Uncertainty Principle

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In quantum physics, the Heisenberg uncertainty principle is the statement that locating a particle in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the position uncertain.

In quantum mechanics, the position and momentum of particles do not have precise values, but have a probability distribution. There are no states in which a particle has both a definite position and a definite momentum. The narrower the probability distribution is in position, the wider it is in momentum.

Physically, the uncertainty principle requires that when the position of an atom is measured with a photon, the reflected photon will change the momentum of the atom by an uncertain amount inversely proportional to the accuracy of the position measurement. The amount of uncertainty can never be reduced below the limit set by the principle, regardless of the experimental setup.

A mathematical statement of the principle is that every quantum state has the property that the root-mean-square (RMS) deviation of the position from its mean (the standard deviation of the X-distribution):

\Delta X = \sqrt{\langle X^2 \rangle-\langle X \rangle ^2 } \,

times the RMS deviation of the momentum from its mean (the standard deviation of P):

\Delta P = \sqrt{\langle P^2 \rangle-\langle P \rangle ^2} \,

can never be smaller than a small fixed multiple of Planck's constant:

\Delta X \Delta P \ge {\hbar \over 2}

The mathematical statement implies the physical statement. Once an observer measures the position of a particle with accuracy \Delta X , the state of the particle immediately after the measurement has \Delta P \ge \hbar /(2\Delta X) .

The uncertainty principle is related to the observer effect, with which it is often conflated. In the Copenhagen interpretation of quantum mechanics, the uncertainty principle is a theoretical limitation of how small this observer effect can be. A precise position measurement must alter the momentum by a large indeterminate amount, and vice-versa.

While this is true in all interpretations, in many modern interpretations of quantum mechanics (many-worlds and variants), the quantum state itself is the fundamental physical quantity, not the position or momentum. Taking this perspective, while the momentum and position are still uncertain, the uncertainty is an effect caused not just by observation, but by any entanglement with the environment.

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