Where did Schrödinger solve the radiating problem of Bohr's model?

Question

One of the problems with Bohr's theory to describe the hydrogen atom, was that the electron orbiting around the nucleus has an acceleration. Therefore it radiates and loses energy, until it would collapse with the nucleus.

Now Schrödinger describes the electron as a wave function. His theory is able to describe all atoms (in contrast to Bohr's model), but how is the radiation problem solved? I understand that the wave has not an exact position in the time anymore. But the electron still "moves", so it has an acceleration anyway (because of vibrations or so).

Why is in this theory the electron not radiating anymore? And if it is, why doesn't the atom collapse?

Answer 1

In any proper quantum mechanical understanding of the atom, a bound electron does not have a position and follow a path (i.e. have a time-varying position) in the sense that it would have in a classical or semi-classical theory.

Instead the electron "has a state" or "occupies an orbital" (an orbital not a orbit!), and because there is not a path there is not an acceleration associated with the path.

This poses a problem when you ask "Well, does it radiate or not?" because at first there is no theory for interaction of electromagnetic fields with "orbitals". You need to develop a new theory (eventually QED).

So the answer is that Schrödinger didn't fully solve the problem. He just said, "it doesn't have an acceleration in a classical sense" and left it at that.

Answer 2

I think the problem here is with E&M, and it is in the assumptions implicit to the question. Here is the assumption:

the electron orbiting around the nucleus has an acceleration. Therefore it radiates and loses energy, until it would collapse with the nucleus.

This statement can be demolished in short order, considering the topic of a non-radiating condition. The definition of the condition is:

the conditions according to classical electromagnetism under which a distribution of accelerating charges will not emit electromagnetic radiation.

The statement that acceleration leads to radiation and therefore collapse must be completely abandoned. We still have room for skepticism - formally we are arguing over whether the Bohr model of an electron fits the radiating condition.

I will argue that there is not a simple answer "yes" or "no". The reason is that we could currently re-interpret it as a model where the answer is that it does not fit a radiating condition. The picture of the Bohr atom I am most familiar with looks like this:

This is, an electron moving around a nucleus doing two things, a) exhibiting orbital behavior and b) fitting the orbital circumference to a mode of its de Broglie wavelength. But is the electron a particle or a wave in this interpretation? Following from that, does the electron have a defined location? If it's a defined particle, then it must radiate. But then why the requirement that the wavelength fit in the first place?

Let's turn to a classical analog. Imagine a ring of wire with no resistance and current flowing. This is an ideal electromagnet. It has a static electric field and a static magnetic field, but nothing is changing. It will sit there and maintain the current forever. (img source)

What's more, we could paint a picture of a simple electron orbital that fits this. Mentally combine the above two images. You'll probably be confused about the peaks and troughs of the wave, since they are not smooth with respect to position along the orbit. But this is an electron wave, not a EM wave. You can not distinguish the peaks and troughs in reality because the probability density is the imaginary wave function squared.

This doesn't describe all orbitals perfectly. For one, the electron in this model has a net angular momentum. But still, this gets surprisingly close to a passable model of nature. I think this is the correct stepping stone to move between the Bohr atom and real QM. The only question not sufficiently answered is why the electron arranges itself in this torus-like state, and the answer is that it doesn't because the actual dynamics are explained by QM, which not only accurately describes the emission lines (better than the orbital calculations confined to wavelength intervals), but also correctly predicts chemical bonds and the entire world around us.

What Schrödinger (really QM in general) did to advance beyond the Bohr model was to give a correct physical justification of the use of electron wavelength - leading to the wave equation. In this regard, the Bohr model was a confused mess, but it could be tweaked to be non-radiating per my above argument, which entails non-localization of the electron along the circumference of orbit. Obviously this is nonsense, because if it's non-local along that line, then why isn't it non-local along other dimensions as well? QM answered this correctly.

Answer 3

As discussed in other answers, the short answer is "he didn't", in the sense you seem to be asking. The electron is simply not a classical object - it's behaviour is something wholly new (as of 1926 that is, when Schrödinger published his equation). The "solution" was that his approach foretold the hydrogen spectrum to an accuracy that was much greater than Bohr's model: in short the "solution" ultimately came experimental verification. You may benefit simply by reading more on the history of QM if you have not already done so so that you understand that great minds have had almost exactly the same kind of misgivings as you seem to be having. To invoke Feynman here (the exact words are on the audio version of his QM lectures) how does one come up with a new theory? - step 1 - you guess it, step 2 you do experiments to test it, step 3, if the experiments tell against the theory, no matter how clever or appealing it is, then it's wrong and you go back to step 1.

