Dirac Bra-Ket Notation | Mathematical Language of Quantum States
Quick Technical Answer:
Introduced by Paul Dirac in 1939, Bra-Ket notation represents quantum state vectors as kets |ψ⟩ (column vectors) and their complex conjugate duals as bras ⟨φ| = |φ⟩† (row vectors). Their inner product ⟨φ|ψ⟩ yields a scalar probability amplitude.
Formula / Unitary:
|\psi\rangle = \begin{pmatrix} c_0 \\ c_1 \end{pmatrix}, \quad \langle\phi| = \begin{pmatrix} d_0^* & d_1^* \end{pmatrix}, \quad \langle\phi|\psi\rangle = d_0^* c_0 + d_1^* c_1
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Frequently Asked Questions
What is an outer product in Dirac notation?
The outer product |ψ⟩⟨φ| forms an operator (matrix). If |ψ⟩ = |φ⟩ with unit norm, |ψ⟩⟨ψ| is a projection operator that projects states onto |ψ⟩.