Dirac Notation Cheatsheet

Paul Dirac introduced a compact shorthand for quantum states and their relationships. Once you internalize it, manipulating quantum mechanics becomes as fluent as algebra. This lesson collects the four core constructs in one place.

Kets — column vectors representing states

A ket $|\psi\rangle$ represents a quantum state as a column vector. The two computational basis kets for a single qubit are

|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.

A general single-qubit state is a superposition

|\psi\rangle = \alpha\,|0\rangle + \beta\,|1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix},

where $\alpha, \beta \in \mathbb{C}$ and the normalization condition $|\alpha|^2 + |\beta|^2 = 1$ ensures that the Born-rule probabilities sum to one.

Bras — row vectors and dual space

Every ket $|\psi\rangle$ has a bra partner $\langle\psi|$ , which is its conjugate transpose (a row vector):

\langle\psi| = \big(\alpha^*,\; \beta^*\big).

The name "bra" and "ket" together form a "bracket" — a pun that hints at the inner product below.

Inner product — overlap and probability

The inner product $\langle\phi|\psi\rangle$ multiplies the bra of $|\phi\rangle$ by the ket of $|\psi\rangle$ :

\langle\phi|\psi\rangle = \phi_0^*\,\psi_0 + \phi_1^*\,\psi_1.

Two key consequences follow immediately. First, basis states are orthonormal:

\langle 0|0\rangle = 1, \quad \langle 1|1\rangle = 1, \quad \langle 0|1\rangle = 0.

Outer product — operators from state pairs

The outer product $|\psi\rangle\langle\phi|$ multiplies the ket of $|\psi\rangle$ by the bra of $|\phi\rangle$ and produces a matrix (an operator), not a scalar. For example,

|0\rangle\langle 0| = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.

The projector $|0\rangle\langle 0|$ projects any state onto $|0\rangle$ , extracting the $\alpha$ component. More generally, any linear operator on an $n$ -dimensional space can be written as a linear combination of at most $n^2$ such rank-1 terms, because the space of $n\times n$ matrices has dimension $n^2$ . A special case is the spectral decomposition: a normal operator (e.g.\ a Hermitian observable) can be diagonalized as $A = \sum_i \lambda_i |e_i\rangle\langle e_i|$ using only $n$ projectors weighted by eigenvalues $\lambda_i$ .

Quick reference

| Symbol | Name | Object type | What it does | |--------|------|------------|--------------| | $|\psi\rangle$ | ket | column vector | represents a quantum state | | $\langle\psi|$ | bra | row vector | dual of a ket (conjugate transpose) | | $\langle\phi|\psi\rangle$ | inner product | scalar (complex number) | overlap; gives probabilities via Born rule | | $|\psi\rangle\langle\phi|$ | outer product | matrix (operator) | maps states to states |

These four pieces underpin everything from gate definitions to measurement theory to algorithm analysis. Keep this cheatsheet handy as a reference while you work through the rest of the module.