Uncertainty Principle

Discovered by Werner Heisenberg in 1927, the Uncertainty Principle states that you cannot simultaneously know both the position and the momentum of a given object to arbitrary precision. It furthermore precisely quantifies the imprecision. It is one of the corner stones of quantum mechanics.

The statement is as follows. If several identical copies of a system in a given state are prepared, measurements of position and momentum will vary according to known probability distributions; this is the fundamental postulate of quantum mechanics. We could measure the standard deviation Δx of the position measurements and the standard deviation Δp of the momentum measurements. Then we will find that

Δx Δp ≥ h / (4π)

where h is Planck's constant and π is Archimedes' constant. (In some treatments, the "uncertainty" of a variable is taken to be the smallest width of a range which contains 50% of the values, which, in the case of normally distributed variables, leads to a lower bound of h/2π for the product of the uncertainties.) Note that this inequality allows for several possibilities: the state could be such that x can be measured with high precision, but then p will only approximately be known, or conversely a state with sharply defined p will not allow precise determinatin of x. In other states, both x and p can be measured with "reasonable" (but not arbitrary high) precision.

In everyday life, we don't observe these uncertainties because the value of h is extremely small.

The uncertainty principle does not just apply to position and momentum. In its general form, it applies to every pair of conjugate variables. Two variables are conjugate if the associated operators do not commute. Examples of pairs of conjugate variables are energy vs. time and x-component of angular momentum (spin) vs. y-component of angular momentum. In general, the lower bound for the product of the uncertainties of two conjugate variables depends on the state the system is in. The uncertainty principle becomes a then theorem in the theory of operators (see functional analysis).

The uncertainty principle is sometimes erroneously explained by claiming that the measurement of position necessarily disturbs a particle's momentum. Disturbance plays no part since the principle even applies if position is measured in one copy of the system and momentum is measured in another, identical one. A better analogy is the following: suppose you have a varying signal such as a sound wave, and you want to know the exact frequencies in your signal at an exact moment in time. This is impossible: in order to determine the frequencies accurately, you need to sample the signal for some time and you thereby lose time precision. Time and frequency are "conjugate variables".

Albert Einstein was not happy with the uncertainty principle, and he challenged Niels Bohr with a famous thought experiment: we fill a box with a radioactive material which randomly emits radiation. The box has a shutter, which is opened and immediately thereafter shut by a clock at a precise time, thereby allowing some radiation to escape. So the time is already known with precision. We still want to measure the conjugate variable energy precisely. No problem says Einstein: just weigh the box before and after. The equivalence between mass and energy from special relativity will allow you to determine precisely how much energy left the box. Bohr countered as follows: should energy leave, then the now lighter box will rise slightly on the scale. That changes the position of the clock. Thus the clock deviates from our stationary reference frame, and again by special relativity, its measurement of time will be different from ours, leading to some unavoidable margin of error. In fact, a detailed analysis shows that the imprecision is correctly given by Heisenberg's relation.

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