Bra–ket notation

Bra–ket notation is a compact way to represent quantum states, dual vectors, and operators. It uses kets such as |ψ⟩, bras such as ⟨ψ|, and inner products such as ⟨φ|ψ⟩.

Hilbert space: A complete inner-product vector space that provides the mathematical setting for quantum states. Kets represent vectors in the Hilbert space of a quantum system.
Quantum state: A mathematical description of a quantum system that determines probabilities for measurement outcomes. A ket commonly represents a pure quantum state.
Schrödinger equation: The equation governing the time evolution of a quantum state under a Hamiltonian. Its state vector can be written as a time-dependent ket.
Matrix representation: A representation of linear transformations and vectors as matrices and coordinate arrays. It makes basis-dependent coordinates explicit where bra–ket notation can remain abstract.
Vector space: A set of vectors with operations for addition and scalar multiplication. Kets obey vector addition and scalar multiplication.
Outer product: A product of a vector and a dual vector that forms a linear operator. The expression |ψ⟩⟨φ| denotes an operator built from a ket and a bra.
Quantum measurement: The physical process that produces outcomes described by probabilities in quantum theory. Bras and kets express outcome amplitudes and measurement probabilities.
Wavefunction: A complex-valued function that represents a quantum state in a position or other continuous basis. A wavefunction gives basis-specific components of the ket.
Dual space: The vector space of linear functionals on a given vector space. Bras belong to the dual space associated with the ket space.
Projection operator: An idempotent linear operator that projects vectors onto a subspace. A normalized ket forms a projector as |ψ⟩⟨ψ|.

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