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While I was reading the derivation of Sommerfeld's model it was stated in the book that an electron moving in ellipse has Two degrees of freedom namely the radial distance $r$ and the azimuthal angle. I couldn't understand how these two are two different degrees of freedom. Does the radial distance not depend on the azimuthal angle? And finally the book said that these degrees of freedom must be quantized separately and further the following equations were written:

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Can anyone explain how these equations are obtained?

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The Bohr problem has just radius, this problem has two radii.

At the moment you decide the system is quantum mechanical, you cannot have a classical trajectory anymore. You cannot see or define a single revolution path anymore, rather you can describe it statistically - e.g. how much motion system has in separate axes. Every measurement trying to reconstruct the ellipse is limited by Heisenberg uncertainty, in that sense there are two degrees of freedom.

  • you strictly stick to the ellipse (you use just one degree of freedom) it will surely not reflect the quantum mechanical motion, but represent some artificial input condition - and you will go the Bohr direction with just one quantum number and probably some like-deformation parameter described by excentricity $\epsilon$. Or you should find a way to quantize $\epsilon$.

Did you check http://uw.physics.wisc.edu/~knutson/phy448/wilson-sommerfeld.pdf ? They speak about the rule that the action integral of a generalized coordinate and its conjugate momentum taken over one cycle of motion is quantized (if cyclic motion).

Equation of motion is from 4.6 - 4.8. where they get the equations you show.

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