As an aside, it might help (I know this too might seem highly artificial) to know that there is a formulation of QM (the Heisenberg picture) wherein the electron is perfectly "still": its state does not change and instead the "observables" - the operators + the special "recipe: which tells you how to interpret them and whose eigenvalues are the possible measurements - are the things that evolve with time. This Heisenberg "matrix mechanics" is made equivalent to the Schrödinger picture through a unitary (i.e. roughly something that preserves probability distributions) transformation that evolves with time. The Heisenberg picture is kind of analogous to doing mechanics in a rotating frame, but it may help to know of this approach and anyway who's to say, without further experimental justification, which is the rotating frame!

The "solution" discussed took a long time to be truly embraced. Schrödinger came up with his famous cat thought experiment because he thought the Copenhagen interpretation of his and Heisenberg's theory (Heisenberg's matrix mechanics and Schrödinger's wave mechanics were shown to be the same) was crazy and needed to be repudiated - he ultimately left the field; Rutherford was aghast at what he thought was the possibility of "half an electron" being somewhere and the other half somewhere else (these thoughts provoked discussion as to whether even our conception of space as being modelled by a smooth manifold were even valid at the atomic level) and Einstein famously fought and thought so hard against what QM seemed to be saying that he (and Podolsky and Rosen) came up with the famous EPR paradox which they thought repudiated the prevailing QM interpretations but instead led to the discovery of quantum entanglement. I find it ironic that even if Einstein had done nothing else than try to tear the probabilistic interpretation of QM down, he would still have been one of the greatest physicists of the 20th century.

Lastly, as animals we have evolved to recognise and understand patterns that arose in our birth home, to wit the wet savannas of Eastern Africa. From a evolutionary biology standpoint, there is absolutely no reason why we should understand things like electrons, let alone why they should fit into our "Wet Savannah World" view of the world studied in classical physics. You just have to accept that the electron is standing still, de-localized and so somehow spread over all the points in its orbital at once and not shifting from one to the next. The weirdness and psychological jarring that you feel in letting go of the idea of an electron as a point buzzing around reflects that your evolutionary forebears came across nothing truly analogous to the delocalised electron in their wet Savannah home. This is not to say we give up because it's too hard - we simply have to heed that we have certain biologically programmed prejudices about the World that sometimes help us in phyics as intuitional insight, but other times they hinder us and point the wrong way.

You may find it helpful not to think of separate electrons as the fundamental things, but rather the oneness of the quantum electron field: the electrons themselves are like the discrete "data packets" whereby this field communicates with and interacts with the other quantum fields in the World and they don't needfully have to be anywhere in particular - but again, you have to be wary of leaning too hard on any analogy. Please see this excellent elementary level video http://www.youtube.com/watch?v=Fxeb3Pc4PA4&list=UUUHW94eEFW7hkUMVaZz4eDg.

Further Reading: Heisenberg very much chose to concentrate on what could be measured rather than what was happenning to give rise to the measurement and this "quantum measurement theory" approach is something some people find appealing. This is where one goes back to a set of experimental observations about "observables" (the so-called "operators") and uses them as axioms without trying to get too caught up in other ideas. I have a set of undergrad notes along these lines by Hideo Mabuchi which I thought were excellent - I couldn't find them on the web but you might like to contact him through the link.

Answer 4

Schrodinger describes the electron in an atom as a standing wave pattern. Essentially it's a particle in a box. A particle in a box has a maximum wavelength, and therefore a minimum kinetic energy. This is the ground state. In an actual atom, the details differ, but you still have a ground state, which is a state of minimum energy.

Due to conservation of energy, the ground state can't radiate. If an electron in the ground state were to radiate, it would be radiating energy, but it doesn't have a state of lower energy that it can go to.

Answer 5

Here is my answer to this very difficult issue. In my opinion the elementary Schroedinger approach does not solve the problem of radiation. The electron still radiates when it changes its energy level and this process is not described in the elementary Schroedinger model based on the Coulomb potential only. Experiments prove that every level is not stable, only the fundamental level is stable. This is the only true difference with respect the classical description, where there is no lower bound to the energy of the electron.

