Dear Ilian,

you are missing orbital angular momentum in the equation, which can be carried by the atom-photon system. That's the resolution to the "paradox".

Angular momentum conservation holds for the *total* angular momentum, which combines both orbital angular momentum and spin contributions.

Furthermore, even in the n=5 shell, there are states with l=0 (s-orbitals), l=1 (p-orbitals), and so on. Those do not even need any other type of angular momentum.

I hope this helps.

Best regards,

Alex

Dear Ilian,

you are missing orbital angular momentum in the equation, which can be carried by the atom-photon system. That's the resolution to the "paradox".

Angular momentum conservation holds for the *total* angular momentum, which combines both orbital angular momentum and spin contributions.

Furthermore, even in the n=5 shell, there are states with l=0 (s-orbitals), l=1 (p-orbitals), and so on. Those do not even need any other type of angular momentum.

I hope this helps.

Best regards,

Alex

Dear Alex,

Thank you for trying to answer this issue. I don't understand exactly what are you stating.

Are you stating that the nucleus takes the missing part of the total angular momentum? (3h in the case)

I haven't read the original Bohr paper but just a retold story in QM books time ago. I don't remember anything said there about orbital momentum of the nucleus. Also didn't find anything about nucleus orbital momentum by rereading a book now. Nobody cares about orbital momentum but they just derive the energy formula for the photon. I also didn't notice this when I first read about Bohr model.

There is no need to make the case more complicated as I consider only the Bohr model. There are no quantum numbers l, m and s in it, except  the spin of the photon which I take into account.

And yes it is a fact that a sole electron can not emit a photon so the role of the nucleus is important. The only probability I see yet is that the nucleus takes some opposite orbital momentum (though I am not sure there can be such thing as nucleus orbital momentum - especially for Hydrogen nucleus). But even if so that will be very very strange as it looks like a nonlocal interaction.

The energy of the photon doesn't have anything to do with its angular momentum, because it's known, already in the Bohr approximation, that the energy levels are degenerate: the energy depends only on the principal quantum number n, not on the orbital quantum number l. So the statement that the angular momentum changes is incorrect, as given.  Angular momentum is conserved, of course.

However since the energy levels are eigenstates of the Hamiltonian, for a transition to take place, an external field must be imposed, in a way consistent with the conservation laws. An electron, initially,  at n=5 will evolve always at n=5 and its wavefunction will just acquire a phase, absent any external field. So there isn't any inconsistency. For a transition to n=1 to take place, an external field is necessary, that must have the appropriate (total) angular momentum, as well as the appropriate energy. 

Dear Stam,

I agree with all what you wrote. But according to the derivation of Bohr 'n' (the principal quantum number) in the formula for the orbital momentum of the electron L=nh is the same as in formula for the energy. E=R/n^2. I has doubt about it after your post and checked it again. There is no doubt. The level with energy E=R/n^2 has orbital momentum L=nh (h is Dirac constant, R Rydberg constant).

Notice this is not my statement but Bohr's.

Sure and the statement is that an electron in such a state, that's an eigenstate of the Hamiltonian of the hydrogen atom, *stays* in that state-upon evolving, with the Hamiltonian, it picks up a global phase.  It cannot realize a transition to any other eigenstate, since the eigenstates are orthogonal, so any transition amplitude to another eigenstate of the Hamiltonian vanishes. Any transition from the state n=5 to the state n=1 requires the contribution of an additional term in the Hamiltonian. And that term must and does describe the transition from all the properties of the n=5 state to those of the n=1 state. The additional term, inevitably,  doesn't commute with the Hamiltonian of the hydrogen atom. It will describe the transition from a state of zero  photons to a superposition of states of a number of photons consistent with the conservation laws.  

I don't how are you refuting my statement at all.

First Heisenberg formalism wasn't invented when Bohr model was presented. So I don't see any necessity to rely on it. Just turn back to that time. And check the angular momentum law. Is it fulfilled or not? Is the nucleus momentum concerned or not?

Secondly from QFT viewpoint what is the field you are talking about? Because EM has spin 1 and can not fit in the picture.

