In atoms, what force or charge, etc. keeps electrons from flying away or into their nucleus? is there a kind of weak-force at work on the atomic scale?

Note I am aware the electron positions are only abstract variables and can be referred to as the electron field and the like. This is not the question.

What is the reason an electron is bound to that nucleus to the point it can sustain an "orbit" or variable probability path, and not fly away or into it's nucleus?


In order to find the possible ways of an how an electron acts in the presence of a proton we solve the Schrodinger equation with a coulomb potential, $\frac{k q}{r}$. At the outset of solving an equation, from a strictly mathematical viewpoint, the equation you are solving might have no solution, 1 solution or infinitely many solutions, and also the solutions might be continuous or discrete. That is, when we solve for the possible energies of system the energy might be able to only be one value, any value, or some particular discrete values. In the case of the Schrodinger equation for a hydrogen atom, it turns out that it only takes on the particular discrete values:

$E = \frac{-13.6 eV}{n^2}$

where n is an integer (and not equal to zero). So the lowest energy level an electron can have in this bound state is $-13.6eV $, it just can't go any lower. Its sort of like a building with definite floors to it, there just isn't anyway to be 'in-between' the levels. I want to emphasize that it is crucial we treat the electron quantum mechanically - 'classically' treating the electron like a point particle in the presence of a coulomb potential the electron spirals inward towards the nucleus. Thus the fact that an electron stays in its orbit is fundamentally different than way the earth stays in its orbit.

Now at this point, many people go on to say something about the uncertainty principal forbidding an electron to have definite position and momentum which forbids the electron from ever being so localized so as to be right at the site of the nucleus. As far as I can tell these arguments are simply false since an electron can actually sometimes fall into a nucleus to be annihilated, see electron capture - http://en.wikipedia.org/wiki/Electron_capture. So in principal you can have a process where the electron does fall into the nucleus to become annihilated. The thing is, the force that mediates this process is not the electromagentic force, its the 'weak' force which got its name for obvious reasons - its hardly noticeable over distances much bigger than $10^{-17}m$ (see http://en.wikipedia.org/wiki/Weak_interaction) which is much much smaller than the Bohr radius at $\sim 10^{-11}$ which is a rough way of thinking of how far the electron is from the nucleus at the lowest energy level ($n=1$).

In short, the dominant force that governs an electron in the presence of a proton is the electromagnetic interaction which only allows particular energy levels which are very stable. There are smaller, subdominant forces that allow other processes, but these are weak which in quantum mechanical terms means the processes governed by the weak force are very rare.

I'm not sure how much background you have, so let me know if you want more information in one direction or another.

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    $\begingroup$ I'd like to also emphasize that not just any solution to the Schrodinger equation will do; the solutions, to be physical, must be normalizable. $\endgroup$ Jun 29 '12 at 2:10
  • $\begingroup$ This is a good point, I wasn't sure how much background he had so I didn't want to go off an a tangent about normalizablity and boundary conditions etc. $\endgroup$
    – DJBunk
    Jun 29 '12 at 2:12

The scale of the atom has Planck's constant in it, so it's quantum. The force that keeps the electrons near the nucleus is the electrostatic attraction between the electron and the nucleus. To understand why the electron doesn't fall into the nucleus formally, you can solve the Schrodinger equation, but there are seat-of-the-pants arguments that are correct to give the right order of magnitude.

  • Uncertainty principle: in order to confine the electron to a box of radius r around the nucleus, you have to give it a momentum of order h/r, which means that it's kinetic energy is roughly ${\hbar^2\over 2m r^2}$, while the potential energy is (negative) ${ke^2\over r}$. The total energy is the difference of these two, and it has a minimum when the size of the box r is of the order of the Bohr radius: $r_B = { \hbar^2\over 2m ke^2}$.
  • Radiation frequency: The closely related original argument by Bohr is that the frequency of the radiation emitted by the atom should be of the order of the classical orbital frequency. Yet the radiation should also be in quanta. The classical orbital frequency is the reciprocal of the time to go around once, and it obeys Kepler's law ${1\over T}={1\over r^{1.5}}$, but the binding energy goes as ${1\over r}$ so that the level spacing at small r scales to eventually be bigger than the energy magnitude, and there should be a lowest energy level.

For a different power law force, if the potential energy goes to minus infinity with a power faster than $1\over r^2$, the particles attract so much that there is no stable bound state, they end up sitting on top of each other.


There are four "fundamental" forces. They are Gravitational (which keeps us stuck to the earth), Electromagnetic (keeps magnets stuck to the 'fridge), "Weak nuclear" (something to do with radioactive decay) and "Strong nuclear" (which keeps protons and neutrons stuck together inside an atom).

In addition to keeping drawings stuck to the refrigerator, the electromagnetic force also keeps electrons in orbit. Electrons have a negative charge, and the nucleus has a positive charge. Opposites attract, and so the atom holds together.


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