A thought today from Pchem:
Infinity is a relative term. One meter away from the nucleus of an atom is infinity, and 10 billion billion kilometers away from the sun is infinity. Since infinity is a general concept, rather than a number, it can be defined anywhere. So, if we consider the probability of finding an electron such-and-such a distance from the nucleus, we can find the probability that it will be from that point inwards, or the probability of finding the electron between two points by doing the same method but subtracting the smaller value. We know that the probability of finding the electron converges to 0 at infinity, but infinity can be anywhere we set it to be. Supposing you want to find the probability of finding the electron on Mars (as was the example given today), you can find the probability between "Nucleus and Mars" (A very high number), and you can then find the probability between"Just beyond Mars and Infinity". Then you can subtract "Nucleus and Mars" probability from "Just beyond Mars and Infinity" to get a real probability of finding the electron on Mars. I think this all arises because we can set infinity anywhere we want (which is necessary for the concept of infinity to be of any use).
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Correct me if I've misunderstood you, but what you've said makes no sense at all. Infinity is not a number (unless you work in some funky field that defines it as an element in the field). We can't "set it to anywhere we want" because it's not anywhere. In the ordinary real line or complex plane, "infinity" is a notion defined via limits.
ReplyDeleteThe probability that the electron of a hydrogen atom is on Mars given that its nucleus is at some origin (say at the centre of the sun), assuming that the electron does not interact with anything in the way, is given by integrating the magnitude squared of the wavefunction of the hydrogen atom over the region that comprises of Mars. Some people think of this region as a sphere. Some people think of the region as a spherical shell centered at the nucleus, of thickness the diameter of Mars. Some people think of the region as a torus tracing out the planet's orbit. Some people think that the region is a point. Other approximations are possible. Depending on choice of what constitutes "Mars", the numbers will be different.
Also if you subtract P(electron found from nucleus to Mars) from P(electron found from Mars and beyond) you'll almost certainly get a negative number. While that is a real number, it doesn't mean much as a probability.
No, I don't think you've misread me.
ReplyDeleteI don't pretend to be a mathematician, and I was certainly thinking in the real field of numbers. That actually helps clear things up for me: if it just arises because of limits, then the reason we can find the electron on Mars, even if it is located at the sun, is just because you integrate the square of the wavefunction over all space. This has to be done because we don't know where the electron is (in a classical sense), but only the probability, and that probability is only equal to one over all space. This means that the probability of finding the electron on Mars is a feature of the electron itself, and not infinity. Which is what was kind of bugging me. Thank you.
And you're right about the Probability subtraction. I meant to say: P(all space) - [P (Sun to Mars) + P (Just past Mars to Infinity)].