61 Bra and Ket

A notation that writes vectors and linear functionals in matching symbols that is used to compute scalar values from vectors and functionals.

definition [d] (Bra and Ket) A ket \(|\psi\rangle\) is an element of a complex Hilbert space \(\mathcal{H}\); a bra \(\langle\phi|\) is an element of its dual space \(\mathcal{H}^{*}\) acting on kets:

\(\langle\phi| : \mathcal{H} \rightarrow \mathbb{C}, \quad |\psi\rangle \mapsto \langle\phi|\psi\rangle\) .
\(\langle\phi| = |\phi\rangle^{\dagger}\) .

where

\(\mathcal{H}\) is a complex Hilbert space, \(\mathcal{H}^{*}\) its dual.
\(\langle\phi|\psi\rangle \in \mathbb{C}\) is the inner product.
\(\dagger\) is the Hermitian conjugate.

61.1 References

Schwichtenberg, J. Physics from Symmetry. Springer, 2018. — Eqs. 8.35–8.37 (Dirac notation, ket, bra, inner product).
Cahill, K. Physical Mathematics. Cambridge University Press, 2019.
Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006.

60 Bohr Radius

62 Conservation Laws