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authorPrefetch2022-10-20 18:25:31 +0200
committerPrefetch2022-10-20 18:25:31 +0200
commit16555851b6514a736c5c9d8e73de7da7fc9b6288 (patch)
tree76b8bfd30f8941d0d85365990bcdbc5d0643cabc /source/know/concept/dirac-notation
parente5b9bce79b68a68ddd2e51daa16d2fea73b84fdb (diff)
Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
Diffstat (limited to 'source/know/concept/dirac-notation')
-rw-r--r--source/know/concept/dirac-notation/index.md22
1 files changed, 11 insertions, 11 deletions
diff --git a/source/know/concept/dirac-notation/index.md b/source/know/concept/dirac-notation/index.md
index 8861505..46cc325 100644
--- a/source/know/concept/dirac-notation/index.md
+++ b/source/know/concept/dirac-notation/index.md
@@ -12,16 +12,16 @@ layout: "concept"
without needing to worry about the space's representation. It is
basically the *lingua franca* of quantum mechanics.
-In Dirac notation there are **kets** $\Ket{V}$ from the Hilbert space
-$\mathbb{H}$ and **bras** $\Bra{V}$ from a dual $\mathbb{H}'$ of the
+In Dirac notation there are **kets** $$\Ket{V}$$ from the Hilbert space
+$$\mathbb{H}$$ and **bras** $$\Bra{V}$$ from a dual $$\mathbb{H}'$$ of the
former. Crucially, the bras and kets are from different Hilbert spaces
and therefore cannot be added, but every bra has a corresponding ket and
vice versa.
Bras and kets can be combined in two ways: the **inner product**
-$\Inprod{V}{W}$, which returns a scalar, and the **outer product**
-$\Ket{V} \Bra{W}$, which returns a mapping $\hat{L}$ from kets $\Ket{V}$
-to other kets $\Ket{V'}$, i.e. a linear operator. Recall that the
+$$\Inprod{V}{W}$$, which returns a scalar, and the **outer product**
+$$\Ket{V} \Bra{W}$$, which returns a mapping $$\hat{L}$$ from kets $$\Ket{V}$$
+to other kets $$\Ket{V'}$$, i.e. a linear operator. Recall that the
Hilbert inner product must satisfy:
$$\begin{aligned}
@@ -29,7 +29,7 @@ $$\begin{aligned}
\end{aligned}$$
So far, nothing has been said about the actual representation of bras or
-kets. If we represent kets as $N$-dimensional columns vectors, the
+kets. If we represent kets as $$N$$-dimensional columns vectors, the
corresponding bras are given by the kets' adjoints, i.e. their transpose
conjugates:
@@ -45,7 +45,7 @@ $$\begin{aligned}
\end{bmatrix}
\end{aligned}$$
-The inner product $\Inprod{V}{W}$ is then just the familiar dot product $V \cdot W$:
+The inner product $$\Inprod{V}{W}$$ is then just the familiar dot product $$V \cdot W$$:
$$\begin{gathered}
\Inprod{V}{W}
@@ -60,7 +60,7 @@ $$\begin{gathered}
= v_1^* w_1 + ... + v_N^* w_N
\end{gathered}$$
-Meanwhile, the outer product $\Ket{V} \Bra{W}$ creates an $N \cross N$ matrix:
+Meanwhile, the outer product $$\Ket{V} \Bra{W}$$ creates an $$N \cross N$$ matrix:
$$\begin{gathered}
\Ket{V} \Bra{W}
@@ -80,9 +80,9 @@ $$\begin{gathered}
\end{bmatrix}
\end{gathered}$$
-If the kets are instead represented by functions $f(x)$ of
-$x \in [a, b]$, then the bras represent *functionals* $F[u(x)]$ which
-take an unknown function $u(x)$ as an argument and turn it into a scalar
+If the kets are instead represented by functions $$f(x)$$ of
+$$x \in [a, b]$$, then the bras represent *functionals* $$F[u(x)]$$ which
+take an unknown function $$u(x)$$ as an argument and turn it into a scalar
using integration:
$$\begin{aligned}