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According to Sommerfeld's atomic model, an electron moving around a central positively charged nucleus is influenced by the nuclear charge. As a result of which, the electron moves in an elliptical path with the nucleus situated at one of the foci. But how the path of electron becomes circular to elliptical? I've trouble understanding this.

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  • $\begingroup$ Well, the model assumed that the area of the ellipse drawn by the orbit of the electron was a quantum property and an integer. While the model did treat the electron-nucleus interaction as a harmonic oscillator, it didn't really model the various forces involved in shaping the orbit - rather reasonably, given that it was known that such motion would result in the electron death spiral anyway. Sommerfeld's electron was known not to behave the way free electrons did - the theory is incomplete and not self-consistent. $\endgroup$ – Luaan Oct 3 '14 at 8:15
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The Sommerfeld model, and the Bohr model from which it is derived, are toy models developed in an attempt to describe spectral lines in the era before modern quantum mechanics. You might be interested to look at the question Is it possible to recover the old Bohr-Sommerfeld model from the QM description of the atom by turning off some parameters? for more on this.

Electrons do not orbit atoms like planets orbiting a star, so to ask whether their trajectory is circular or elliptical is a meaningless question.

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    $\begingroup$ So what does it mean when they say that atoms near a magnetar are squeezed into long thin cylinders? Wouldn't that imply an elliptical shape? (Not an orbit exactly I know, but still an ellipse rather than a circle.) $\endgroup$ – Ariel Oct 3 '14 at 0:40
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    $\begingroup$ Why is it meaningless to ask "in this, now outdated, model the path was described as an ellipse. How, in this outdated model, would one explain a tranisition from circular to elliptical?". That doesn't strike me as meaningless, it's just a thought experiment that doesn't happen to apply to the world as we know it. $\endgroup$ – GreenAsJade Oct 3 '14 at 1:10
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Further to John Rennie's answer, which I wholeheartedly agree with (insofar that the planetary orbit picture is outdated by nearly a century), you might like find the following interesting. If the electron really were like a planetary orbit, then it would generally be an ellipse rather than a circle. The force field has the same functional form as the gravitationl problem: both feature bodies moving in force field directed at one unmoving point (assuming the Sun or the nucleus to be so massive that its movement can be neglected) and the force dwindles with the square of the distances in both cases. If a body is gravitationally (or electrostatically) bound, that is, its kinetic energy is less than that needed to raise its potential energy to the that at a point infinitely far away from the centre, then its path must be an ellipse and a circle is simply one kind of ellipse. A cirular orbit only follows from the very special initial condition that the speed at periapsis (perihelion) is precisely that which makes the gravitational acceleration owing to the Sun precisely equal to the centrepetal force needed to maintain a circular orbit; if otherwise, the orbit is an ellipse.

There is an amusing piece of history that sheds a poignant light on Isaac Newton's "lone wolf" personality: Christopher Wren, Edmund Halley and Robert Hooke were wrestling with problem of how to find what the shape of an orbit in an inverse square, centrally directed force field would be. Christopher Wren had offered a cash prize for the mathematical characterisation and a proof. The scientific world was abuzz with activity trying to answer this question, and Newton had probably known the answer for years, but chose to sit on his discovery. Finally, when Halley visited Newton (one version of this story that I have read had Halley having to bring special white rose plants and carefully chosen wine to coax the irascible, grumpy, cranky genius into even holding a conversation) and asked Newton what he thought the path would be, Newton quipped, utterly without hesitation and with unshaking certainty, that it would be an ellipse. Halley was stunned, and asked Newton how on Earth did he know. Again quipped Newton: "Why, I have calculated it, of course!". Halley asked for a proof. Newton answered with a bit more than that, and it was this very request that prompted Newton to write and publish his famous Principia (at Halley's expense!). Newton would not have published when he did would this meeting not have happened.

See here for one version of the story.

Newton claimed that the inverse square law was "obvious" from his Principia, but in fact this is far from true, at least for non Newtonian mortals like me. The Feynman Lost Lecture (see also the Feynman-biographical Feynman's Lost Lecture: The Motion of Planets around the Sun) was Feynman's attempt to make Newton's assertion obvious, partially by reviewing and "translating" the relevant parts Principia into modern geometric ideas: Feynman found the Principia hard going, for geometric knowledge that was second nature in Newton's time has sunken into obscurity, thus demonstrating how starkly and swiftly ways of thinking about problems shifts with time.

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Considering the way matter waves are associated with all moving particles, it seems inconceivable to me that electrons cannot move in other than elliptical orbits. Close examination of the harmonics & resonance effects of phase waves & matter waves, it becomes apparent that they need to move in elliptical orbits in order to harmonize the way they do. Group waves it should be remembered are made up of many thousands if not millions of phase waves, these too have to harmonize, unless electrons travel in elliptical orbits, they cannot do that.

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  • $\begingroup$ Can you please provide an explanation, or atleast summarize. $\endgroup$ – Cicero Jun 10 '15 at 23:12

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