This isn't history of physics, but physics. But, in any case, the statement, that is valid in the Bohr model, but, also, in quantum mechanics, is that an electron, in a stationary state, does not radiate, doesn't emit any photon. That was a postulate of the Bohr model, it can be proven in quantum mechanics. The states of fixed principal quantum number are such states. So an electron in a state with n=5 will stay in that state forever-which is consistent with angular momentum conservation, also. It will never realize a transition to any other state. To realize the transition to the state n=1 requires coupling the atom to an external field. Such a field, since it will affect the number of photons, is an electromagnetic field. And it's a standard exercise to show that this term has all the correct properties. This isn't quantum field theory, since the external field is classical, so the number of photons doesn't fluctuate and the number of electrons is, also, fixed. All these statements are consistent with the Bohr model-quantum mechanics provides the framework for realizing that some of Bohr's postulates can be proven and the transition probabilities can be calculated.  So, before proclaiming to the world that there's a possible inconsistency, it might be useful to study quantum mechanics-not from a history book, but from a textbook. 

Dear Ilan,

it is not the nucleus itself, it is rather the system consisting of the nucleus and the photon. This situation is not more complicated than necessary, on the contrary, it is required by QM because of angular momentum conservation (which the system must have in the first place because of the form of its Hamlitonian).

The issue with the Bohr model is indeed that it is a semi-classical model (in fact nearly fully classical, in all points other than Bohr's postulate that electrons on certain orbits would not constantly lose their energy by radiating off light). This furthermore involves that the statements the Bohr model makes about angular momentum are not at all accurate. In fact, the Bohr model just "predicts" the maximally possible angular momentum, but it does not even regard the electron as having spin - although by now we know well that it does, and that its spin is crucial for the shape of the atomic shells.

In a nutshell, nowadays we would regard the Bohr model as pretty bad description of the nature of atoms, because it disregards certain physical fact. Hence no wonder it leads to unphysical predictions and/or cannot fully explain atoms.

Of course this judgement is unfair, given that Bohr himself could not possibly have known all the stuff we know today. Nevertheless, from a modern physics point of view, the Bohr model is incomplete and this is why it does not yield very accurate descriptions of Nature.

Hope this helps.

Alex

I think there is no orbital momentum going to the atom-photon system as a whole or to the nucleus neither. This can be only considered if the Bohr model is right.

According to QM the orbital momentum for energy level n can take any whole number values from n-1 to 0. (already here one can see that the state Bohr totally relies on n, l=n lacks in QM). So further in the example regarded in the question n=4 it can l take values 3,2,1, 0. From this four states to level n=1 l=0 can reach only levels 4,1 and 4,0. The transition for levels 4,3 and 4,2 are forbidden from the angular momentum conservation law, because photon has s= 1,0,-1 and 4,1 must emit s=-1 and 4,0 emit s=0.

If the Bohr state 4,4 existed it also couldn't emit a photon and get to 1,0. So the Bohr calculation of right formula for E is based on a happy coincidence.

The model of Bohr has indeed historical importance but is studied even at school and it is worth considering how it has to be taught.

Speaking strictly about of the Bohr model only, I have to say that you are unnecessarily mixing up orbital momentum of electron with photon spin 1. When you talk about Bohr model, there is no need to talk about photon spin at all. Secondly, there is no need to talk about Quantum mechanics of Dirac and Heisenberg. Thirdly, there is no need to talk about Hamiltonian of Schrodinger Wave mechanics. These later developments of atomic model have thrown out of the window, couple of basic formula of Bohr model. Bohr model is based on the wave property of electron orbit which puts following restriction on the orbital radii.

                                                2*pi*r = n*Lambda.

where Lambda is the De Broglie wavelength of the electron wave, n is the principal quantum number. In Bohr model, orbital velocity of electron is given by

                                            v = e^2*k*Z / (h_bar*n).

h is the Planck constant.

From the first formula you can see that for n=1 state, there is only one standing wave. For n=5. the circumference of the orbit is made up of five de Broglie waves. From second formula, you can see that the velocity of electron is five times greater for n =1 than for n = 5. Therefore photon spin has nothing to do with the transition from n =5 to n = 1. If you put the velocity formula in Rydberg constant, you will find that                                     

                                             R = m*v^2*n^2 / 2.

Therefore quantized energy level are simply the kinetic energy

                                              E = m*v^2 / 2.

