How can one describe electron motion around hydrogen atom?

I remember from introductory Quantum Mechanics, that hydrogen atom is one of those systems that we can solve without too much ( embarrassing ) approximations.

After a number of postulates, QM succeeds at giving right numbers about energy levels, which is very good news.

We got rid of the orbit that electron was supposed to follow in a classical way ( Rutherford-Bohr ), and we got orbitals, that are the probability distribution of finding electron in space.

So this tiny charged particle doesn't emit radiation, notwithstanding its "accelerated motion" ( Larmor ), which is precisely what happens in real world.

I know that certain "classic questions" are pointless in the realm of QM but giving no answers it makes people asking the same questions over and over.

• If the electron doesn't follow a classic orbit, what kind of alternative "motion" we can imagine?
• Is it logical that while the electron is around nucleus it has to move in some way or another?
• Is it correct to describe electron motion as being in different places around nucleus at different instants, in a random way?
• Related: physics.stackexchange.com/q/2860/2451 and links therein. – Qmechanic Apr 9 '12 at 16:08
• – voix Apr 9 '12 at 16:35
• I'm more curious how the electron moves without producing EM radiation. But someone will tell me that it doesn't have a lower ground state to decay to. I know... but it's still a moving charge. I think a more satisfactory model would be that coherent states aren't "moving" in the classical sense, because the concept of moving is a limit-case approximation of QM to begin with. – Alan Rominger Apr 10 '12 at 13:15

The problem is that you're thinking of the electron as a particle. Questions like "what orbit does it follow" only make sense if the electron is a particle that we can follow.

But the electron isn't a particle, and it isn't a wave either. Our current best description is that it's an excitation in a quantum field (philosophers may argue about what this really means; the rest of us have to get on with life). An electron can interact with its environment in ways that make it look like a particle (e.g., a spot on a photographic plate) or in ways that make it look like a wave (e.g., the double slits experiment) but it's the interaction that is particle-like or wave-like, not the electron.

If we stick to the Schrödinger equation, which gives a good description of the hydrogen atom, then this gives us a wavefunction that describes the electron. The ground state has momentum zero, so the electron doesn't move at all in any classical sense. Excited states have a non-zero angular momentum, but you shouldn't think of this as a point like object spinning around the atom. The angular momentum is a property of the wavefunction as a whole and isn't concentrated at any particular spot.

• Why do you say "momentum" zero. At n=1 l=o m=0 it is the angular momentum that is zero. There is still energy in the orbital which I think can be interpreted as momentum in some manner. – anna v Apr 9 '12 at 16:10
• I'm aware that some classical "prejudice" has to be dropped; but given the excited states, even if we don't have a trajectory over time for the electron, can we conjecture a kind of non-accelerated non-classical, ( weird ) "motion"? Or the wave-particle duality is unbalanced towards waves? – Marco De Lellis Apr 9 '12 at 20:33

You might be helped by reading carefully the wikipedia article on the hydrogen atom particularly the figures.

The electron described in the orbital has not only a specific energy but also momentum and angular momentum, though it is only the operators of energy angular momentum and spin that give the eigenvalues for n l and m.

So what is random is not the electron per se but the probability of finding it when you try to measure it in some way . It is moving with 1/137 of the velocity of light according to the linked article.

If the electron doesn't follow a classic orbit, what kind of alternative "motion" we can imagine?

as given in the pictures of the orbitals. such a fast moving particle will look like a cloud anyway, even if possible classically.

Is it logical that while the electron is around nucleus it has to move in some way or another?

Yes, we just cannot pin it, think of the uncertainty principle organized by a solution to Schrodinger's equation.

Is it correct to describe electron motion as being in different places around nucleus at different instants, in a random way?

No, not random. It is organized by the probabilities of the orbital it happens to be in.

