---
title: "Slater determinant"
date: 2021-02-22
categories:
- Quantum mechanics
- Physics
layout: "concept"
---

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.