To get the probability from the wave function, all you have to do is square the wave function. For example, the “position wave function” can tell you the probability of finding a particle at any position. The square of a quantum wave function is the probability of finding it in a particular state. If you know what a “Fourier transform” is, and are at least a little comfortable with them, then this could be interesting to you. What follows is answer gravy.Īnswer gravy: This gravy has some lumps. So, unless the basic assumptions of physics are completely wrong (and they’ve held up to some serious scrutiny), the uncertainty principle is in the company of things like “you can’t go faster than light”, “energy and mass are conserved”, and “modern mathematicians don’t have beards” (has anyone else noticed this?).
Rather than being something that’s merely very challenging like, “you can’t break the sound barrier”, “what goes up must come down”, and “you can’t be the world’s best kick-boxer and be the world’s most handsome physicist”, the uncertainty principle is a mathematical absolute. Maybe we could arrange things so that the high frequency photon hit softer or something? There was a lot of back and forth for a long time (still is in some circles) about overcoming the uncertainty principle, but in the end it can never be violated. So far this seems more like an engineering problem than a problem with the universe. High frequency photons (like sunlight) are terrible at measuring momentum, but can tell you position well. Low frequency photons (like radio waves) don't tell you much about where a particle is, but they doesn't knock it around much either (so you can measure its momentum better). But then you won’t be able to figure out the particle’s position very well. Any method you can come up with to measure the momentum will require you to use low-frequency, low-energy, gentle photons. So you’ll know where it is pretty exactly, but it’ll go flying off with some unknown amount of momentum. After all, how can you trust the results from a wandering, low frequency photon? But, if you use a precise, high-frequency, and thus, high-energy photon, you’ll end up smacking the hell out of the particle you’re trying to measure. So, if you want to see where something is with precision you’ll need to use a high frequency photon. For example, you can still pick up a radio station even when you can’t see it directly.
That’s why sunlight (high frequency) seems to go in a perfectly straight line, but radio waves can spread out around corners. A general rule of thumb for light (waves in general really) is high frequency waves propagate in straight lines, and low frequency waves spread out. The uncertainty principle (the “position/momentum uncertainty principle”) is generally presented like this: you have some particle floating along and you’d like to know its position and its momentum (velocity, whatever) so you bounce a photon off of it (“Bounce a photon off of it” is just another way of saying “try to see it”). So overcoming the uncertainty principle is exactly as hard as overcoming that whole “1+1=2” thing. The Heisenberg uncertainty principle, while normally presented in physics circles, is actually a mathematical absolute.
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