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authorPrefetch2022-10-20 18:25:31 +0200
committerPrefetch2022-10-20 18:25:31 +0200
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--- a/source/know/concept/slater-determinant/index.md
+++ b/source/know/concept/slater-determinant/index.md
@@ -9,13 +9,13 @@ layout: "concept"
---
In quantum mechanics, the **Slater determinant** is a trick
-to create a many-particle wave function for a system of $N$ fermions,
+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
+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
+function $$\Psi(x_1, ..., x_N)$$ that satisfies the
[Pauli exclusion principle](/know/concept/pauli-exclusion-principle/).
Enter the Slater determinant:
@@ -34,13 +34,13 @@ 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.
+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 $\Psi$.
+$$\psi_n(x)$$ 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