Categories: Physics, Quantum mechanics.

Dirac notation

Dirac notation enables us to do calculations in a general Hilbert space without needing to worry about the space’s representation. It is the lingua franca of quantum mechanics.

In Dirac notation there are kets ∣V⟩\ket{V} from the Hilbert space H\mathbb{H} and bras ⟨V∣\bra{V} from its dual space H′\mathbb{H}'. 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 ⟨V∣W⟩\inprod{V}{W}, which returns a scalar, and the outer product ∣V⟩⟨W∣\ket{V} \bra{W}, which returns a linear operator that maps kets ∣V⟩\ket{V} to other kets ∣V′⟩\ket{V'}. Recall that by definition the Hilbert inner product must satisfy:

⟨V∣W⟩=⟨W∣V⟩∗\begin{aligned} \inprod{V}{W} = \inprod{W}{V}^* \end{aligned}

So far, nothing has been said about the actual representation of bras or kets. If we represent kets as NN-dimensional columns vectors, the corresponding bras are given by the kets’ adjoints, i.e. their transpose conjugates:

∣V⟩=[v1⋮vN]  ⟹  ⟨V∣=[v1∗⋯vN∗]\begin{aligned} \ket{V} = \begin{bmatrix} v_1 \\ \vdots \\ v_N \end{bmatrix} \qquad \implies \qquad \bra{V} = \begin{bmatrix} v_1^* & \cdots & v_N^* \end{bmatrix} \end{aligned}

The inner product ⟨V∣W⟩\inprod{V}{W} is then just the familiar dot product V⋅WV \cdot W:

⟨V∣W⟩=[v1∗⋯vN∗]⋅[w1⋮wN]=v1∗w1+...+vN∗wN\begin{gathered} \inprod{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 ∣V⟩⟨W∣\ket{V} \bra{W} creates an N×NN \cross N matrix, which can be thought of as applying an operation to any vector it multiplies:

∣V⟩⟨W∣=[v1⋮vN]⋅[w1∗⋯wN∗]=[v1w1∗⋯v1wN∗⋮⋱⋮vNw1∗⋯vNwN∗]\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 continuous functions f(x)f(x) of x∈[a,b]x \in [a, b], then the bras are functionals F[u(x)]F[u(x)] that take an arbitrary function u(x)u(x) as an argument and return a scalar:

∣f⟩=f(x)  ⟹  ⟨f∣=F[u(x)]=∫abf∗(x) u(x)dx\begin{aligned} \ket{f} = f(x) \qquad \implies \qquad \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:

⟨f∣g⟩=F[g(x)]=∫abf∗(x) g(x)dx\begin{gathered} \inprod{f}{g} = F[g(x)] = \int_a^b f^*(x) \: g(x) \dd{x} \end{gathered}

However, the outer product is then rather abstract: a continuous analogue of a matrix:

∣f⟩⟨g∣=f(x) G[u(x)]=f(x)∫abg∗(ξ) u(ξ)dξ\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 maybe makes more sense if we surround it by a bra ⟨u∣\bra{u} and a ket ∣w⟩\ket{w} and rearrange:

⟨u∣ ⁣( ⁣∣f⟩⟨g∣ ⁣) ⁣∣w⟩=U[f(x) G[w(x)]]=U[f(x)∫abg∗(ξ) w(ξ)dξ]=∫abu∗(x) f(x) (∫abg∗(ξ) w(ξ)dξ)dx=(∫abu∗(x) f(x)dx)(∫abg∗(ξ) w(ξ)dξ)=⟨u∣f⟩⟨g∣w⟩\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) \\ &= \inprod{u}{f} \inprod{g}{w} \end{aligned}

References

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