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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">\(\mathbb{H}'\)</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">\(\ket{V'}\)</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">\(\ket{V'}\)</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">\[\begin{aligned} @@ -68,9 +68,9 @@ v_1^* & \cdots & v_N^* \end{bmatrix} \end{aligned}\]</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">\[\begin{gathered} - \braket{V | W} + \braket{V}{W} = \begin{bmatrix} v_1^* & \cdots & v_N^* @@ -109,7 +109,7 @@ \end{aligned}\]</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 @@ \\ &= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) \\ - &= \braket{u | f} \braket{g | w} + &= \braket{u}{f} \braket{g}{w} \end{aligned}\]</span></p> <hr> © "Prefetch". Licensed under <a href="https://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA 4.0</a>. |