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Mathematics for Physics

128 Operators

A linear mapping on a Hilbert space that is used to represent a physical observable acting on quantum states, where a Hilbert space is a complete inner-product space of states.

The quantization map to self-adjoint operators. Each classical real function on phase space is associated with a self-adjoint operator on the quantum Hilbert space. A self-adjoint operator is an operator equal to its adjoint. This principle is used to promote \(x\) and \(p\) to \(\hat{x}\) and \(\hat{p}\).

Hermitian observables. Observables act on kets. A Hermitian operator has real eigenvalues. This principle is used to identify measured values with those eigenvalues.

Expectation values. The expectation value of an operator \(A\) in a normalized state is \(\langle A\rangle=\langle\alpha|A|\alpha\rangle\). This principle is used to compute the mean of many measurements.

The expectation value is

\[ \langle A\rangle = \langle\alpha|A|\alpha\rangle \]

where

  • \(A\) is an observable operator.
  • \(|\alpha\rangle\) is a normalized state.

The position-representation operators. In the position representation the position operator multiplies by \(x\) and the momentum operator differentiates. This principle is used to write explicit operators on wavefunctions.

The position and momentum operators are

\[ (\hat{x}\psi)(x) = x\psi(x),\qquad (\hat{p}\psi)(x) = -i\hbar\dfrac{d\psi}{dx} \]

where

  • \(\psi\) is a wavefunction.
  • \(\hbar\) is the reduced Planck constant.

Note: Also called quantum operators.

128.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — Axiom 2: classical \(f\) associated with self-adjoint \(\hat{f}\).
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — observables represented by Hermitian operators on kets.