Editing Indistinguishable particles (section)

=== Symmetrical and antisymmetrical states ===
[[Image:Asymmetricwave2.png|right|thumb|Antisymmetric wavefunction for a (fermionic) 2-particle state in an infinite square well potential]]
[[Image:Symmetricwave2.png|right|thumb|Symmetric wavefunction for a (bosonic) 2-particle state in an infinite square well potential]]

What follows is an example to make the above discussion concrete, using the formalism developed in the article on the [[mathematical formulation of quantum mechanics]].

Let ''n'' denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the [[particle in a box]] problem, take ''n'' to be the quantized [[wave vector]] 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 ''n''<sub>1</sub>, and the other is in the state ''n''<sub>2</sub>. The quantum state of the system is denoted by the expression
: <math> | n_1 \rang | n_2 \rang </math>
where the order of the tensor product matters ( if <math> | n_2 \rang | n_1 \rang </math>, then the particle 1 occupies the state ''n''<sub>2</sub> while the particle 2 occupies the state ''n''<sub>1</sub>). This is the canonical way of constructing a basis for a [[tensor product]] space <math>H \otimes H</math> of the combined system from the individual spaces. This expression is valid for distinguishable particles, however, it is not appropriate for indistinguishable particles since <math> |n_1\rang |n_2\rang</math> and <math>|n_2\rang |n_1\rang </math> as a result of exchanging the particles are generally different states.
* "the particle 1 occupies the ''n''<sub>1</sub> state and the particle 2 occupies the ''n''<sub>2</sub> state" ≠ "the particle 1 occupies the ''n''<sub>2</sub> state and the particle 2 occupies the ''n''<sub>1</sub> state".

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 suggests that a state for two indistinguishable (and non-interacting) particles is given by following two possibilities:<ref>Haynes, P. [http://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/node14.html Linear-scaling methods in ab initio quantum-mechanical calculations]. Diss. University of Cambridge, 1998. Section 2.3 Identical particles</ref><ref>{{harvtxt|Tuckerman|2010|p=385}}</ref><ref>{{Cite book|title=Introductory Quantum Mechanics|last=Liboff|first=Richard|publisher=Addison-Wesley|year=2003|isbn=978-0805387148|pages=597}}</ref>
: <math> |n_1\rang |n_2\rang \pm |n_2\rang |n_1\rang </math>

States where it is a sum are known as '''symmetric''', while states involving the difference are called '''antisymmetric'''. More completely, symmetric states have the form
: <math> |n_1, n_2; S\rang \equiv \mbox{constant} \times \bigg( |n_1\rang |n_2\rang + |n_2\rang |n_1\rang \bigg) </math>
while antisymmetric states have the form
: <math> |n_1, n_2; A\rang \equiv \mbox{constant} \times \bigg( |n_1\rang |n_2\rang - |n_2\rang |n_1\rang \bigg) </math>

Note that if ''n''<sub>1</sub> and ''n''<sub>2</sub> 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 [[Pauli exclusion principle]], and it is the fundamental reason behind the [[chemistry|chemical]] properties of atoms and the stability of [[matter]].