From 6fb3b28a2ce91b8f12683692cbe32d9a4d35fc9d Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 21 Feb 2021 10:46:28 +0100
Subject: Add "Slater determinant" + improvements to knowledge base
---
.../concept/pauli-exclusion-principle/source.md | 10 ++---
latex/know/concept/probability-current/source.md | 2 +-
latex/know/concept/slater-determinant/source.md | 43 ++++++++++++++++++++++
.../time-independent-perturbation-theory/source.md | 2 +-
4 files changed, 50 insertions(+), 7 deletions(-)
create mode 100644 latex/know/concept/slater-determinant/source.md
(limited to 'latex/know/concept')
diff --git a/latex/know/concept/pauli-exclusion-principle/source.md b/latex/know/concept/pauli-exclusion-principle/source.md
index d1c2149..f95541f 100644
--- a/latex/know/concept/pauli-exclusion-principle/source.md
+++ b/latex/know/concept/pauli-exclusion-principle/source.md
@@ -3,8 +3,8 @@
# Pauli exclusion principle
-In quantum mechanics, the **Pauli exclusion principle** is a theorem that
-has profound consequences for how the world works.
+In quantum mechanics, the **Pauli exclusion principle** is a theorem with
+profound consequences for how the world works.
Suppose we have a composite state
$\ket*{x_1}\ket*{x_2} = \ket*{x_1} \otimes \ket*{x_2}$, where the two
@@ -34,8 +34,8 @@ $$\begin{aligned}
As it turns out, in nature, each class of particle has a single
associated permutation eigenvalue $\lambda$, or in other words: whether
-$\lambda$ is $-1$ or $1$ depends on the species of particle that $x_1$
-and $x_2$ represent. Particles with $\lambda = -1$ are called
+$\lambda$ is $-1$ or $1$ depends on the type of particle that $x_1$
+and $x_2$ are. Particles with $\lambda = -1$ are called
**fermions**, and those with $\lambda = 1$ are known as **bosons**. We
define $\hat{P}_f$ with $\lambda = -1$ and $\hat{P}_b$ with
$\lambda = 1$, such that:
@@ -109,7 +109,7 @@ $$\begin{aligned}
= 0
\end{aligned}$$
-At last, this is the Pauli exclusion principle: **fermions may never
+And this is the Pauli exclusion principle: **fermions may never
occupy the same quantum state**. One of the many notable consequences of
this is that the shells of atoms only fit a limited number of
electrons (which are fermions), since each must have a different quantum number.
diff --git a/latex/know/concept/probability-current/source.md b/latex/know/concept/probability-current/source.md
index bb84121..7c29111 100644
--- a/latex/know/concept/probability-current/source.md
+++ b/latex/know/concept/probability-current/source.md
@@ -59,7 +59,7 @@ $$\begin{aligned}
= - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}}
\end{aligned}$$
-By removing the integrals, we thus arrive at the *continuity equation*
+By removing the integrals, we thus arrive at the **continuity equation**
for $\vec{J}$:
$$\begin{aligned}
diff --git a/latex/know/concept/slater-determinant/source.md b/latex/know/concept/slater-determinant/source.md
new file mode 100644
index 0000000..5d13c56
--- /dev/null
+++ b/latex/know/concept/slater-determinant/source.md
@@ -0,0 +1,43 @@
+% 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.
diff --git a/latex/know/concept/time-independent-perturbation-theory/source.md b/latex/know/concept/time-independent-perturbation-theory/source.md
index 48504f4..d2b83e2 100644
--- a/latex/know/concept/time-independent-perturbation-theory/source.md
+++ b/latex/know/concept/time-independent-perturbation-theory/source.md
@@ -297,7 +297,7 @@ The trick is to find a Hermitian operator $\hat{L}$ (usually using
symmetries of the system) which commutes with both $\hat{H}_0$ and $\hat{H}_1$:
$$\begin{aligned}
- \comm*{\hat{L}}{\hat{H}_0} = \comm{\hat{L}}{\hat{H}_1} = 0
+ \comm*{\hat{L}}{\hat{H}_0} = \comm*{\hat{L}}{\hat{H}_1} = 0
\end{aligned}$$
So that it shares its eigenstates with $\hat{H}_0$ (and $\hat{H}_1$),
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