Dirac Bra-Ket Notation

Dirac notation writes quantum states as kets and their duals as bras, making inner products, outer products, and operators compact.

A compact language

Dirac notation, or bra-ket notation, is the standard shorthand of quantum mechanics. A state is a ket |psi>, a column vector. Its conjugate transpose is a bra

Inner products

Joining a bra and a ket gives a bracket , the inner product — a single complex number measuring overlap. If it is zero the states are orthogonal (perfectly distinguishable); its squared magnitude gives a transition probability via the Born rule. Normalisation is = 1.

• Ket |psi>: a state, column vector
• Bra
• Bracket : inner product, a complex number
• Outer product |psi>

Outer products and operators

Reversing the order, |psi>outer product — a matrix, i.e. an operator. The projector onto |psi> is |psi>density matrix is exactly this. The identity resolves as sum over a basis of |x>

Worked expressions

With |psi> = a|0> + b|1>, we have <0|psi> = a and <1|psi> = b, so amplitudes are just brackets. The Born probability of 0 is |<0|psi>|^2 = |a|^2. An expectation value is . The notation makes these standard patterns readable at a glance.

Why it endures

Dirac notation is basis-independent and mechanical: expressions can be manipulated by moving bras and kets past operators without choosing coordinates. This is why nearly all quantum computing literature uses it, and fluency in it is a prerequisite for reading algorithms.