Bra–ket notation

Wave functions of the electron in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations. The brighter areas represent a higher probability of finding the electron.

Used ubiquitously to denote quantum states.

- Bra–ket notation

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Dual space

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

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

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

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

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).

Geometric interpretation of a 1-form α as a stack of hyperplanes of constant value, each corresponding to those vectors that α maps to a given scalar value shown next to it along with the "sense" of increase. The zero plane is through the origin.
Linear functionals (1-forms) α, β and their sum σ and vectors u, v, w, in 3d Euclidean space. The number of (1-form) hyperplanes intersected by a vector equals the inner product.

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.

Wave functions of the electron in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations. The brighter areas represent a higher probability of finding the electron.

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.

An example of curly brackets used to group sentences together
Angle brackets, less-than/greater-than signs and single guillemets in fonts Cambria, DejaVu Serif, Andron Mega Corpus, Andika and Everson Mono

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.

Spontaneous parametric down-conversion process can split photons into type II photon pairs with mutually perpendicular polarization.
Article headline regarding the Einstein–Podolsky–Rosen paradox (EPR paradox) paper, in the May 4, 1935 issue of The New York Times.
The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state. When the eigenvalue has value .5, von Neumann entropy is at a maximum, corresponding to maximum entanglement.

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.

Wave functions of the electron in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations. The brighter areas represent a higher probability of finding the electron.
Fig. 1
Position space probability density of a Gaussian wave packet moving in one dimension in free space.
1-dimensional potential energy box (or infinite potential well)
Some trajectories of a harmonic oscillator (i.e. a ball attached to a spring) in classical mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a wave (called the wave function), with the real part shown in blue and the imaginary part shown in red. Some of the trajectories (such as C, D, E, and F) are standing waves (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy.
Schematic of a Mach–Zehnder interferometer.
Max Planck is considered the father of the quantum theory.
The 1927 Solvay Conference in Brussels was the fifth world physics conference.

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.

Various examples of physical phenomena

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

Outer product

Matrix.

In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Bra–ket notation for outer product

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.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

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