0.3 Bras and kets

An inner product on a vector space V (over the complex numbers) is a function that assigns to each pair of vectors |u\rangle,|v\rangle\in V a complex number \langle u|v\rangle, and satisfies the following conditions:

  • \langle u|v\rangle=\langle v|u\rangle^\star;
  • \langle v|v\rangle\geqslant 0 for all |v\rangle;
  • \langle v|v\rangle= 0 if and only if |v\rangle=0.

The inner product must also be linear in the second argument but antilinear in the first argument: \langle c_1u_1+c_2u_2|v\rangle = c_1^\star\langle u_1|v\rangle+c_2^\star\langle u_2|v\rangle for any complex constants c_1 and c_2.

With any physical system we associate a complex vector space with an inner product, known as a Hilbert space3 \mathcal{H}. The inner product between vectors |u\rangle and |v\rangle in {\mathcal{H}} is written as \langle u|v\rangle.

For example, for column vectors |u\rangle and |v\rangle in \mathbb{C}^n written as |u\rangle = \begin{bmatrix}u_1\\u_2\\\vdots\\u_n\end{bmatrix} \qquad |v\rangle = \begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix} their inner product is defined by \langle u|v\rangle = u_1^\star v_1 + u_2^\star v_2+\ldots + u_n^\star v_n. Following Dirac, we may split the inner product into two ingredients: \langle u|v\rangle \longrightarrow \langle u|\,|v\rangle. Here |v\rangle is a ket vector, and \langle u| is called a bra vector, or a bra, and can be represented by a row vector: \langle u| = \begin{bmatrix}u_1^\star,&u_2^\star,&\ldots,&u_n^\star\end{bmatrix}. The inner product can now be viewed as the result of the matrix multiplication: \begin{aligned} \langle u|v\rangle &= \begin{bmatrix}u_1^\star,&u_2^\star,&\ldots,&u_n^\star\end{bmatrix} \cdot \begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix} \\&= u_1^\star v_1 + u_2^\star v_2 + \ldots + u_n^\star v_n. \end{aligned}

Bras are vectors: you can add them, and multiply them by scalars (which, here, are complex numbers), but they are vectors in the space {\mathcal{H}}^\star which is dual to \mathcal{H}. Elements of {\mathcal{H}}^\star are linear functionals, that is, linear maps from \mathcal{H} to \mathbb{C}. A linear functional \langle u| acting on a vector |v\rangle in \mathcal{H} gives a complex number \langle u|v\rangle.

All Hilbert spaces of the same dimension are isomorphic, so the differences between quantum systems cannot be really understood without additional structure. This structure is provided by a specific algebra of operators acting on \mathcal{H}.

  1. The term “Hilbert space” used to be reserved for an infinite-dimensional inner product space that is complete, i.e. such that every Cauchy sequence in the space converges to an element in the space. Nowadays, as in these notes, the term includes finite-dimensional spaces, which automatically satisfy the condition of completeness.↩︎