\braket{\phi\vert\psi} - LaTeX Symbol

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⟨ ϕ ∣ ψ ⟩

\braket{\phi\vert\psi}

The \braket{\phi\vert\psi} notation is used in quantum mechanics to denote the inner product of two states, typically represented by the wave functions \phi and \psi. This notation is part of the bra-ket notation introduced by Paul Dirac, where the 'bra' \langle \phi | represents a row vector and the 'ket' | \psi \rangle represents a column vector. The inner product \braket{\phi\vert\psi} is a complex number that provides information about the probability amplitude of transitioning from state \phi to state \psi.

Examples

⟨ ϕ ∣ ψ ⟩ = \int ϕ^{*} (x) ψ (x) d x

Inner product of wave functions \phi and \psi over continuous space.

\braket{\phi\vert\psi} = \int \phi^*(x) \psi(x) \, dx

⟨ 0∣1 ⟩ = 0

Inner product of orthogonal quantum states, such as the basis states of a qubit.

\braket{0\vert 1} = 0

⟨ ψ ∣ ψ ⟩ = 1

Normalization condition for a quantum state \psi, ensuring it has unit probability.

\braket{\psi\vert\psi} = 1