Applications of
Quantum Physics
Implications of
Quantum Physics

35. The Uncertainty Principle

Summary
The uncertainty principle says that a small uncertainty in the measured momentum of the wave function implies a large uncertainty in the measured position of the wave function, and vice versa.

If one assumes matter is made up of very small particles, then the uncertainty principle is, from a classical point of view, quite surprising. It says one cannot measure both the position and momentum (mass times velocity) of the particle with arbitrary precision. If you know position very accurately, then there will be a large uncertainty in the momentum. But from the wave-function-only view (see No Evidence for Particles) this principle is not surprising or mysterious; it is just a property of wave functions.

The uncertainty principle has a technical statement and derivation along with an interpretation or explanation of what the technical statement implies. To illustrate, suppose we have either a light-like wave function or an electron-like wave function that goes through a narrow slit. The particle-like wave function is initially traveling in the z direction and the slit is in the y direction. After the wave goes through the slit, it spreads out in the x-direction. Because of this, if one measures the x position many times, one will get a spread in values, even if the measurements are exact. This is just the nature of waves. The same conclusion holds for momentum; if one measures the momentum in the x direction many times, one will get a spread in values, even if the measurements are exact.

It is remarkable that, by the use of standard vector-space arguments, one can get a relation between the spreads in position and momentum. It is:  (35-1)
(where the comes from the fact that [px,x]=-i ). Or, if we take the square root of this relation and use root-mean-square quantities,  (35-2)
This relation is often interpreted as follows: There exists a particle embedded in the wave function whose position and momentum cannot be simultaneously measured exactly. But we have seen that there is No Evidence for Particles, so this interpretation is not warranted.

What the uncertainty principle tells us is the following: First, even if we make exact measurements of, say, position on many systems identically prepared, there will inevitably be a spread in the measurements. And second if we make many exact measurements of position on a set of runs, and then we make many exact measurements of momentum on a separate set of runs, the product of the rms spreads of the two quantities must be greater than /2. Again, this result is simply a consequence of the nature of waves; if there are no particles, it is not so mysterious.

One common way the uncertainty principle is often used is to say that if the wave function is confined to a very small region in space, it will have a large spread in momentum. This is a valid translation of the import of equations (35-1) and (35-2).  