The Heisenberg Uncertainty Principle
One of the most striking principles of quantum mechanics is the Heisenberg uncertainty principle.
Quantum mechanics is the name given to the theory, developed in the early part of the 20th century, of how matter and light interact. It was a theory that united the study of light, previously thought to be a non-particle, wave phenomenon with the study of matter, previously thought to be a non-wave, particle phenomenon. Quantum mechanics is physically, mathematically, and even philosophically quite deep, and a complete, understandable description of it is beyond the topic of this chapter.
The nub of quantum mechanics is that everything, from a large object to the small atoms making up that object, and even subatomic particles making up the atoms, can be understood by a mathematical model that makes those things look like some kind of local disturbance, or wave, where the particle is located. One can then much more easily understand the interaction of light with matter as the interaction of two waves, rather than the interaction of a wave (light) with a particle (like an electron).
However, this wave model for everything has a consequence that even many of the very smart people (Einstein, Fermi, Born, Planck, among others) who were working out this theory found very uncomfortable to live with. That is, that all matter described in this way had to be, at least to some extent, delocalized. For example, say you know there is an electron some very small region of space, and you set about trying to look for it. For this purpose, the electron is so very small that we can consider it to be at a single point. However, if you observe it to be at a particular point, and then look again, you will find it at a different point. And, regardless of how close together in time these observations were, the distance between the two points where the electron was located can be relatively large. An interpretation of this is that, given the physical environment an electron finds itself in (like what other charged particles are nearby), the most you can say about its location is that there is a certain probability that, when you look, you will find it in a particular, very small region of space. Other nearby regions will have their probabilities for being found to contain the electron. So you can only say where an electron is probabilistically. Another way to look at that is that you can specify the location of an electron, but there will always be an uncertainty in that location.
The German Physicist, Werner Heisenberg, showed that this wave-like nature of matter implies that there is always some uncertainty not only in the location, but also in the velocity of a particle. If we call the uncertainty in position x and the uncertainty in velocity v , then the product of these two uncertainties is always bigger than some constant (call it k) divided by the particle's mass:
x v Æ k/m
What this means is that, for a single particle, the better you know its position (i.e. the smaller the x), the worse you know its velocity (the larger v has to be). Or vice-versa.
What does the Heisenberg Uncertainty Principle have to do with
Atoms?
To be specific, let's think of a hydrogen atom, consisting of a single proton and a single electron. The proton is about 2000 times as heavy as the electron. The proton and the electron are bound to each other by the electrostatic attraction between the positively-charged proton and the negatively-charged electron. Why doesn't the electron just fall into the nucleus and snuggle up good and tight to the proton?
If this atom is moving through space, to stay together the electron and the proton always have to have very nearly the same velocity, on average. So they have approximately the same uncertainty in their velocities, too. Now, each particle must itself obey the Heisenberg uncertainty principle. So we have two inequalities:
xp v Æ k/mp for the proton
xe v Æ k/me for the electron.
Note that we have left the subscript off the v, since it is the same
for both proton and electron. If we substitute mp = 2000 me , and
rearrange both equations, we get the equations
2000 xp Æ k'
xe Æ k'
where k' = k/(me v) , which is constant.
The above equations imply that the uncertainty in the position of the electron is 2000 times greater than the uncertainty in the position of the proton. That is, if we know that the proton is in a very small region of space (about 10-15.5 m3 ), the electron will be in a much larger region of space. It cannot be observed very often right in the vicinity of the proton, since that would mean it would have a x that was comparable to the x of the proton.