I agree with W. Zageri. Strickly speaking Bohr theory is based on the quantization of the  orbital momentum and its modulus is l (h_bar). The shrödinger equation gives the orbital momentum equals to (l(l+1))**1/2(h_bar). When l --> infinity this tends towards l (h_bar). So the Bohr model is "false", it gives a good expression of the energy  and a good expression of the modulus of the orbital for high l. So it is a coincidence that this model gives a good expression of quantized energy but this was a good idea and  alowed to have a simple model in order to calculate energy. So "the old quantics" cannot provide correct selection rule since the orbital momentum is "false". Moreover forget the spin.....

Bohr's model is at best incomplete.  We have known for a long time that it could not be exact this is just one very striking example of the incomplete picture that we as scientist have had to deal with in other areas of science.  As a chemist and experimentalist I see things that are just not quite right and just have to live with the fact that we do not know.  

I some years back came up with a better model of the atom that more explains the inconsistencies that I have seen in the atom but have not published it for fear of being labeled crazy.  

Good luck in getting by the sensors which it seems we have in the science community.  I think the only way to make chance is to not publish in the places that do not care about the truth but only in preserving the past.  Then publish in some places that scientist like myself will still see it but do not have to worry that some censorship has changed the work and thereby made it wrong.

To be clear In 2 words: Schrödinger equation and Bohr model are uncompatible from the point of view of the quantized orbital momenta unless tending towards the classical limit (high l) . 

Marie-Françoise Politis

Ground state (l=0) energy levels of Bohr atom and Schrodinger theory are same. Schrodinger theory provides additional energy levels for l not equal to 0.

Indeed any claim that the Bohr model is suprseded by subsequent work isn't relevant, because the question refers to consistency, not experiment. The Bohr model is consistent, with respect to the issue raised, because the states with definite n are stationary states, in that model, so there isn't any transition possible from n=5 to n=1 at all, in the absence of an electric field. And, in the presence of an electric field, such a transition is possible. It was, indeed, measured, is known as the Stark effect and was explained, in detail, by Bohr and Sommerfeld (that explained how to describe the states with non-zero angular momentum)-and, apparently, Schwarzschild was, also, involved. Cf. http://arxiv.org/pdf/1404.5341.pdf

The best understanding of Bohr Model was presented by Einstein in 1917, not very knowing paper which insired Schrodinger and De Broglie and was at the origin of their  waves mechanics. I atached stone paper who explain little bit Eisntein insights. I'll attatched the english version of Einstein paper per se later when I'l arrive at my office.

The paper contained an elegant reformulation of the Bohr–Sommerfeld quantization rules of the old quantum theory, a rethinking that extended and

clarified their meaning. Even more impressive, the paper

offered a brilliant insight into the limitations of the old

quantum theory when applied to a mechanical system that

is nonintegrable—or in modern terminology, chaotic. It was at the origin also of the work around Geometric Quantization by Kirrilov and Souriau.

Best,

Joseph

http://lptms.u-psud.fr/nicolas_pavloff/files/2010/03/Stone-phys_today1.pdf

thanks to you, Joseph , very interesting paper, to be studied ....

Best

MF

Ilian,

You mix things, a couple of things.

You say,

"The first postulate of Bohr theory is that the Orbital momentum of the electron is quantized L=mvr=nh  (where h  means the Dirac constant)."

Let's make order in the balagan.

Besides the fact that h is Planck's constant (not Dirac's), n is the quantum number of the energy level, not of the angular momentum, see Wikipedia

https://en.wikipedia.org/wiki/Hydrogen_spectral_series

The Lyman (not Balmer) series contains all the falls from higher energy levels, n=2, n=3, n=4, n=5 as you mentioned, etc., to n=1, see figure 2 in the reference.

Now, on each such level, the angular momentum (which is quantized as you say) can take values as follows. We denote this quantum number by ℓ. For the n-th energy level one has ℓ = n-1, n-2, . . ., 0. The number ℓ is never negative. The absolute value |L| of the orbital angular momentum is given by the following relation:  L2 = ℓ(ℓ +1)ħ, where ħ = h/2π.

Assume now that we place the atom in a magnetic field of direction, say, z. The projections of L on this direction, Lz, will take one of the values:  ℓħ, (ℓ-1)ħ, . . . ħ, 0, -ħ, . . . -(ℓ-1)ħ, -ℓħ. However, the electron has also spin, which is equal to ħ/2, and its projection on the z axis can be either ħ/2, -ħ/2. We denote the projections of the total angular momentum on the axis z, by Jz. So, the values that Jz can take are: (ℓ+1/2)ħ, (ℓ-1/2)ħ, . . . , ħ/2, -ħ/2, . . . , (ℓ-1/2)ħ, (ℓ-1/2)ħ.

