Question

In elementary electrostatics questions, a common situation is an electron and a proton in close proximity and calculating the attractive force between them (Coulomb's Law, etc.)
If the force is attractive, then eventually the particles will collide. What is the ultimate consequence of this? I'm sure it depends on the momentum of the collision, but what might happen?

Answer 1

Neutralisation?

Answer 2

if ultimately electron and proton collided by attraction then ,in general all electron would be inside the nucleus..they behave as waves and due to uncertainity principle they dont collide..

Answer 3

They are extremely unlikely to collide. Generally, they will have at least a small relative velocity off the axis of the line between them (a little "sideways" velocity, that is). That will mean they will not run straight into each other, but will instead orbit each other.

Now, if we want to apply classical mechanics, then the orbiting implies acceleration, which implies radiation and loss of energy, so they will actually spiral in toward each other. However, classically they are both point particles, so they can never actually collide, they just spiral closer and closer, faster and faster, radiating more and more energy until you reach some absurd state of ridiculously fast motion and large amounts of radiation. At some point relativity comes into play...

But this is all fantastical, because an electron and proton are highly nonclassical. If we apply the correct quantum mechanics, there is a limit to how much energy can be radiated away, and at some point they will fall into the lowest possible orbital energy state, and form a stable hydrogen atom. That will happen even if their initial orbital momentum is exactly zero (there is no "sideways" motion), because the ground state of the hydrogen atom also has orbital momentum of zero, strangely enough.

Now, there is nevertheless some finite probability that the electron and proton will in fact find themselves so close together that the weak nuclear force has a chance to act on both (it has a very short range). The probability of that happening is greatly increased if the initial closing momentum is large, and the orbital momentum nearly zero, that is, you fire one particle at very high speed into the other.

If this happens, the weak force may allow an "inverse" beta decay reaction to occur instead of the formation of an H atom, and a neutron will be formed. (Probably some neutrinos are involved, too, to balance the spin books.) However, a free neutron is not stable. It has a half life of about 10 minutes, as I recall, so even if a neutron formed, pretty soon it would decay back into a proton and electron. Generally there'd be a lot of energy released, so the electron would rocket away from the proton, without being able to form an atom. Hence, overall, it would just look like an electron scattered off of a proton, with possibly some nuetrinos and other bric-a-brac spraying off.

Answer 4

wow :)

Answer 5

Thanks, Carl. Those were the possibilities I was considering: either a hydrogen atom would form, or they would merge into a neutron, but I wasn't sure what other details were involved that would make the difference.
I've been delving more into quantum physics these days, and I should probably go back and review some basic nuclear physics too.

Question about this, though: "Generally, they will have at least a small relative velocity off the axis of the line between them . . . That will mean they will not run straight into each other,"
- Where would this small sideways velocity come from though? If we are considering the contrived situation of nothing else existing in the universe except for a single electron and a single proton, and the only forces present are the electrostatic attraction, the gravitational attraction (and maybe some weak nuclear force?). The electric and gravity forces are both acting in the same line, so the only acceleration should be of the particles heading straight toward each other.
This is still treating the particles classically, but isn't that how they should be treated until the electron is close enough to occupy an atomic energy level?

Answer 6

So you're assuming the particles start off at rest with respect to each other, and furthermore with zero angular momentum? Sure, in that situation, they will gradually accelerate toward each other in a straight line.

At least classically, that is. Unfortunately, this situation cannot be set up quantum mechanically. If you want to put them in a stationary state (known fixed energy with respect to each other) and l=0 (zero angular momentum), that's fine, but what you have now is actually a bizarre state of an H atom, with the electron in a very, very very high energy state, extremely far from the nucleus. These states actually exist, and are called "Rydberg states." They have some very interesting dynamical properties, because of how closely spaced the stationary states are. This is frontier quantum chemistry stuff.

Alternatively, if you want to specify their precise separation, you can do that, too, but then -- uncertainty principle again -- you can't put them in a stationary state with known angular momentum, you'll have to put them in a superposition of H-atom states, so that the energy and relative angular momentum are not known precisely, but can take on a range of values. In that case, there will be some (very small) probability that the system will act as if it has l=0, but generally we can expect it to act as if has some angular momentum.

Even classically, I should point out that starting two particles off very far from each other and with l = 0 exactly is very hard. The reason is that l = w r, where w is the angular velocity of one around the other. The larger r gets, the more precisely w must equal zero to give you l = 0. This is why in planetary or stellar dynamics, you almost never see one body collide with another if they are in a good two-body situation. Comets starting essentially from rest, with respect to the Sun, in the Oort Cloud, when they accelerate toward the Sun essentially always end up in orbits, and not colliding head on with the Sun. Because even a teeny tiny sideways velocity initially, starting from 200 AU out, means you have a pretty big angular momentum and will miss the Sun. (To get collisions, you generally need third bodies that perturb the orbits into collision courses.)