In quantum mechanics, 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. This statement has been interpreted in two different ways. According to Heisenberg its meaning is that it is impossible to determine simultaneously both the position and velocity of an electron or any other particle with any degree of accuracy or certainty. According to others (for instance Ballentine) this is not a statement about the limitations of a researcher's ability to measure particular quantities of a system, but it is a statement about the nature of the system itself as described by the equations of quantum mechanics.
In quantum mechanics, a particle is described by a wave. The position of the particle is regarded as being where the wave amplitude is greatest and the momentum is determined by the wavelength. The position is uncertain to the degree that the wave is spread out (well-defined wavelength), but the momentum is certain (well-defined) only to the degree that the wavelength is well-defined. Thus, position and momentum for a particle have opposite requirements for good definition, so that both position and wavelength cannot simultaneously be well-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):
times the RMS deviation of the momentum from its mean (the standard deviation of P):
can never be smaller than a fixed fraction of Planck's constant:
Any measurement of the position with accuracy collapses the quantum state making the standard deviation of the momentum larger than .