Bohr's quantization of angular momentum

Question

I cannot seem to find a derivation for $L=\frac{nh}{2\pi}$ I do not understand what led Bohr to quantize angular momentum in units of Planck's constant and how he was sure it works. I understand that the motivation was because in classical mechanics the electron would crash into the atom due to it emitting electromagnetic radiation, but again I do not see how quantizing solves this issue and why quantizing in units of h is the correct way to do it. It seems that in every book I read, the equation is just given with some verbal justification rather than a rigorous mathematical one. We are currently using Thornton, S. T., & Rex, A. F. (2006). Modern physics for scientists and engineers. Belmont, CA: Thomson, Brooks/Cole. and it really does not give the best motivation for this quantization.

Any help would be appreciated.

Answer 1

the equation is just given with some verbal justification rather than a rigorous mathematical one.

Think: can there be a rigorous proof of Newton's laws of mechanics?

Why not?

Because they are extra "axioms" particularly developed in order to pick up from the mathematical solutions to the differential equations, those that fit the data.

Physics is about modeling observations and data mathematically. Unless extra axioms, called "laws", "principles" "postulates" .... are used, there is no way to connect abstract mathematical equations to numbers measured.

It is an associative process, how models in physics develop. Bohr had to fit the spectra of atoms, which were completely incomprehensible, and he knew of the photoelectric effect, that implied specific energies for getting electrons our of surfaces, and of the black body radiation formula that would only fit the data if one postulated quantization of the photon energy ( that is where the $h$ comes from).

It was a brilliant guess to set $L=\frac{nh}{2\pi}$ , a fixed orbit, as an "axiom"and when the spectrum series came out as a solution, it was a confirmation of the guess.

Now the model has been superseded by the theory of quantum mechanics, and in this theory where the "axioms" are the postulates of quantum mechanics, this $L=\frac{nh}{2\pi}$ comes out from the mathematics of formal quantum mechanics, and can be considered as "proof" of existence. This does not diminish the brilliance of the hypothesis set up in the Bohr model, which is still used as an approximation to quantum mechanical calculations.

Answer 2

I concur with @anna_v that it was a brilliant guess. What was known at the time, in a nutshell, was the concept of quantum of energy (Planck) and "quantum of radiation" (Einstein), which are rooted in thermodynamics. Also, that atoms only emit and absorb at certain discrete frequencies (Lyman, Balmer, and other "spectroscopists"...)

The brilliant step by Bohr (highly praised by Einstein in his autobiographical notes) was to contemplate that, when in certain orbits, electrons are forbidden to radiate even though they are subject to acceleration. A precursor of this idea seems to have been implied --though not explicitly formulated-- by Paul Langevin when trying to explain "permanent magnetism."

Just to complement the excellent explanation above, and because you seem to be interested in bibliographic sources, the best one I know you can find in:

The Conceptual Development of Quantum Mechanics,

by Max Jammer. (Chapters 1 & 2)

Answer 3

I find the discussion of the Bohr atom by Michael Fowler very interesting.

The following is not meant as an answer, it is a description of Michael Fowler's approach.

It appears Michael Fowler has made use of historical documentation extensively. In two places in his discussion Micheal Fowler refers to written communication from Bohr to Rutherford. The discussion is quantitative; the concepts are discussed, and then expressed in formula.

The development of the ideas of Bohr is described as a two stage process.

The first stage was Bohr trying to find a viable model for a stable state of a multi-electron atom, on the assumption that the model must involve Planck's constant.

Quote:
"Planck's constant plays a role in restricting allowed orbital changes in the oscillators in black body radiation—and these oscillators, although not very clearly understood, were of the same general size as atoms."

During this stage Bohr was guided by experimental data of atoms being bombarded with high velocity electrons, and giving of characteristic x-rays. The data appeared to corroboratre an expected correlation between electron speed (in the lowest energy state), and the total charge of the nucleus.

Quote:
"In February 1913, Bohr was surprised to find out in a casual conversation with the spectroscopist H. R. Hansen that some patterns had been discerned in the apparent chaos of spectral lines. In particular, Hansen showed him Balmer's formula for hydrogen."

Balmer's formula had been discovered earlier, but Bohr was unaware of that.

Bohr said later:
"As soon as I saw Balmer's formula, the whole thing was immediately clear to me."

Bohr realized that the existence of a series in the spectrum shows that for Hydrogen, with its single electron there isn't just a single stable ground state, there is a sequence of stationary states.

Cautionary remark:
As so often happens when a physicist does history of physics, probably this narrative has been much streamlined. No doubt there were many blind alleys, and only the succesful path is retained.

Still, we do have that quote from Bohr himself on his realization when he learned about the Balmer series. Clearly there was a big jump in his confidence level that he was on the right track.

Answer 4

I think Bohr did not make the "brilliant guess" of $L=\frac{nh}{2\pi}$. Instead, his guess was that there were multiple stable states, each has a fixed energy level; and the difference between two of the energy levels gives a spectrum line (a guess that would come directly and naturally from Balmer's formula), whose frequency was related to the energy difference by Planck's frequency-energy relationship $\Delta E=h\nu$ (a readily available relationship at that time). The quantization of angular momentum was then derived.

It is like when we work on a math problem, we often work from the result, backwards, to the conditions given. Then, when we write down the answer, we write from the conditions, forwardly, to the result, as if we know how to solve the problem the first time we see it.

With the same spirit, we can see that De Broglie did not make the "brilliant guess" of mass wave equation $\lambda=\frac{h}{p}$ and end up with being confirmed by its consistency with Bohr's orbits; but rather he tried to fit mass wavelength to Bohr's orbits and end up with the equation we have just mentioned.