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authorPrefetch2024-07-17 10:01:43 +0200
committerPrefetch2024-07-17 10:01:43 +0200
commit075683cdf4588fe16f41d9f7b46b9720b42b2553 (patch)
treef200b341b35e9c89aa1030a2f6da94cd0e1e958b /source/know/concept/dirac-notation
parentc4d597e8d695eb145755464cffbf88a68fd0c88a (diff)
Improve knowledge base
Diffstat (limited to 'source/know/concept/dirac-notation')
-rw-r--r--source/know/concept/dirac-notation/index.md16
1 files changed, 10 insertions, 6 deletions
diff --git a/source/know/concept/dirac-notation/index.md b/source/know/concept/dirac-notation/index.md
index 2830a33..bbf31e5 100644
--- a/source/know/concept/dirac-notation/index.md
+++ b/source/know/concept/dirac-notation/index.md
@@ -27,7 +27,8 @@ that maps kets $$\ket{V}$$ to other kets $$\ket{V'}$$.
Recall that by definition the Hilbert inner product must satisfy:
$$\begin{aligned}
- \inprod{V}{W} = \inprod{W}{V}^*
+ \inprod{V}{W}
+ = \inprod{W}{V}^*
\end{aligned}$$
So far, nothing has been said about the actual representation of bras or kets.
@@ -36,12 +37,14 @@ the corresponding bras are given by the kets' adjoints,
i.e. their transpose conjugates:
$$\begin{aligned}
- \ket{V} =
+ \ket{V}
+ =
\begin{bmatrix}
v_1 \\ \vdots \\ v_N
\end{bmatrix}
- \quad \implies \quad
- \bra{V} =
+ \qquad \implies \qquad
+ \bra{V}
+ =
\begin{bmatrix}
v_1^* & \cdots & v_N^*
\end{bmatrix}
@@ -88,8 +91,9 @@ then the bras are *functionals* $$F[u(x)]$$
that take an arbitrary function $$u(x)$$ as an argument and return a scalar:
$$\begin{aligned}
- \ket{f} = f(x)
- \quad \implies \quad
+ \ket{f}
+ = f(x)
+ \qquad \implies \qquad
\bra{f}
= F[u(x)]
= \int_a^b f^*(x) \: u(x) \dd{x}