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---
title: "Slater determinant"
firstLetter: "S"
publishDate: 2021-02-22
categories:
- Quantum mechanics
- Physics
date: 2021-02-22T21:38:03+01:00
draft: false
markup: pandoc
---
# Slater determinant
In quantum mechanics, the **Slater determinant** is a trick
to create a many-particle wave function for a system of $N$ fermions,
with the necessary antisymmetry.
Given an orthogonal set of individual states $\psi_n(x)$, we write
$\psi_n(x_n)$ to say that particle $x_n$ is in state $\psi_n$. Now the
goal is to find an expression for an overall many-particle wave
function $\Psi(x_1, ..., x_N)$ that satisfies the
[Pauli exclusion principle](/know/concept/pauli-exclusion-principle/).
Enter the Slater determinant:
$$\begin{aligned}
\boxed{
\Psi(x_1, ..., x_N)
= \frac{1}{\sqrt{N!}} \det\!
\begin{bmatrix}
\psi_1(x_1) & \cdots & \psi_N(x_1) \\
\vdots & \ddots & \vdots \\
\psi_1(x_N) & \cdots & \psi_N(x_N)
\end{bmatrix}
}\end{aligned}$$
Swapping the state of two particles corresponds to exchanging two rows,
which flips the sign of the determinant.
Similarly, switching two columns means swapping two states,
which also results in a sign change.
Finally, putting two particles into the same state makes $\Psi$ vanish.
Not all valid many-fermion wave functions can be
written as a single Slater determinant; a linear combination of multiple
may be needed. Nevertheless, an appropriate choice of the input set
$\psi_n(x)$ can optimize how well a single determinant approximates a
given $\Psi$.
In fact, there exists a similar trick for bosons, where the goal is to
create a symmetric wave function which allows multiple particles to
occupy the same state. In this case, one needs to take the **Slater
permanent** of the same matrix, which is simply the determinant, but with
all minuses replaced by pluses.
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