• I have read the linked article, and thanks for your answer. However something sounds uncomfortable. The electron is moving at 1/137 c, so it shows a classical property, speed. If we consider the wave part from the wave-particle duality, we can imagine this wave that travels at that speed in space around the nucleus, drawing a weird pattern ( in places where orbital is non-zero ). However no traces of this moving wave are found in Schrödinger solutions ( the wave functions! ) for the hydrogen atom, why? – Marco De Lellis Apr 9 '12 at 21:19
• I believe that it is the solutions of the Schrödinger equation that predict those patterns. Why do you say no traces? The probability functions are highly osclillatory in theta and phi except the n=o m=0. These are probability functions. One can only see waves by their interference after all. Or think of them as "standing waves". – anna v Apr 10 '12 at 4:08
• "Standing waves": this is really interesting. Could be the electron described as a kind of stationary wave? And wave of what? This wave describes only the probability to find it, or something more deep, like a wave of its properties like mass, charge, ... ? Thanks for your patience with me. – Marco De Lellis Apr 10 '12 at 5:45
• Just the probability of finding it. Once in a potential well the electron itself is in a virtual state. Virtual means that it is not possible to measure mass or charge except collectively with the atom as energy and charge conservation.One does not have a moment by moment snapshot of the electron, or the nucleus at that. Only of the probability of what you will find if you take a snapshot. – anna v Apr 10 '12 at 6:08
• This is a difficult concept to swallow when one's intuition comes from classical physics which is our everyday experience, but it is true because it has been experimentally checked in very many cases, not just the hydrogen atom. The uncertainty principle and the probabilistic nature of nature is the corner stone of modern physics. Not random, there are envelopes to the uncertainty, but probabilistic. – anna v Apr 10 '12 at 6:10

Is it logical that while the electron is around nucleus it has to move in some way or another?

That probably depends of what exactly you call motion, but I would highly recommend an excellent book And Yet It Moves by Mark P. Silverman, and the chapter #3 in particular. If you replace an electron (which is a stable particle, that is a particle without age and individual history) in a simple atom with a negative muon (which decays quickly, its lifetime being some 2 microseconds in its rest frame) you would expect that measured lifetime (in the atom or lab rest frame) will be longer if the muon moves at relativistic velocities due to time dilation, exactly as experiments confirm.

• Well, this quite interesting. I knew of "older" muons from cosmic rays, but if I understand correct, they made a "setup" with muon Moving at relativistic speed nearby some nucleus. To experience longer life, it has to move in some semi-classical way, is it correct? – Marco De Lellis Apr 10 '12 at 16:47
• @Marco Yes, you are right. Atoms where muon replaces electron is prepared and muon lifetime mesured. Its length corresponds to the expected semi-classical velocity of muon in such an exotic atom and special relativistic time dilatation. – Leos Ondra Apr 10 '12 at 20:25

Think of an electron as of non-point particle. In a hydrogen atom it is "smeared" around proton. Its total momentum is zero – it is neither moving (in total) nor accelerating – hence in a classical limit it does not emit radiation.

If an electron in an atom is a "cloud" rather than a point, it is in different points at the same time. That means that there is a non-zero distribution of "electron density" smeared around proton.

An electron is not "moving" as a whole, but we can say that "parts of the cloud" are moving, since they carry non-zero momentum resulting in total angular momentum. This is a consequence of the fact that integration of the electron's momentum density over limited volume in space is non-zero.

• Smeared electron sounds good. A cloud that stays in the same place over time, only changing shape, not a fast moving electron drawing a cloud-like shape. The classical speed goes away, so we don't care anymore. – Marco De Lellis Dec 17 '14 at 11:42
• Anyway, as Leos Ondra has pointed out, yet it moves in a semi-classical way, since the relativistic effect accounts for longer muons' life ( delayed decay ). It moves faster in space, and slower in time, basic relativity. We have to cope with this classical behavior. Has anyone predicted longer muon life using Schrödinger equation? Why should electron be at rest while muon is not? State equation is a good model like an ideal gas, but has to be adjusted to work in real life. – Marco De Lellis Dec 17 '14 at 13:52