Bra–Ket (Dirac) Notation, Operators, and More

Author

Daniel Fischer

Ket-vectors

A ket vector \(\left\lvert \alpha \right\rangle\) denotes an element of a vector space, more specifically a Hilbert space \(\mathscr{H}\).
It represents a quantum state, for example a wave function or a spin state.

Bra-vectors

A bra vector \(\left\langle \beta \right\rvert\) denotes an element of the dual space of \(\mathscr{H}\).
It is a linear form that maps each vector in \(\mathscr{H}\) into the complex plane \(\mathbb{C}\) by means of the inner product:

\[ \left\langle \beta \middle| \alpha \right\rangle\in \mathbb{C}. \]

For example, a wave function \(\psi(\vec{r})\) corresponding to a state \(\left\lvert \psi \right\rangle\) is obtained by mapping it into coordinate space using the inner product:

\[ \left\langle \vec{r} \middle| \psi \right\rangle= \psi(\vec{r}). \]

Inner product for wave functions

For two wave functions \(\psi(x)\) and \(\phi(x)\), the inner product is defined as

\[ \left\langle \phi \middle| \psi \right\rangle= \int_{-\infty}^{\infty} \phi^\ast(x)\,\psi(x)\,dx, \]

where \(\phi^\ast(x)\) is the complex conjugate of \(\phi(x)\).
This definition guarantees that inner products are complex numbers, and it provides the mathematical foundation for probability amplitudes in quantum mechanics.

Linear operators

An operator \(\hat{O}\) acting on a ket creates another ket:

\[ \hat{O}\left\lvert \psi \right\rangle= \left\lvert \phi \right\rangle. \]

A linear operator satisfies the two additional conditions:

  1. \(\hat{O}(\left\lvert \psi \right\rangle+\left\lvert \phi \right\rangle) = \hat{O}\left\lvert \psi \right\rangle+ \hat{O}\left\lvert \phi \right\rangle\)
  2. \(\hat{O}c\left\lvert \psi \right\rangle= c\hat{O}\left\lvert \psi \right\rangle\)

Additional notes:

  • The operator can also be written as an outer product:
    \[ \hat{O} = \frac{1}{\left\langle \psi \middle| \psi \right\rangle} \left\lvert \phi \right\rangle\!\!\left\langle \psi \right\rvert. \]
  • If \(\{\left\lvert \phi_i \right\rangle\}\) is a complete basis of \(\mathscr{H}\), a linear operator \(\hat{O}\) can be written as a matrix with entries
    \[ O_{ij} = \left\langle \phi_j \middle| \hat{O} \middle| \phi_i \right\rangle. \]

Hermitian conjugates

The Hermitian conjugate of a vector or operator is denoted by a \(\dagger\) (called dagger).
The rules are:

  • The Hermitian conjugate of a bra is the corresponding ket, and vice versa.
  • The Hermitian conjugate of a complex number is its complex conjugate (denoted with a superscript \(^\ast\)).
  • The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself.
  • For any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators in bra–ket notation, its Hermitian conjugate is computed by reversing the order and taking the conjugate of each component.

Examples:

  • \(\left\lvert \psi \right\rangle^\dagger = \left\langle \psi \right\rvert\) and \(\left\langle \psi \right\rvert^\dagger = \left\lvert \psi \right\rangle\)
  • \(\big(c_1\left\lvert \psi_1 \right\rangle+ c_2\left\lvert \psi_2 \right\rangle\big)^\dagger = c_1^\ast \left\langle \psi_1 \right\rvert+ c_2^\ast \left\langle \psi_2 \right\rvert\)
  • \(\big(\left\langle \psi \middle| \phi \right\rangle\big)^\dagger = \left\langle \psi \middle| \phi \right\rangle^\ast = \left\langle \phi \middle| \psi \right\rangle\)
  • \(\big(\left\langle \phi \middle| \hat{A}^\dagger \middle| \psi \right\rangle\big)^\ast = \left\langle \psi \middle| \hat{A} \middle| \phi \right\rangle\)
    and
    \(\big(\left\langle \phi \middle| \hat{A}^\dagger \hat{B}^\dagger \middle| \psi \right\rangle\big)^\ast = \left\langle \psi \middle| \hat{B}\hat{A} \middle| \phi \right\rangle\)

Observables

Observables are represented by Hermitian operators. “Hermitian” means \(\hat{A} = \hat{A}^\dagger\).
They are associated with measurable physical quantities (e.g., energy, momentum, angular momentum, spin).

Properties

  • The eigenvalues \(a\) of a Hermitian operator \(\hat{A}\), satisfying \(\hat{A}\left\lvert \alpha \right\rangle= a\left\lvert \alpha \right\rangle\), are real numbers.
  • For every observable, a complete and orthonormal basis can be found:
    • The basis is given by eigenvectors solving
      \(\hat{A}\left\lvert n \right\rangle= a_n\left\lvert n \right\rangle\) (discrete spectrum) or \(\hat{A}\left\lvert k \right\rangle= k\left\lvert k \right\rangle\) (continuous spectrum).
    • Orthonormality: \(\left\langle n' \middle| n \right\rangle= \delta_{n'n}\) or \(\left\langle k_i \middle| k_j \right\rangle= \delta(k_i-k_j)\).
    • Completeness: every state \(\left\lvert \psi \right\rangle\) in \(\mathscr{H}\) can be expressed as
      \(\left\lvert \psi \right\rangle= \sum_n c_n\left\lvert n \right\rangle\) or \(\left\lvert \psi \right\rangle= \int c(k)\left\lvert k \right\rangle\,dk\).
    • The identity operator can be written as \(\mathbb{1} = \sum_n \left\lvert n \right\rangle\!\!\left\langle n \right\rvert\) or \(\mathbb{1} = \int \left\lvert k \right\rangle\!\!\left\langle k \right\rvert\,dk\).

Additional notes

If \(\left\lvert \alpha \right\rangle\) is a normalized eigenvector of \(\hat{A}\) and the system is in the normalized state \(\left\lvert \psi \right\rangle\), then:

  • The probability of finding the system in state \(\left\lvert \alpha \right\rangle\) is
    \[ P = \left|\left\langle \alpha \middle| \psi \right\rangle\right|^2. \]
  • The expectation value of \(\hat{A}\) is
    \[ \langle \hat{A} \rangle = \left\langle \psi \middle| \hat{A} \middle| \psi \right\rangle. \]
  • The variance of \(\hat{A}\) is
    \[ \sigma_A = \langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2. \]