Dirac notation is a notation to do calculations in a Hilbert space 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 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 only be combined in two ways: the inner product \(\braket{V | W}\), 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:
+Bras and kets can only be combined in two ways: the inner product \(\braket{V}{W}\), 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:
\[\begin{aligned} - \braket{V | W} = \braket{W | V}^* + \braket{V}{W} = \braket{W}{V}^* \end{aligned}\]
So far, nothing has been said about the actual representation of bras or kets. If we represent kets as \(N\)-dimensional columns vectors, the corresponding bras are given by the kets’ adjoints, i.e. their transpose conjugates:
\[\begin{aligned} @@ -68,9 +68,9 @@ v_1^* & \cdots & v_N^* \end{bmatrix} \end{aligned}\]
-The inner product \(\braket{V | W}\) is then just the familiar dot product \(V \cdot W\):
+The inner product \(\braket{V}{W}\) is then just the familiar dot product \(V \cdot W\):
\[\begin{gathered} - \braket{V | W} + \braket{V}{W} = \begin{bmatrix} v_1^* & \cdots & v_N^* @@ -109,7 +109,7 @@ \end{aligned}\]
Consequently, the inner product is simply the following familiar integral:
\[\begin{gathered} - \braket{f | g} + \braket{f}{g} = F[g(x)] = \int_a^b f^*(x) \: g(x) \dd{x} \end{gathered}\]
@@ -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}\]