# Bra–ket notation

Used ubiquitously to denote quantum states.

- Bra–ket notation65 related topics

## Dual space

In mathematics, any vector space , together with the vector space structure of pointwise addition and scalar multiplication by constants.

This gives rise to the bra–ket notation used by physicists in the mathematical formulation of quantum mechanics.

## Self-adjoint operator

Infinite-dimensional complex vector space V with inner product

In physics, particularly in quantum mechanics, the spectral theorem is expressed in a way which combines the spectral theorem as stated above and the Borel functional calculus using Dirac notation as follows:

## Linear form

Linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).

For more information see bra–ket notation.

## Wave function collapse

In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world.

This can be expressed as a vector using Dirac or bra–ket notation :

## Bracket

Either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings.

This is known as Dirac notation or bra–ket notation, to note vectors from the dual spaces of the Bra.

## Quantum entanglement

Physical phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance.

The following subsections are for those with a good working knowledge of the formal, mathematical description of quantum mechanics, including familiarity with the formalism and theoretical framework developed in the articles: bra–ket notation and mathematical formulation of quantum mechanics.

## Quantum mechanics

Fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles.

Ghirardi, GianCarlo, 2004. Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra–ket notation can be passed over on a first reading.

## Schrödinger picture

Formulation of quantum mechanics in which the state vectors evolve in time, but the operators are mostly constant with respect to time ( an exception is the Hamiltonian which may change if the potential.

. More abstractly, the state may be represented as a state vector, or ket,, or both.

## Spectral theory

Inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.

This topic is easiest to describe by introducing the bra–ket notation of Dirac for operators.