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Mathematics for Physics

97 Bra and Ket

A notation that writes vectors and linear functionals in matching symbols that is used to compute inner products and matrix elements, where a linear functional is a map from states to complex numbers.

The ket as a state vector. A ket \(|\psi\rangle\) is a vector in a complex Hilbert space. A Hilbert space is a complete inner-product space of states. This principle is used to write quantum states as kets.

The bra as a linear functional. A bra \(\langle\phi|\) is the linear functional that maps a ket to the inner product \(\langle\phi|\psi\rangle\). This principle is used to compute amplitudes.

The action of a bra on a ket is

\[ \langle\phi| : \mathcal{H} \rightarrow \mathbb{C},\qquad |\psi\rangle \mapsto \langle\phi|\psi\rangle \]

where

  • \(\mathcal{H}\) is the Hilbert space.
  • \(\langle\phi|\psi\rangle\) is the inner product.

The dagger relation. The bra is the Hermitian conjugate of the corresponding ket. This principle is used to pass from \(|\phi\rangle\) to \(\langle\phi|\).

The bra-ket relation is

\[ \langle\phi| = |\phi\rangle^{\dagger} \]

where

  • \(\dagger\) is the Hermitian conjugate.

Dirac matrix elements. The matrix element of an operator is \(\langle\phi|A|\psi\rangle\). This principle is used to compute amplitudes and expectation values.

Note: Also called Dirac notation.

97.1 References

  1. Schwichtenberg, J. Physics from Symmetry. Springer, 2018. — Eqs. 8.35–8.37 (Dirac notation, ket, bra, inner product).
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — bras, kets, and inner products.
  3. Cahill, K. Physical Mathematics. Cambridge University Press, 2019.
  4. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006.