Let me be more clear. If you consider the classical model neglecting electrodynamics, but only considering Coulomb interaction, even the classical model is stable. The same happens to the Coulomb-Schroedinger model neglecting the interaction with the quantized electromagnetic field (in the vacuum state in general). The energy levels seem to be stable, but they are not as experiments show. In fact, also theoretically, once you switch on the interaction with the E.M. field, exactly as in classical electrodynamics, all levels but the fundamental one become unstable in accordance with experiments. The only true difference form the classical description including all electrodynamics (barring the fact that the allowed energies are now discrete) is the existence of the ground state in the quantum picture.

So a more right question would be instead "why, differently form classical physics, there is a ground state for the Coulomb attractive potential in the Schroedinger model?"

Answer 6

One of the problems with Bohr's theory to describe the hydrogen atom, was that the electron orbiting around the nucleus has an acceleration. Therefore it radiates and loses energy, until it would collapse with the nucleus.

This is a common error in physicist's history of physics. The problem was not with Bohr's model, but (as Bohr thought) with Rutherford's (planetary) model. The problem was brought up by Bohr and he proposed a solution - Bohr's model - where certain orbits are assumed to be special in that the electron can move on them without radiating - it has a kind of exception from the laws of electromagnetic theory there.

Why is in this [Schroedinger's] theory the electron not radiating anymore? And if it is, why doesn't the atom collapse?

Actually, both in Rutherford's and Schroedinger's models electrons do not radiate. The reason is that these models do not include enough of electromagnetic theory to describe radiation. They only use its non-relativistic version - Coulomb's formula for electric force (potential energy).

Electromagnetic radiation is a phenomenon that does not exist in a model where EM forces are described this way - the Coulomb formula describes instantaneous interaction with no radiation. To include radiation, one may attempt to use Maxwell's equations instead of Coulomb's formula or some explicit formulae for the EM fields. However, this is difficult to analyze accurately both in classical and quantum theory and there are many possibilities for the choice of appropriate EM fields (the free field part) and the resulting behaviour of the system depends greatly on this choice.

The great advantage of Schroedinger's model as opposed to Bohr's is not in stability, but in the fact that Schroedinger's model does not need special assumption about preferred orbits - the eigenfunctions and eigennumbers follow naturally from one general differential equation. Also, it is applicable to more general cases with many more charged particles (atoms, molecules) where it is difficult to see how the appropriate Bohr orbits should be chosen.

Answer 7

The Schroedinger model does much more than people here give it credit for. It doesn't just solve the problem of why the ground state doesn't radiate. It solves the problem of why and how the excited states radiate, and it does so by using nothing other than Maxwell's equations.

In the Schroedinger model, the states that don't radiate are the states with a stationary charge distritubtion, and the ones that do are the states with an oscillating charge distribution. You get an oscillating distribution by taking the superposition of two eigenstates. The amount of radiation you get from such a state is exactly the amount you calculate from classical antenna theory using Maxwell's equations.

I don't know why this isn't more well-known. I have a series of blogposts starting here where I explain this in more detail and show how to do the semi-classical calculation applying Maxwell's equations to the superposition of the s and p states of a hydrogen atom.

Answer 8

To add yet another voice, I believe that Bohr got around the radiation problem by simply stipulating that, for reasons not yet understood, electrons in certain special orbits simply did not radiate.

His theory certainly allowed radiation as when an electron moved from one orbit to another. But the predicted orbits in Bohr theory were to be "special".

Answer 9

I will put in my two cents:

The Bohr model as such can be saved by postulating standing waves for the electrons. The contrast with the Schrodinger formalism lies not only, as observed by others , that the solutions of the Schrodinger equations are more accurate and can be generalized to complicated potentials, but that the Bohr model is only one step higher than the numerology/data-fitting of the Lyman and Balmer series and the Rydberg formula. It is not a theory.

What made Schrodinger's equation lead to a theory, was the postulate that the solutions of his wave equation were to be squared and interpreted as a probability distribution of the possible position of the electron. This is that led to a theory of quantum mechanics, rather than a model of quantized states. The position of the electron was interpreted as a probable one, leading to orbitals, and not orbits. In addition within this theory the eigenstates of these orbitals were stable, and thus not radiating.

Answer 10

Actually Bohr's model solved the radiation problem by setting a minimal value for the angular momentum of the electron. Hence, in his model, radiation or not, the system cannot go lower in energy.

Now Bohr was wrong in his model mainly because for instance it was completely incompatible with a zero angular momentum which incindently happens to be the ground state of the hydrogen atom. In this respect it was an epic fail but otherwise it was a very nice model that solved many problems.