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Mathematics for Physics

126 Momentum Operator

An operator equal to \(-i\hbar\) times differentiation that is used to represent the momentum observable in the position representation.

\(P=-i\hbar d/dx\). In the position representation the momentum operator is \(-i\hbar\) times the derivative. This principle is used to compute \(\langle p\rangle\) and kinetic energy \(p^{2}/2m\).

The momentum operator is

\[ P = -i\hbar\dfrac{d}{dx} \]

where

  • \(P\) is the momentum operator.
  • \(\hbar\) is the reduced Planck constant.
  • \(x\) is the position coordinate.

Momentum eigenfunctions. Plane waves are eigenfunctions of \(P\). This principle is used to identify states of definite momentum.

The momentum eigenvalue equation is

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

where

  • \(\psi_{p}(x)\) is a momentum eigenfunction.
  • \(p\) is the momentum eigenvalue.

Momentum as the generator of translations. Momentum generates translations. This principle is used to write an infinitesimal shift as \(1-(i/\hbar)p\,dx\).

The infinitesimal translation operator is

\[ \mathcal{J}(dx') = 1 - \dfrac{i}{\hbar}\,p\,dx' \]

where

  • \(\mathcal{J}(dx')\) is the translation by \(dx'\).
  • \(p\) is the momentum operator.
  • \(\hbar\) is the reduced Planck constant.

Note: Also denoted \(P\). Also denoted \(\hat{p}\). Also denoted \(p\).

126.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — Proposition 3.6, Definition 3.7: \((P\psi)(x)=-i\hbar\dfrac{d\psi}{dx}\).
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — Equations 1.214, 1.248, 1.249: \(\langle x'|p|\alpha\rangle=-i\hbar\dfrac{\partial}{\partial x'}\langle x'|\alpha\rangle\).
  3. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — Equations 24.30, 24.31: \(P=-i\hbar D\).