Paul Dirac Notation

Last updated Jun 23, 2022

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• Ket is a column vector, denoted as $|v \rangle$.
• Bra is a row vector, denoted as $\langle v |$.
• The names bra and ket come from Bracket.
• They are related to one another: $|\psi\rangle^\dagger = \langle \psi|$
• The length of $\psi\rangle$ is defined as: $|\psi\rangle|^2 = \langle \psi| |\psi\rangle$.
• We usually omit the $|$ from $\langle \psi| |\psi\rangle$ and use this notation: $\langle \psi| \psi \rangle$.
• If $M$ is a matrix and $|\psi\rangle$ is a ket, then $$(M|\psi\rangle)^\dagger = \langle \psi|M^\dagger$$