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. 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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