Heisenberg’s Area ( about 10-34 Joule-second) is small, one ten-millionth of the explosive action in a single molecule of TNT. OK, that’s maybe important for sub-atomic physics, but it’s way too small to have any implications for anything bigger, right? Well, it could be responsible for shaping our Universe.
Quick recap: The Heisenberg Uncertainty Principle (HUP) says that certain quantities (for instance, position and momentum) are linked in a remarkable way. We can’t measure either of them perfectly accurately, but we can make repeated more-or-less sloppy measurements that give us average values. The linkage is in that sloppiness. Each repeated measurement lands somewhere in a range of values around the average. HUP says that even with very careful measurement the product of those two spans must be greater than Heisenberg’s Area.
So now let’s head out to empty space, shall we? I mean, really empty space, out there between the galaxies, where there’s only about one hydrogen atom per cubic meter.
Here’s a good cubic meter … sure enough, it’s got exactly one hydrogen atom in it.
For practice using Heisenberg’s Area, what can we say about the atom? (If you’re checking my math it’ll help to know that the Area, h/4π, can also be expressed as 0.5×10-34 kg m2/s; the mass of one hydrogen atom is 1.7×10-27 kg; and the speed of light is 3×108 m/s.) On average the atom’s position is at the cube’s center. Its position range is one meter wide. Whatever the atom’s average momentum might be, our measurements would be somewhere within a momentum range of (h/4π kg m2/s) / (1 m) = 0.5×10-34 kg m/s. A moving particle’s momentum is its mass times its velocity, so the velocity range is (0.5×10-34 kg m/s) / (1.7×10-27 kg) = 0.3×10-7 m/s.
With really good tools we could determine the atom’s velocity within plus or minus 0.000 000 03 m/s. Pretty good.
Now zoom in. Dial that one-meter cube down a billion-fold to a nanometer (10-9 meters, which is still about ten times the atom’s width). Yeah, the atom’s still in the box, but now its velocity range is 300 m/s. The atom could be just hanging out at the center, or it could zoom out of the cube a microsecond after we looked — we just can’t tell which.
All of which illuminates the contrast between physics Newton-style and the physics that has bloomed since Einstein’s 1905 “miracle year.” If Newton were in charge of the Universe, Heisenberg’s Area would be zero. We could determine that atom’s position and momentum with complete accuracy. In fact in principle we could accurately determine everything’s position and momentum and then calculate where everything would be at any time in the future. But he isn’t and it’s not and we can’t.
Theorists and experimenters use the word “measurement” in different ways. A measurement done by a theoretician is generally based on fundamental constants and an elaborate mathematical structure. If the measurement is a quantum mechanical result, part of that structure is our familiar bell-shaped curve. It’s an explicit recognition that way down in the world of the very small, we can’t know what’s really going on. Most calculations have to be statistical, predicting an average and an expected range about that average. That prediction may or may not pan out, depending on what the experimentalists find.
By contrast, when experimenters measure something, even as an average of multiple tests, it’s an estimate of the real distribution. The research group (usually it’s a group these days) reports a distribution that they claim overlaps well with a real one out there in the Universe. Then another group dives in to prove they or the theoreticians or both are wrong. That’s how Science works.
So there could be a collection of bell-curves gathered about the experimental result. Remember those extra dimensions we discussed earlier? One theory that’s been floated is that along those extra dimensions the fundamental constants like h might take on different values. Maybe further along “Dimension W” the value of h is bigger than it is in our Universe, and quantum effects are even more important than they are here.
Now how can we test that?
BTW, Heisenberg will be 114 on Dec 5. Alles Gute zum Geburtstag, Werner!
~~ Rich Olcott