As I mentioned in the previous post, you need to solve Schrödinger Equation: Through separation of variable time t and space r, you will have time independent Schrödinger Equation:
−
ħ2
2 m
∇2 ψn(r) + V(r) ψ(r) = En ψn(r)
where n is all collection of quantum number, e.g., spin, angular momentum, and energy.
For your question, let me take an example of Hydrogen Atom which has one electron. It's potential energy can be written as:
V(r) = −
e2
4πϵ0r2
Now if you solve the Schrödinger Equation which is nothing but a linear differential equation, you will find the answer very nicely. One thing to remember, in quantum physics, is that the solution of Schrödinger Equation should be interpreted probabilitically. For example, the expectation value of position of electron with n state, ψn(r), can be obtained as:
< rn > =
⌠ ⌡
ψ*n r ψn d3r
Anyway, you will find how the Schrödinger Equation is powerful and beautiful is to understand subatomic world. Your solution will give you electron's stationary state without collapsing into the nucleus.
You may apply Newton's equation of motion for a certain rotational velocity to remain a certain radius but problem is the electromagnetic radiation which means that electron has to keep losing energy by electromagnetic radition because a rotational motion involves acceleration. When a charged particle is accelerated, there are always electromagnetic radiations.
N.B.: The above one line Schrödinger Equation gives most answers to your quantum physics problems although you need to understand some basic concepts.
Sorry But I am off now. If I have a spare time, I can add more.
