Editing Indistinguishable particles (section)

=== Wavefunction representation ===

So far, the discussion has included only discrete observables. It can be extended to continuous observables, such as the [[position (vector)|position]]&nbsp;''x''.

Recall that an eigenstate of a continuous observable represents an infinitesimal ''range'' of values of the observable, not a single value as with discrete observables. For instance, if a particle is in a state |''ψ''⟩, the probability of finding it in a region of volume ''d''<sup>3</sup>''x'' surrounding some position ''x'' is
: <math> |\lang x | \psi \rang|^2 \; d^3 x </math>

As a result, the continuous eigenstates |''x''⟩ are normalized to the [[dirac delta function|delta function]] instead of unity:
: <math> \lang x | x' \rang = \delta^3 (x - x') </math>

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:
: <math>\begin{align}
  |x_1 x_2 \cdots x_N; S\rang &= \sqrt{\frac{\prod_j n_j!}{N!}} \sum_p \left|x_{p(1)}\right\rang \left|x_{p(2)}\right\rang \cdots \left|x_{p(N)}\right\rang \\
  |x_1 x_2 \cdots x_N; A\rang &= \frac{1}{\sqrt{N!}} \sum_p \mathrm{sgn}(p) \left|x_{p(1)}\right\rang \left|x_{p(2)}\right\rang \cdots \left|x_{p(N)}\right\rang
\end{align}</math>

A many-body [[wavefunction]] can be written,
: <math>\begin{align}
  \Psi^{(S)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \ldots, x_N)
    & \equiv \lang x_1 x_2 \cdots x_N; S | n_1 n_2 \cdots n_N; S \rang \\[4pt]
    & = \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) \\[10pt]
  \Psi^{(A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \ldots, x_N)
    & \equiv \lang x_1 x_2 \cdots x_N; A | n_1 n_2 \cdots n_N; A \rang \\[4pt]
    & = \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{align}</math>
where the single-particle wavefunctions are defined, as usual, by
: <math>\psi_n(x) \equiv \lang x | n \rang </math>

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:
: <math>\begin{align}
  \Psi^{(S)}_{n_1 \cdots n_N} (\cdots x_i \cdots x_j\cdots) =
     \Psi^{(S)}_{n_1 \cdots n_N} (\cdots x_j \cdots x_i \cdots) \\[3pt]
  \Psi^{(A)}_{n_1 \cdots n_N} (\cdots x_i \cdots x_j\cdots) =
    -\Psi^{(A)}_{n_1 \cdots n_N} (\cdots x_j \cdots x_i \cdots)
\end{align}</math>

The many-body wavefunction has the following significance: if the system is initially in a state with quantum numbers ''n''<sub>1</sub>, ..., n<sub>N</sub>, and a position measurement is performed, the probability of finding particles in infinitesimal volumes near ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''N''</sub> is
: <math> N! \; \left|\Psi^{(S/A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \ldots, x_N) \right|^2 \; d^{3N}\!x </math>

The factor of ''N''! comes from our normalizing constant, which has been chosen so that, by analogy with single-particle wavefunctions,
: <math> \int\!\int\!\cdots\!\int\; \left|\Psi^{(S/A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \ldots, x_N)\right|^2 d^3\!x_1 d^3\!x_2 \cdots d^3\!x_N = 1 </math>

Because each integral runs over all possible values of ''x'', each multi-particle state appears ''N''! times in the integral. In other words, the probability associated with each event is evenly distributed across ''N''! 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.

Finally, antisymmetric wavefunction can be written as the [[determinant]] of a [[Matrix (mathematics)|matrix]], known as a [[Slater determinant]]:
: <math>\Psi^{(A)}_{n_1 \cdots n_N} (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|
</math>