## Wednesday, June 10, 2009

### Heisenberg's uncertainty principle

Werner Heisenberg

In quantum physics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known. It is impossible to measure simultaneously both position and velocity of a microscopic particle with any degree of accuracy or certainty. This is not a statement about the limitations of a researcher's ability to measure particular quantities of a system, but rather about the nature of the system itself and hence it expresses a property of the universe.

In quantum mechanics, a particle is described by a wave. The position is where the wave is concentrated and the momentum is the wavelength. The position is uncertain to the degree that the wave is spread out, and the momentum is uncertain to the degree that the wavelength is ill-defined.

The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength. Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there are no states that describe a particle with both a definite position and a definite momentum. The more precise the position, the less precise the momentum.

The uncertainty principle can be restated in terms of measurements, which involves collapse of the wavefunction. When the position is measured, the wavefunction collapses to a narrow bump near the measured value, and the momentum wavefunction becomes spread out. The particle's momentum is left uncertain by an amount inversely proportional to the accuracy of the position measurement. The amount of left-over uncertainty can never be reduced below the limit set by the uncertainty principle, no matter what the measurement process.

This means that the uncertainty principle is related to the observer effect, with which it is often conflated. The uncertainty principle sets a lower limit to how small the momentum disturbance in an accurate position experiment can be, and vice versa for momentum experiments.

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):

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

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

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

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

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

Any measurement of the position with accuracy $\scriptstyle \Delta X$ collapses the quantum state making the standard deviation of the momentum $\scriptstyle \Delta P$ larger than $\scriptstyle \hbar/2\Delta x$.

Taken from Wikipedia: http://en.wikipedia.org/wiki/Uncertainty_principle