How does the speed of electron are
of the light speed when they revolve around the nucleus of the atom .
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Don't copy from google and please give full explanation.
Answers
Explanation:
The quantum world is confusing, and much of this arises from the fact that there are some questions we cannot get answers to. What one needs to focus on is the facts, and just the facts, as Sgt. Joe Friday always said on Dragnet.
In the classical world, the velocity can be found by measuring subsequent positions over smaller and smaller time intervals and taking the limit for the ratio of the change in position to the change in time as the change in time goes to zero. For some of us, that is a hard enough concept to understand. The problem is, even if you understand that, it does not help you in the quantum world. For, if you measure the position at different times and try to take the limit as the time interval goes to zero, you find no such limit exists. The particle positions you find will vary over a wide range given by the probability distribution, and there is no way to make the conventional (classical) definition of velocity make sense.
So, now we have choices to make, and many of the people answering have made their own choice. One choice would be to measure how the probability distribution changes with time. Then you would get zero, because it is not changing with time when you are in a quantum energy eigenstate. But, I think that is not the best way to proceed, because we could do the same thing for earth orbiting the sun, if we asked for the probability distribution over millions of years, we would find it lies on the earth’s orbit and (ignoring all other perturbations) is constant with time. But, we would like to acknowledge that the earth moves over its orbit and find that average speed.
One can examine superpositions of energy eigenstates and find probability patterns that move in time, but those are not a very good definition of velocity either in my opinion, because they suffer from similar issues as above. (For example, a plane wave, which is normally thought to move with velocity \(\hbar k/m\) would be given zero velocity, independent of k, for any definition of velocity coming from a probability distribution, since the probability distribution for a plane wave is a constant over all space).
So, in quantum mechanics we need to ask for measurable quantities which come from the expectation value of a quantum operator in the state it is in at a given time. For the bound states of a hydrogen atom, we can measure the average momentum operator, and find that it vanishes in all energy eigenstates, and that its average value vanishes for superpositions, because the electron always remains bound to the atom. This is similar to the motion of the earth around the sun. What is its average velocity about the sun, if I average over a year? The answer is zero, because it goes as much this way as that and it remains in the orbit around the sun.
OK, so we have established three facts from the quantum world so far: (1) the probability distribution of an electron in an energy eigenstate is constant in time; (2) defining velocity in terms of time derivatives of probability distributions is not the best choice for defining an average speed; and (3) the average momentum, and hence average velocity is zero for an electron bound inside an atom.
But, we don’t have to give up just yet. What about the kinetic energy? Here, we find that the classical and quantum calculations agree, and the average kinetic energy is equal to the absolute value of the total energy. It is nonzero for the electron in any eigenstate. We can take that average kinetic energy, multiply by 2/m and take the square root to get the root mean square speed of the electron. This is probably the best choice for the “speed of an electron in an atom”. As we go to higher energy bound states, we find the root mean square speed goes to zero and it matches up with the speed one would assign to a free, or nearly free electron that is unbound, but moving in the field of the nucleus.
So, by sticking to the facts for what we can measure in a quantum system, and looking at the different possibilities—-measured positions versus time, probability distributions, average momentum, and average kinetic energy—- we find that the one that makes the most sense is the root mean square speed. Others are free to keep to their choices if they want, but they are then likely to have problems with relating their methods to simpler systems. Note that nowhere in this description did we need to say “the electron orbits the nucleus”. We don’t know precisely what the electron does, although orbiting might be a good word choice to describe that behavior. Asking for the path the electron takes as it moves in a bound state of an atom is a quantum question that we have no answer for. So we need to look elsewhere for our answers. This is where the confusion often springs up. Ask the right questions, and stick to the facts, and the confusion goes away!