Quantum Mechanics/Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that the product of uncertainties in related physical quantities (e.g. position and momentum, energy and time, etc.) has a finite lower bound. This arises from the fact that the momentum and position operators do not commute. A common misunderstanding is that in simple terms this means that a measurement for position disturbs the momentum of a particle, and a measurement for the momentum of a particle disturbs its position. All of this information is false.

In other words, if you know the position of a particle and you measure the momentum, it disturbs the position - thus you are less certain of its position. This is incorrect.

In actuality, the uncertainty arises from fundamental underlying physical laws governing physical systems which can be measured. Particles have a fundamental dual nature and may be considered either a point source or a probability distribution of position depending upon what one is trying to accomplish by interacting with the particle.

Repeated measurements of particles in known/delivered states will give a distribution range governed by a probability function for either of the two conjugate properties, position (x) or momentum (p). The Uncertainty Principle gives a mathematically provable lower bound of the product of the uncertainty (error or deviation from a precise center of the probability wave law) in the measured conjugate properties.

$$\Delta x\Delta p \ge\frac{\hbar}{2}=\frac{\frac{h}{2\pi}}{2}\, $$ where $$\pi \approx 3.14 $$ and h is Planck's constant with $$\hbar$$ being Planck's reduced constant.

Thus the precision with which one can know one property is related mathematically to what is known of the conjugate property. If position is known with great precision for a specific particle or wave then the conjugate momentum is not known with a precision greater than that allowed by the lower bound provided by the Uncertainty Principle equation shown above.

Here is the derivation for the general uncertainty principle for two operators, A and B, that do not commute.

Let iC be the commutator of A and B:

[A,B] = iC

Furthermore, A and B are Hermitian operators. Recall the definition of the uncertainties:

$$\left(\Delta A\right)^{2} = \langle\left(A- \langle A\rangle\right)^{2}\rangle $$

$$\left(\Delta B\right)^{2} = \langle\left(B- \langle B\rangle\right)^{2}\rangle $$

To do:

Find a mathematically simple derivation of uncertainty principle and provide it here.

Find an intuitive example and provide here.

Provide a variety of equivalent derivations using different techniques to allow understanding by widest possible audience.

External links: Wikipedia:Uncertainty Principle