Dirac notation

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 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:

\[\begin{aligned} \braket{V}{W} = \braket{W}{V}^* \end{aligned}\]

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:

\[\begin{aligned} \ket{V} = \begin{bmatrix} v_1 \\ \vdots \\ v_N \end{bmatrix} \quad \implies \quad \bra{V} = \begin{bmatrix} v_1^* & \cdots & v_N^* \end{bmatrix} \end{aligned}\]

The inner product \(\braket{V}{W}\) is then just the familiar dot product \(V \cdot W\):

\[\begin{gathered} \braket{V}{W} = \begin{bmatrix} v_1^* & \cdots & v_N^* \end{bmatrix} \cdot \begin{bmatrix} w_1 \\ \vdots \\ w_N \end{bmatrix} = v_1^* w_1 + ... + v_N^* w_N \end{gathered}\]

Meanwhile, the outer product \(\ket{V} \bra{W}\) creates an \(N \cross N\) matrix:

\[\begin{gathered} \ket{V} \bra{W} = \begin{bmatrix} v_1 \\ \vdots \\ v_N \end{bmatrix} \cdot \begin{bmatrix} w_1^* & \cdots & w_N^* \end{bmatrix} = \begin{bmatrix} v_1 w_1^* & \cdots & v_1 w_N^* \\ \vdots & \ddots & \vdots \\ v_N w_1^* & \cdots & v_N w_N^* \end{bmatrix} \end{gathered}\]

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 using integration:

\[\begin{aligned} \ket{f} = f(x) \quad \implies \quad \bra{f} = F[u(x)] = \int_a^b f^*(x) \: u(x) \dd{x} \end{aligned}\]

Consequently, the inner product is simply the following familiar integral:

\[\begin{gathered} \braket{f}{g} = F[g(x)] = \int_a^b f^*(x) \: g(x) \dd{x} \end{gathered}\]

However, the outer product becomes something rather abstract:

\[\begin{gathered} \ket{f} \bra{g} = f(x) \: G[u(x)] = f(x) \int_a^b g^*(\xi) \: u(\xi) \dd{\xi} \end{gathered}\]

This result makes more sense if we surround it by a bra and a ket:

\[\begin{aligned} \bra{u} \!\Big(\!\ket{f} \bra{g}\!\Big)\! \ket{w} &= U\big[f(x) \: G[w(x)]\big] = U\Big[ f(x) \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big] \\ &= \int_a^b u^*(x) \: f(x) \: \Big(\int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) \dd{x} \\ &= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) \\ &= \braket{u}{f} \braket{g}{w} \end{aligned}\]

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