Categories: Physics, Quantum mechanics.

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. 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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