This is the full story. Therefore, if you consider a Lyman transition from n=5, you have to specify from which value of Jz to which value of Jz the transition occurs. Note that for n=1 one has only ℓ=0, (recall that the highest value of ℓ on the energy level n, is n-1). Thus, on the energy level with n=1, Jz can be either ħ/2, or -ħ/2.

As you said, the angular momentum of the photon is ħ, s.t. a transition with emission of one photon from a value Jz can be only to a value Jz - ħ, or Jz + ħ. In the 1st case the projection of the angular momentum of the photon on the axis z is ħ, and in the 2nd case is -ħ.

With this information in mind, you can judge whether Bohr was wrong about something.

Sofia,

I don't know why are you retelling the QM version of the energy levels. This is NOT what Bohr proposed in his model. Read again the Bohr model especially the derivation of the energy level from the angular moment. Than tell me what is the angular moment for E=R/n*n in his proposal (not in QM).

Then please tell me how the angular conservation law is applied in Bohr's model (not to mix with QM) for great difference between two levels.  When the difference is more than two the transition is forbidden.

I'm really very surprised this was omitted by so many people (also including me before a long time) for almost 100 years. I recently was again interested in it and found this surprising fact.

I wrote h for the Dirac constant because I don't know how to write hbar. (this was clear to everyone). Now I will already copy it from your post.

Thank you for correction for the Balmer. You are right here it is Lyman.

Angular momentum conservation is realized in Bohr's model by the statements that an energy eigenstate is a stationary state-so the electron neither emits nor absorbs a photon spontaneously-along with the statement that angular momentum, also, has a definite value in such a state, an integer multiple of hbar, the same integer that appears in the expression for the energy levels. There's no spin in Bohr's model and no degeneracy, either. Therefore an electron in such a state remains in that state and its angular momentum doesn't change, either. Energy and angular momentum are conserved. What Bohr's postulates predict is the frequency and thus the energy of the *applied* electromagnetic field required for any transition between the energy levels to be *possible*. These frequencies did, of course, correspond to what was known at the time, already. There's no inconsistency whatsoever-that's different from the statement that it is incomplete.

 For a transition to take place an external field is required and how to take this into account was found by Epstein, Schwarzschild and Sommerfeld, in the paper linked to in a previous message, in order to explain the measurements by Zeeman (for a magnetic field) and Stark (for an electric field). There it was found that the notion of degenerate energy levels was required and the orbital quantum number was introduced. Apparently the additional degeneracy, due to the spin of the electron was noticed by Heisenberg in his senior thesis, on the Zeeman effect, in 1921 or so; but the idea of fractional quantum numbers was considered as incongruous and was rejected at the time.

@M. F. Politis

Dear Marie Françoise,

Thank you for your kind word. As I said Please find attached a copy of Einstein 1917 paper about quantization (Model Bohr). It's the most profound conception of Bohr's model. This paper is not very weel knowing but it's an indication of the topological conception which has Einstein concerning the quantization. As you know today we can show that the duality waves-particles and Heisenberg incertainty can be derived from Lie algebra of Poincaré group (which is the symmetry group of the space-time). More precisely Quantization is the central extension of the classical symmetry or what we call today Cohomology of the group of symmetry. what is overwhelming, when we read Einstein Paper it concern this cohomology where the quantization is the indication that the second class of cohomology is trivial. It's the most profound conception of the quantization or what quantization is after all.

If some one are interest to understand this way of thinking of quantization I'll ne happy to send a paper treating and intrducing all those aspects.

Best,

Joseph

http://astro1.panet.utoledo.edu/~ljc/EinstEBK.pdf

Stam,

I do not state that the angular momentum is not conserved on the orbit itself, but just when there is a transition from n+k to n (where k is bigger than 1), because l=n in Bohr model. There is not any point in what triggers the process. This is an electromagnetic transition and for it the permitted (just by momentum conservation law) change in l is Δℓ=±1.

