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Mathematics for Physics

99 Conjugate Variable

A variable paired with another variable through a derivative that is used to replace that other variable in a Legendre transform, where a Legendre transform is a change from velocity variables to momentum variables.

The definition of conjugate momentum. The momentum conjugate to a coordinate \(q_{i}\) is the derivative of the Lagrangian with respect to the corresponding velocity. This principle is used to define \(p_{i}\) from \(L\).

The conjugate momentum is

\[ p_{i} = \dfrac{\partial L}{\partial\!\left(\dfrac{dq_{i}}{dt}\right)} \]

where

  • \(L\) is the Lagrangian.
  • \(q_{i}\) is the \(i\)-th generalized coordinate.
  • \(\dfrac{dq_{i}}{dt}\) is the \(i\)-th generalized velocity.
  • \(p_{i}\) is the momentum conjugate to \(q_{i}\).

Conjugate pairs. A coordinate and its conjugate momentum form a conjugate pair. Position and momentum are the standard example. This principle is used to label phase-space coordinates \((q,p)\).

The canonical commutation relation. In quantum mechanics a conjugate pair does not commute. This principle is used to write the canonical commutation relation and the uncertainty principle.

The canonical commutation relation is

\[ [\hat{q},\hat{p}] = i\hbar \]

where

  • \(\hat{q}\) is the position operator.
  • \(\hat{p}\) is the momentum operator.
  • \(\hbar\) is the reduced Planck constant.

Note: Also called a canonically conjugate variable. Also called conjugate momentum when paired with a coordinate.

99.1 References

  1. Cahill, K. Physical Mathematics. Cambridge University Press, 2019. — \(p_{i}=\partial L/\partial(\mathrm{d}q_{i}/\mathrm{d}t)\) as momentum canonically conjugate to \(q_{i}\).
  2. Park, D. Introduction to the Quantum Theory. Dover, 2005. — coordinates and momenta are conjugate to each other.
  3. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \([x,p]=i\hbar\).