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Mathematics for Physics

153 Uncertainty Principle

An inequality relating position and momentum spreads that is used to bound how sharply both can be known in the same quantum state.

The Heisenberg relation. It is impossible to prepare a particle in a state in which position and momentum along one axis are both exactly known. This principle is used to replace a classical phase-space point by a minimum product of spreads.

The Heisenberg uncertainty relation is

\[ \Delta x\,\Delta p \geq \dfrac{\hbar}{2} \]

where

  • \(\Delta x\) is the uncertainty in position.
  • \(\Delta p\) is the uncertainty in momentum.
  • \(\hbar\) is the reduced Planck constant.

The variance form. The same bound is written in terms of mean-square deviations. This principle is used to compute the product of variances from expectation values.

The variance form of the uncertainty relation is

\[ \langle(\Delta x)^{2}\rangle\langle(\Delta p_{x})^{2}\rangle \geq \dfrac{\hbar^{2}}{4} \]

where

  • \(\langle(\Delta x)^{2}\rangle\) is the mean-square deviation in position.
  • \(\langle(\Delta p_{x})^{2}\rangle\) is the mean-square deviation in \(x\)-momentum.

The Gaussian minimum-uncertainty packet. A Gaussian wave packet saturates the bound. This principle is used to identify the minimum-uncertainty state.

The tradeoff of localization. A tighter localization in \(x\) forces a larger spread in \(p\). This principle is used to explain diffraction of a tightly collimated beam.

Note: Also called the Heisenberg uncertainty principle. Also called the position-momentum uncertainty relation.

153.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \((\Delta_{\psi}X)(\Delta_{\psi}P)\geq\hbar/2\).
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \(\langle(\Delta x)^{2}\rangle\langle(\Delta p_{x})^{2}\rangle\geq\hbar^{2}/4\).
  3. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(\Delta x\,\Delta p\geq\hbar/2\).