To Joseph Kouneiher. Joseph the article by Einstein you had attached is NOT AT ALL about Bohr model of Hydrogen. It is about the following facts

1. Hydrogen model is quantizable ala Bohr only because in the corresponding Hamilton Jacobi equation for hydrogen Hamiltonian there is a complete separation of variables. Because of this, 

2. 3 dimensional problem is reduced to 3 one dimensional ones exactly solvable. For each of these the Bohr-Sommerfel'd quantization condition is applied then. Therefore

3. Bohr model is having 3 quantum lumbers:n, l and m.

4. There is no room for spin in Bohr model as much that there is no room for spin in the Chrodinger's equation. Spin can be introduced either by hands, as it was done by Pauli. or via Dirac equation with subsequent transition to the non relativistic limit.

5. In 1917 Einsten, being aware of  exact solution for hydrogen atom, rased the following issue. Since Hamiltonian for H admits complete separation of variables, Bohr-Sommerfeld quantization prescription is applicable.BUT, what is the solution protocol , say, for the He atom. In this case the Hamiltonian DOES NOT admit separation of variables and yet, spectroscopically,  He ehxibits well defined spectrum.  So, Einstein raised the question: How one can extend the Bohr Sommerfeld quantization beyond H atom? And he came up with his quantization postulate which should not at all to be confused with the usual Bohr Sommerfel'd condition. What Einstein is using is known in mathematics literature as Poincare-Cartan invariant. He proposed that this invariant is quantized. This fact is being widely used in the theory of Quantum Chaos.  Representative applications to chemistry you can find by reading:  Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. Semiclassical calculation of eigenvalues for a three-dimensional system. J. Chem. Phys. 73 (1980), no. 1, 391–395.  Good luck to you and to Illian Peruhov

Dear Arkady,

I DIDN'T SAY THAT EINSTEIN'S PAPER IS ABOUT Bohr model of HYDROGEN. I said that Einstein try to analyse the geometric aspect of Bohr Model and signification of quantization.

Cheers,

JK

Not really! He was not thinking about Bohr model in this paper.He was thinking about quantization of nonintegrable dynamical systems. Nowadays, if you would  look into literature you would notice that EVERY BOOK ON QUANTUM CHAOS cites exactly this paper as the initial seed for quantum chaos problematics

Indeed, for this I mentionned Stone paper from the begining. But that still concern the concetual foundation of quantization. This paper, inspired Schrodinger and De Broglie waves mechanics and geomeric quantization later also.

Thanks

@Arkady Kholodenko:

Following article shows how spin and relativity can be introduced in Schrodinger model.

Article Quarkonium and hydrogen spectra with spin-dependent relativi...

There are 100...000 papers showing how spin could be introduced into Schrodinger model. This has nothing to do with the Bohr model,  calculations for this model or with the original paper by Schrodinger. 

Joseph Kouneiher. Joseph, I was trying to find in your comments the reference to Stone paper and was unable to find such reference. Also, I do not understand why you are not willing to look at the books on quantum chaos. All of them have reference to Einstein's 1917 paper . As for uses of this paper by de Broigle, I have all works by De Broigle and cannot find among them usage of Einstein's 1917 paper.I also have all works by Scrodinger , including Schrodinger's correspondence with Einstein, and again, I cannot find any trace of Einstein's 1917 paper impact on Schrodinger's work. 

Dear Arkay,

It was in my previous answer, here is the likn again:

http://lptms.u-psud.fr/nicolas_pavl...files/2010/03/Stone-phys_today1.pdf

Concerning the De Broglie and Schrodinger references, I'll send you the adequat informations. Here in France is still 5AM in the morning, I'll send you the references when I'll be at my office let us say 10AM.

Best as ever,

JK

http://lptms.u-psud.fr/nicolas_pavl...files/2010/03/Stone-phys_today1.pdf

Dear Arkady,

here is the link again, the link in my previous answer  does not work somehow. If you found some difficulties to download it I can send you a  copy by e-mail.

PS: Already in Stone paper you can find the two references:

L. de Broglie, PhD thesis, reprinted in Ann. Found. Louis de Broglie 17, 22 (1992).

 E. Schrödinger, Ann. Phys. (Leipzig)489, 79 (1926)

A suivre

Best,

JK

http://lptms.u-psud.fr/nicolas_pavloff/files/2010/03/Stone-phys_today1.pdf

A transition from level n+k to level n does not take place, at all, in the absence of an external field, in the Bohr model. The states are stationary states. Such a transition violates energy conservation and angular momentum conservation. So there's no point in asking what happens, when something can't occur-and doesn't occur.

