% Slater determinant


# Slater determinant

In quantum mechanics, the **Slater determinant** is a trick to create an
antisymmetric wave function for a system of $N$ fermions.

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
in the matrix, which flips the sign of the determinant. Similarly,
exchanging two columns means swapping two states, which also results in
a sign change. Finally, putting two particles into the same state makes
$\Psi$ vanish.

Note that not all valid many-particle fermionic 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.