# Quantum Mechanics: The Uncertainty Principle

Sat, Jun 30, 2012In 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 great 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 physics, a particle is described by a wave packet, which gives rise to this phenomenon. Consider the measurement of the absolute position of a particle. It could be anywhere the particle’s wave packet has non-zero amplitude, meaning the position is uncertain - it could be almost anywhere along the wave packet.

To obtain an accurate reading of position, this wave packet must be ‘compressed’ as much as possible, meaning it must be made up of increasing numbers of sine waves added together. The momentum of the particle is proportional to the wavelength of one of these waves, but it could be any of them. So a more accurate position measurementby adding together more wavesmeans the momentum measurement becomes less accurate (and vice versa).

The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength (and therefore an indefinite momentum). 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 can be 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.