Ilan,

You are right, what I described in my answer was what QM says. I didn't claim that Bohr's model says those things. I considered to make clear what QM says for allowing a good comparison with Bohr's model.

Joseph.Oh, yes, I do have Stone's paper for quite some time. I just forgot who wrote the  paper. I've got interested at the time  in who he is and looked at the math.sci.net for his records.Research Gate is not a good place to look for truly serious records. So, I found that he is not a serious mathematical physicist. The source which is adequate is the book by Maurice  de Gosson http://www.amazon.com/Principles-Newtonian-Quantum-Mechanics/dp/1860942741/ref=sr_1_3?ie=UTF8&qid=1449153703&sr=8-3&keywords=maurice+gosson  Surely, the book by Gutzviller is a very good source too http://www.amazon.com/Classical-Quantum-Mechanics-Interdisciplinary-Mathematics/dp/0387971734/ref=sr_1_3?ie=UTF8&qid=1449153837&sr=8-3&keywords=quantum+chaos

Stam,

Did you notice that Ilan asked what is wrong with Bohr's model? Why there is no point in asking what is wrong there? Do you believe that people shouldn't ask questions about things unclear to them? Why discourage questions? You are supposed to explain things (and no doubt, you offered an explanation).

Kind regards,

Sofia

My answer, that I explained in detail, is that the alleged problem in Bohr's model isn't there. But this known and the answer is in the question itself, since energy and angular momentum are expressed in terms of the principal quantum number. On the other hand, beyond the history of physics, there's simply no point in dwelling on Bohr's model as such, beyond stressing that it is fully *consistent* with the later formations of quantum mechanics, though incomplete. Bohr's postulates are now theorems as other ideas have been revealed as being of more general scope.

Dear Arkady,

Thanks for the references. In fact my self I'm  not a lecturer of Stone work per se, In fact I explore the cohomological aspect of the theory and the fact that there is a conditions of integrability hidden always behind the conservations laws, non linear equations or non linear physics and we should find the mathematical objects which verify those conditions of integrability. and the dynamics are a constraintes which should be verified by those mathematical objects.

Cheers,

Joseph

The hydrogen atom, not only as described by the Bohr model, but, also, by the later formulations of quantum mechanics, is an integrable system-this was noticed by Fock and, also, by Pauli.  It was, indeed, Fock's  work that proved that quantum effects led to a stable ground state for this system. 

Correct, Stam. Fock's  method of solution of Hydrogen atom  is absolutely amazing and has many far reaching consequences. http://math.umn.edu/~karl0163/docs/fock.pdf

Fock's solution and its generalizations are reviewed in two papers by Bander and Itzykson: http://inspirehep.net/record/48744 and http://inspirehep.net/record/48601

Sure, these are wonderful papers.I am using them in my own paper http://arxiv.org/abs/1006.4650, e.g. see ref.52, in  rather nontraditional context

Stam,

I can not see what is your point in stating:

"A transition from level n+k to level n does not take place, at all, in the absence of an external field, in the Bohr model."  Here n is the only quantum number in Bohr model and n+k is just a higher value. How does it not take place in Bohr model when Bohr postulates that there is transition  between them? In the QM model it may not take place in absence of EM field. But I'm not discussing QM at all but just Bohr model.

I think you mixing QM and Bohr model.

Of course Bohr hadn't known about the spin of photon s=1. But nevertheless he could notice that in his model light can have any angular momentum in different transitions.

The point is that there aren't any transitions in the Bohr model, in the absence of an external field-NONE are described and NONE are observed, experimentally, under such circumstances, either. The observation of such transitions requires placing hydrogen in an external field. Anything else is irrelevant to the topic. (The spin, mentioned in a previous message, was that of the electron, not the photon, so it's useful to read carefully.The conservation of angular momentum is that of the electron.)

The transition from the level n=5 to the level n=1 implies a change in principal quantum number by 4. This can be realized, *in the presence of an appropriate external field*, for instance, by the emissions 5->4, 4->3, 3->2, 2->1, i.e. by the emission of 4 photons, whose frequencies are predicted by the Bohr model-they're not all the same. 

Bohr's postulates from a physics course University of Oxford!!!!

