How did Sommerfeld arrive at his quantum condition?

[Half Infinity]

Wherever I read, they just simply state the Bohr-Sommerfeld or the Wilson-Sommerfeld quantum condition to be

integral (p.dx) over closed path = nh

But nowhere can I find the insight using which Sommerfeld arrived at this equation. I mean, he could'nt have just guessed that. There must have been some logical thought process for it. What was that?

 

[dextercioby]

I think the major breakthrough was Bohr's motion quantization rule of 1913. Sommerfeld was just a professor of classical mechanics who took the next logical steps after Bohr (elipsis instead of circles, special relativity and not Galilean relativity) and added "motion invariants" to it. I mean I don't want to belittle his work, but if Bohr had not described the atom 2 years before him, it would be unlikely to think he would have produced his work.

 

[Half Infinity]

What are "motion invariants"?

How exactly do L = nh/2π and integral(p.dx) = nh connect? I know the way de Broglie explained it, but that was much later. How did Sommerfeld connect those?

And at last, how did Bohr himself got his quantum condition? Wikipedia states that William Nicholson was the person who first introduced quantization of angular momentum. How did then he arrive at it? And why do we connect Bohr's name to it instead of Nicholson?

 

[kurt101]

Wherever I read, they just simply state the Bohr-Sommerfeld or the Wilson-Sommerfeld quantum condition to be

integral (p.dx) over closed path = nh

But nowhere can I find the insight using which Sommerfeld arrived at this equation. I mean, he could'nt have just guessed that. There must have been some logical thought process for it. What was that?

E/(frequency)=nh is the area of an ellipse that describes a phase plane for harmonic motion of an object with mass like an electron. It is my understanding that Sommerfeld noticed this correspondence. He started with the equation for harmonic motion E = (p^2) / (2 * m) + (k / 2) q^2 (where p is momentum and q is position) and rearranged it in a form that looks like the equation for an ellipse 1=(x^2)/(a^2) + (y^2)/(b^2). Then he pulled out a and b from this equation and plugged them into the equation for the area of ellipse area=pi*a*b and that can be shown to be equal to nh.

I am paraphrasing this explanation from the book "what is quantum mechanics a physics adventure" which goes through the math and explains it in a simple straight forward manner.

 

[Half Infinity]

I think the major breakthrough was Bohr's motion quantization rule of 1913. Sommerfeld was just a professor of classical mechanics who took the next logical steps after Bohr (elipsis instead of circles, special relativity and not Galilean relativity) and added "motion invariants" to it. I mean I don't want to belittle his work, but if Bohr had not described the atom 2 years before him, it would be unlikely to think he would have produced his work.

What are "motion invariants"?

How exactly do L = nh/2π and integral(p.dx) = nh connect? I know the way de Broglie explained it, but that was much later. How did Sommerfeld connect those?

And at last, how did Bohr himself got his quantum condition? Wikipedia states that William Nicholson was the person who first introduced quantization of angular momentum. How did then he arrive at it? And why do we connect Bohr's name to it instead of Nicholson?

 

[vanhees71]

I think the heuristics behind this is the Liouville equation, i.e., that the invariants of motion are phase-space volumes. Then there was the example of the harmonic oscillator (originally in Planck's description of the em. field and Einstein's subsequent reinterpretation as "light quanta", i.e., the description of the free em. field as a collection of independent harmonic oscillators).

The harmonic-oscillator example was then postulated to generalize to all kinds of (quasi-)periodic motions.

 

[gentzen]

And at last, how did Bohr himself got his quantum condition? Wikipedia states that William Nicholson was the person who first introduced quantization of angular momentum. How did then he arrive at it? And why do we connect Bohr's name to it instead of Nicholson?

The current German version of his wikipedia entry puts the facts in a plausible order:

1911 schlug er – unabhängig von Ernest Rutherford und anderen – einen dem Planetensystem ähnliches Modell des Atoms vor, mit dem Atomkern im Zentrum. Sein Modell hatte aber auch noch Elemente des Atommodells von J. J. Thomson, bei dem die positive Ladung über das Atom verteilt war und die Elektronen darin verteilt. ...

Nach Eric Scerri war er der Erste, der die Quantisierung des Drehimpulses (in Einheiten der reduzierten Planck-Konstante) vorschlug, in diesem Fall von Elektronen in Atomen. Das beeinflusste wahrscheinlich auch Niels Bohr in der Entwicklung seines eigenen Modells. Bohr betrachtete die Theorie von Nicholson einerseits als ziemlich verrückt, benutzte sie aber auch als ein Modell von dem er sich mit seiner eigenen Theorie absetzte.[3]

3. ↑ Scerri,A tale of seven scientists, 2016, S. 34

In 1911, independently of Ernest Rutherford and others, he proposed a model of the atom similar to the planetary system, with the atomic nucleus at the center. But his model also had elements of J. J. Thomson's atom model, in which the positive charge was distributed over the atom and distributed the electrons within it. ...

According to Eric Scerri, he was the first to propose the quantization of angular momentum (in units of the reduced Planck constant), in this case of electrons in atoms. This probably also influenced Niels Bohr in the development of his own model. On the one hand, Bohr viewed Nicholson's theory as pretty crazy, but he also used it as a model from which he set himself apart with his own theory.

The English version of his wikipedia entry does not contradict that version, but still gives the story quite a different spin:

Nicholson is noted as the first to create an atomic model that quantized angular momentum as h/2π.[2][3] Nicholson was also the first to create a nuclear and quantum theory that explains spectral line radiation as electrons descend toward the nucleus, identifying hitherto unknown solar and nebular spectral lines.[4][5] Niels Bohr quoted him in his 1913 paper of the Bohr model of the atom.[6]

Career​

Based on the results of astronomical spectroscopy of nebula he proposed in 1911 the existence of several yet undiscovered elements. ...

One thing I did notice is that his motivation was interpretation of astronomical data, instead of better controlled laboratory data. Apparently this made it easier to come up with completely new ideas with a grain of truth, but harder to come up with a sufficiently plausible model that would be accepted for some time, and still remembered later.