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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function Collapse
Experiments Bell's inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Local Superdeterminism Many-worlds Objective collapse Quantum logic Relational Transactional Von Neumann–Wigner
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Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.

Bra–ket notation was created by Paul Dirac in his 1939 publication A New Notation for Quantum Mechanics. The notation was introduced as an easier way to write quantum mechanical expressions.^[1] The name comes from the English word "bracket".

Quantum mechanics

In quantum mechanics, bra–ket notation is used ubiquitously to denote quantum states. The notation uses angle brackets, ${\displaystyle \langle }$ and ${\displaystyle \rangle }$, and a vertical bar ${\displaystyle |}$, to construct "bras" and "kets".

A ket is of the form ${\displaystyle |v\rangle }$. Mathematically it denotes a vector, ${\displaystyle {\boldsymbol {v}}}$, in an abstract (complex) vector space ${\displaystyle V}$, and physically it represents a state of some quantum system.

A bra is of the form ${\displaystyle \langle f|}$. Mathematically it denotes a linear form ${\displaystyle f:V\to \mathbb {C} }$, i.e. a linear map that maps each vector in ${\displaystyle V}$ to a number in the complex plane ${\displaystyle \mathbb {C} }$. Letting the linear functional ${\displaystyle \langle f|}$ act on a vector ${\displaystyle |v\rangle }$ is written as ${\displaystyle \langle f|v\rangle \in \mathbb {C} }$.

Assume that on ${\displaystyle V}$ there exists an inner product ${\displaystyle (\cdot ,\cdot )}$ with antilinear first argument, which makes ${\displaystyle V}$ an inner product space. Then with this inner product each vector ${\displaystyle {\boldsymbol {\phi }}\equiv |\phi \rangle }$ can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: ${\displaystyle ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi |}$. The correspondence between these notations is then ${\displaystyle ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\equiv \langle \phi |\psi \rangle }$. The linear form ${\displaystyle ⟨<!--\; ⟨\; -->\mathrm{\varphi <!--\; \varphi \; -->}|}$Zdroj:https://en.wikipedia.org?pojem=Bra–ket_notation
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