• Quantized angular momentum: L = mvr = nhbar.

• Radiation is only emitted when an atom makes transitions between

stationary states: Eph = Em - En

What I state AGAIN is s(photon)=(m-n)hbar for that model of Bohr. Which is not physically right because s is equal to 1.

Stam what you writing has nothing to do with this.

No inconsistency there, since nowhere is it stated that there's one photon emitted. IF m-n = 1, THEN one photon is emitted. IF m-n>1 *more* than one  photons are emitted, with the appropriate energies, as described in the message just above. And, of course, these transitions are only possible in an external field-that's what the term ``stationary state'' means. It's not useful to cut and paste and appealing to authority is pointless. The content of Bohr's model is well known. 

The mistake lies in assuming that only one photon is emitted. The statement is that energy is absorbed or emitted in countable quantities, that can't be less than a given value-therefore, that, in any transition, at least one photon is emitted, whose energy is, necessarily, at least, equal to the difference of energies of consecutive levels. 

 The algebra is consistent with conservation laws. 

This is absolutely no true. From En to Em only one photon is emitted. This way Bohr explains the spectrum of Hg which is his achievement. This can not be true if the energy is carried away in the cascade mode you propose.

The point is that Bohr didn't knew that the photon can not has spin higher that 1. The value of spin was found 10-15 years later.

Hydrogen  (H) is different from mercury (Hg), which is, hopefully, a typo. The rest are, simply, factually, wrong statements in any case, since electron spin doesn't have anything to do with the subject-it is regrettable that available information (e.g. linked papers) is not taken into account, cf. the paper by Duncan and Janssen.

It's not surprising that the claim is made that the model is inconsistent, if an inconsistent premise is assumed. What's surprising is persisting, in spite of evidence, to assert something wrong. There is absolutely nothing that *prohibits* the emission of more photons when the transition from one level to a non-consecutive level occurs. What's not prohibited by a conservation law occurs-what doesn't occur is prohibited by a conservation law. And the Bohr model does express this, in this case. 

It might be useful to actually *read* Bohr's papers, easily available through search:

http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Bohr_1913.pdf

in particular the discussion around eq. (14) of the first paper;

 http://www.fisica.ufpb.br/~jgallas/CURSOS/Estrutura02/bohr_part02_PM1913_14786441308634993.pdf in some detail. 

Of course I meant Hydrogen H not Hg. So you are right that this was a typo.

Here is the citation from Bohr paper that clearly shows you are wrong:

''If we put t2 = 3, and let t1 vary, we get a series which includes 2 of the series observed by Fowler...''

Quoting half sentences is meaningless, as is the quote provided. In addition, first of all technical papers aren't Scripture and second it's the content that matters. So read the paper-the *whole* paper-and then *think*. It's, also, useful to keep in mind that any theory is quite independent of the persons that formulate it. So even if there were any errors in Bohr's papers (which there aren't regarding this issue), the consistency of the model can be described independently of what any particular person may say.

Of course there is a mistake. As you point out when the angular momentum is not conserved the transition can not happen.

Secondly everybody knows that Bohr wanted to account for the Rydberg formula. How the cascade you suppose is in the Bohr paper acount for this, Its obvious that you can not get the Rydberg formula with members where n-m is higher than 2.

Bohr eventually taught that the particles of light can take any angular momentum. Such transitions of course happen but l is not equal to n as he postulates.

The correct answer to this question was given by Alexander Merle in the first post of this thread. However, even after a follow-up post, his answers do not seem to have been understood/appreciated by all.

What is the angular momentum of the earth? It should be well known to all that this question has no unique answer. Angular momentum with respect to which point in space? We may take this point to be the center of the earth, in which case the answer may be referred to as the intrinsic spin of the earth. Or we may take the center of the sun, in which case we must add the orbital angular momentum due to earth's motion around the sun. I.e., the total angular momentum can be the sum of intrinsic spin and orbital angular momentum.

The same is true of photons: Although its intrinsic spin is 1, it can in addition have (essentially) any integer amount of orbital angular momentum about the center of the Bohr atom^*. Hence, as already indicated by Alex, the premises for the inconsistency are not correct.

As pointed out by Stam, a description of spontaneous emission of photons is not really possible within the Bohr model. Nevertheless, some description existed long before the construction of Quantum Mechanics proper. Cf. the introduction of the Einstein coefficients, which I believe happened around 1916.

