Fix "Dirac notation"

1 files changed, 6 insertions, 6 deletions

diff --git a/static/know/concept/dirac-notation/index.html b/static/know/concept/dirac-notation/index.html
index 74aa0b4..5da8c1b 100644
--- a/static/know/concept/dirac-notation/index.html
+++ b/static/know/concept/dirac-notation/index.html
@@ -52,9 +52,9 @@
 <h1 id="dirac-notation">Dirac notation</h1>
 <p><em>Dirac notation</em> is a notation to do calculations in a Hilbert space without needing to worry about the space’s representation. It is basically the <em>lingua franca</em> of quantum mechanics.</p>
 <p>In Dirac notation there are <em>kets</em> <span class="math inline">\(\ket{V}\)</span> from the Hilbert space <span class="math inline">\(\mathbb{H}\)</span> and <em>bras</em> <span class="math inline">\(\bra{V}\)</span> from a dual <span class="math inline">\(\mathbb{H}&#39;\)</span> 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.</p>
-<p>Bras and kets can only be combined in two ways: the <em>inner product</em> <span class="math inline">\(\braket{V | W}\)</span>, which returns a scalar, and the <em>outer product</em> <span class="math inline">\(\ket{V} \bra{W}\)</span>, which returns a mapping <span class="math inline">\(\hat{L}\)</span> from kets <span class="math inline">\(\ket{V}\)</span> to other kets <span class="math inline">\(\ket{V&#39;}\)</span>, i.e. a linear operator. Recall that the Hilbert inner product must satisfy:</p>
+<p>Bras and kets can only be combined in two ways: the <em>inner product</em> <span class="math inline">\(\braket{V}{W}\)</span>, which returns a scalar, and the <em>outer product</em> <span class="math inline">\(\ket{V} \bra{W}\)</span>, which returns a mapping <span class="math inline">\(\hat{L}\)</span> from kets <span class="math inline">\(\ket{V}\)</span> to other kets <span class="math inline">\(\ket{V&#39;}\)</span>, i.e. a linear operator. Recall that the Hilbert inner product must satisfy:</p>
 <p><span class="math display">\[\begin{aligned}
-    \braket{V | W} = \braket{W | V}^*
+    \braket{V}{W} = \braket{W}{V}^*
 \end{aligned}\]</span></p>
 <p>So far, nothing has been said about the actual representation of bras or kets. If we represent kets as <span class="math inline">\(N\)</span>-dimensional columns vectors, the corresponding bras are given by the kets’ adjoints, i.e. their transpose conjugates:</p>
 <p><span class="math display">\[\begin{aligned}
@@ -68,9 +68,9 @@
         v_1^* &amp; \cdots &amp; v_N^*
     \end{bmatrix}
 \end{aligned}\]</span></p>
-<p>The inner product <span class="math inline">\(\braket{V | W}\)</span> is then just the familiar dot product <span class="math inline">\(V \cdot W\)</span>:</p>
+<p>The inner product <span class="math inline">\(\braket{V}{W}\)</span> is then just the familiar dot product <span class="math inline">\(V \cdot W\)</span>:</p>
 <p><span class="math display">\[\begin{gathered}
-    \braket{V | W}
+    \braket{V}{W}
     =
     \begin{bmatrix}
         v_1^* &amp; \cdots &amp; v_N^*
@@ -109,7 +109,7 @@
 \end{aligned}\]</span></p>
 <p>Consequently, the inner product is simply the following familiar integral:</p>
 <p><span class="math display">\[\begin{gathered}
-    \braket{f | g}
+    \braket{f}{g}
     = F[g(x)]
     = \int_a^b f^*(x) \: g(x) \dd{x}
 \end{gathered}\]</span></p>
@@ -129,7 +129,7 @@
     \\
     &amp;= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big)
     \\
-    &amp;= \braket{u | f} \braket{g | w}
+    &amp;= \braket{u}{f} \braket{g}{w}
 \end{aligned}\]</span></p>
 <hr>
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