on the system's wavefunction (because by our assumptions the particles are in the same state). Therefore, the wavefunction before interchanging similar states equals the wavefunction after interchanging similar states.
Combining (or adding, literally speaking) the above statement with the fundamental asymmetry of the Fermi-Dirac system leads us to conclude that the wavefunction of the system before interchanging equals zero.
This shows that in Fermi-Dirac statistics, more than one particle cannot occupy a single state accessible to the system. This is called Pauli's exclusion principle.
It is found that particles with half-integral spin (or fermions) obey the Fermi-Dirac statistics. This includes, electrons, protons, Helium-3 etc.