Categories: Physics, Quantum mechanics.

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}

References

1. R. Shankar, Principles of quantum mechanics, 2nd edition, Springer.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.