The Uncertainty Principle

Learning Objective

• Summarize the uncertainty principle of quantum mechanics.

Key Points

• The uncertainty principle is written as ${ \sigma }_{ x }{ \sigma }_{\rho }\quad\ge\quad\frac{ \hbar }{ 2 }$ .
• The position of an object cannot be known simultaneously with its momentum.
• The more precisely one quantity is known, the less precisely the other is known.

Terms

• momentumThe product of the mass and velocity of a particle in motion.
• uncertaintyA parameter that measures the dispersion of a range of measured values.

In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position (x) and momentum (p), can be known simultaneously. The more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. The original heuristic argument that such a limit should exist was given by Werner Heisenberg in 1927, after whom it is sometimes named as the Heisenberg Uncertainty Principle. The reasoning was derived from considering the uncertainty in both the position and the momentum of an object. Roughly, the uncertainty in the position of a particle is approximately equal to its wavelength (λ). The uncertainty in the momentum of the object follows from de Broglie’s equation as h/λ. Therefore, to a first approximation the Heisenberg Uncertainty Principle gives that the product of these two uncertainties is on the order of Planck’s constant (h).

A more formal inequality relating the standard deviation of position (${ \sigma }_{ x }$) and the standard deviation of momentum (${ \sigma }_{ \rho }$) was derived by Earle Hesse Kennard later that year (and independently by Hermann Weyl in 1928):

${ \sigma }_{ x }{ \sigma }_{\rho }\quad\ge\quad\frac{ \hbar }{ 2 }$

Historically, the uncertainty principle has been confused with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems. Heisenberg offered such an observer effect at the quantum level as a physical explanation of quantum uncertainty. It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems and that it arises in quantum mechanics simply due to the matter-wave nature of all quantum objects.

Scope of the Uncertainty Principle and Applications

The uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology. It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their main research program. These include, for example, tests of number-phase uncertainty relations in superconducting or quantum optics systems. Applications are for developing extremely low noise technology, such as that required in gravitational-wave interferometers.