^* Details of quantization of the electromagnetic field in spherical coordinates can f.i. be found in the first QFT volume of Landau and Lifshitz. It is related to the expansion of the classical field in vector spherical harmonics.

Ilian> The model of Bohr has indeed historical importance but is studied even at school and it is worth considering how it has to be taught.

It should not be taught at all (beyond anecdotic remarks)!

Dear Kare,

please look at selection rules for hydrogen atom. By example:

http://farside.ph.utexas.edu/teaching/qmech/Quantum/node121.html

equation 1149 in particular.

It is obvious that if orbital momentum l=n as it is postulated in Bohr model (not in QM though!!! where l=n-1,...,0), when there is some difference n-m>2 this selection rule is not applied.

In QM you really have transition  n->m  (with big difference) but l can be for example l=n-3 and easily fulfill the TOTAL angular momentum selection rule.

I believe you point that the photons emitted from an atom can have also intrinsic orbital moment. But that can not happen for a single atom.

Yes, in fact photons can possess orbital moment (not just spin associated with polarization) but under very different circumstances with use of spiral phase plates, spatial light modulators and q-plates forming a helix front. This was discovered by around year 2000 and has nothing in common with Bohr hydrogen atom.

Ilian> orbital momentum l=n

This is a red herring! Although the labeling of angular momentum states by a small L is natural, the choice of n for the main quantum number is essentially a convention: We could just as well have defined n=0 for the ground state (as is done for the harmonic oscillator), with En proportional to 1/(n+1)2 . 

Ilian> But that can not happen for a single atom.

You could flunk Quantum Mechanics with such a statement. Find a good book on intermediate quantum mechanics, and look up the section on multipole expansion of spontaneous emission. The probability of emission to higher order multipoles is small (hence perhaps of little practical interest), but not zero. 

Kare your points are just irrelevant.

l=n: in the Bohr model

No matter how redefine n this is a fact. Its very simple and doesn't need knowledge of QM. Just read the derivation of Bohr and you'll see it.

multipole expansion of spontaneous emission

Bohr theory is essentially about single isolated atom! So it is unnecessary and misleading  to look for higher order effects.

Bohr theory is very simple. You don't need to invoke QM or QFT here. Bohr just missed to check the total angular momentum conservation law. Or he thought that light can take the missing momentum. (as you suggested in the previous post). In fact it has been proven about 10-15 years later that light (photon in the case) carries only angular momentum - spin 1 under these simple circumstances. 

QM in contrary brings l=n-1,...,0. And matrix elements  are not zero for   hence angular momentum conservation law in QM is not a problem.

Ilian, I hope I'm not repeating what others have already mentioned. Bohr's model reted on hypotheses (postulates), and there are issues one could raise.

(1) The lowest energy state in the Bohr model has angular momentum \hbar. Quantum mechanically we know that the electron angular momentum is zero, however.

(2) Also Bohr knew, that classically the planetary model was bound to be somehow wrong, since it is unstable against radiation energy loss.

What the Bohr model delivered, was sort of an explanation of the Ritz combination principle (combining the Lyman, Paschen, Balmer etc. series, i.e. the (approximate) energies of light emission. Angular momentum conservation was not in the game. Since we know that the Bohr model is obsolete there is actually no point in trying to fiddle that in. When the magnetic splittings of spectral lines was observed it was immediately clear that things were not as simple as they looked in the first place and it took a while to get that "fixed"...

[As an anecdote: One reason why the Bohr model did have a chance to work "well" is the sheer fact that the electrostatic potential of point objects is 1/r - like. This is the reason why the states in each shell are all degenerate. Lots of physics would be very different, of course, if that were not the case. And the hydrogen spectrum would be much more complex...]

Kai, my point of view is, that it is out of question, to publish a paper where "Angular momentum conservation was not in the game." Well, Bohr somehow managed to pass between the drops. And he does it still. Apparently everyone (including Einstein, Dirac et.al.) were so blinded by this 'explanation'.

Bohr model is taught everywhere and nobody notices this apparent mistake. I also didn't noticed it when I learned it many years ago. I revisited the model recently and was surprised. But it has been retold (in the book) so many times that is it is already believed to be true, or at least close to truth. To try to alter this is like fighting windmills.