Dirac Notation

Prerequisites: Linear Algebra, Hilbert Space.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

In quantum mechanics the state of a system is a vector in a Hilbert space. We can describe an arrow in ordinary space without choosing axes, and only write down three numbers once we have picked a coordinate system. Dirac notation does the same for quantum states. A state is written as a "ket", and this symbol stands for the vector itself, independent of any basis.

The second ingredient is the "bra". A bra is a machine that eats a ket and returns a complex number. Putting a bra and a ket together gives a "bracket", the overlap of the two states. Its squared modulus is the probability to find the system in the state of the bra when it was prepared in the state of the ket.

The notation is useful because the same few symbols work for every system. A spin with two states, an atom with infinitely many energy levels and a particle moving along a line are all handled with kets, bras and operators acting on them. Changing from one description to another, for example from position to momentum, becomes the insertion of a different sum over basis states. Much of the bookkeeping that would otherwise need explicit matrices and integrals is done by the shape of the symbols.

Concrete

• A particularly nice introduction can be found in Chapter 1 in "Principles of Quantum Mechanics" by Shankar, where he reformulates linear algebra using the Dirac notation. This way the sometimes quite abstract notation can be easily understood using familiar notions from linear algebra.

Abstract

Let $\mathcal{H}$ be a complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$, linear in the second and antilinear in the first argument (the physics convention). A ket $|\psi\rangle$ is an element $\psi\in\mathcal{H}$. A bra $\langle\phi|$ is the continuous linear functional $\psi\mapsto\langle\phi,\psi\rangle$, an element of the dual space $\mathcal{H}^*$. By the Riesz representation theorem every continuous linear functional on $\mathcal{H}$ arises this way from a unique vector, so the map $|\phi\rangle\mapsto\langle\phi|$ is an antilinear isometric bijection $\mathcal{H}\to\mathcal{H}^*$. The bracket $\langle\phi|\psi\rangle$ is the evaluation of the functional on the vector.

The outer product $|\psi\rangle\langle\phi|$ is the rank-one operator $\chi\mapsto\langle\phi|\chi\rangle\,\psi$. For an orthonormal basis $\{|n\rangle\}$ of a separable Hilbert space the resolution of the identity $$\sum_n |n\rangle\langle n| = \mathbb{1}$$ holds with the sum converging in the strong operator topology. More generally, for a self-adjoint operator $A$ the spectral theorem gives a projection-valued measure $E_A$ with $A=\int \lambda\, dE_A(\lambda)$ and $\int dE_A(\lambda)=\mathbb{1}$.

Expressions such as $|x\rangle$ and $|p\rangle$ with $\langle x|x'\rangle=\delta(x-x')$ do not denote elements of $\mathcal{H}=L^2(\mathbb{R})$, because the position and momentum operators have purely continuous spectrum and no normalizable eigenvectors. The standard rigorous framework is a rigged Hilbert space (Gelfand triple) $$\Phi\subset\mathcal{H}\subset\Phi^\times,$$ where $\Phi$ is a dense nuclear subspace with a finer topology, for example the Schwartz space $\mathcal{S}(\mathbb{R})\subset L^2(\mathbb{R})$, and $\Phi^\times$ is its space of continuous antilinear functionals, here the tempered distributions. The nuclear spectral theorem (Gelfand and Maurin) states that a self-adjoint operator which maps $\Phi$ continuously into itself has a complete set of generalized eigenvectors in $\Phi^\times$. In this setting $|x\rangle$ is the functional $\varphi\mapsto\overline{\varphi(x)}$ and $\langle x|\psi\rangle=\psi(x)$, and $\int dx\,|x\rangle\langle x|=\mathbb{1}$ holds as an identity of maps $\Phi\to\Phi^\times$.

Why is it interesting?

The Dirac notation is a convenient way to describe the basic quantities in quantum mechanics.