Uncertainty Principle

In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known. In layman’s terms, the more precisely one property is measured, the less precisely the other can be controlled, determined, or known.

In his Nobel Laureate speech, Max Born said:

…To measure space coordinates and instants of time, rigid measuring rods and clocks are required. On the other hand, to measure momenta and energies, devices are necessary with movable parts to absorb the impact of the test object and to indicate the size of its momentum. Paying regard to the fact that quantum mechanics is competent for dealing with the interaction of object and apparatus, it is seen that no arrangement is possible that will fulfill both requirements simultaneously…

The wave function of an initially very localized particle

Published by Werner Heisenberg in 1927, the uncertainty principle was a monumental discovery in the early development of quantum theory. It implies that it is impossible to simultaneously measure the present position while also determining the future motion of a particle, or of any system small enough to require quantum mechanical treatment. Intuitively, the principle can be understood by considering a typical measurement of a particle. It is impossible to determine both momentum and position by means of the same measurement, as indicated by Born above. Assume that its initial momentum has been accurately calculated by measuring its mass, the force applied to it, and the length of time it was subjected to that force. Then to measure its position after it is no longer being accelerated would require another measurement to be done by scattering light or other particles off of it. But each such interaction will alter its momentum by an unknown and indeterminable increment, degrading our knowledge of its momentum while augmenting our knowledge of its position. So Heisenberg argues that every measurement destroys part of our knowledge of the system that was obtained by previous measurements. The uncertainty principle states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.

The principle states specifically that the product of the uncertainties in position and momentum is always equal to or greater than one half of the reduced Planck constant ħ, which is defined as the re-scaling h/(2π) of the Planck constant h. Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding bases are Fourier transforms of one another (i.e., position and momentum are conjugate variables). In the mathematical formulation of quantum mechanics, any non-commuting operators are subject to similar uncertainty limits.