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I have found the following interesting article: http://arxiv.org/abs/0706.0924

The authors examine the radial momentum operator in detail, in particular its time evolution due to the forces acting on the electron in the Hydrogen atom. Their key result is equations (31), (32) and (33). Here they derive explicit expressions for the centripetal force (resulting from the Coulomb interaction between electron and nucleus) and the centrifugal force (which appears to be the result of conservation of angular momentum).

The authors then demonstrate that, upon averaging, the effects of the centripetal force and the centrifugal force become identical. Hence on average there is no force, and therefore the radial momentum remains constant -- precisely as expected for a time-independent state.

Unfortunately the authors do not comment on the special case of the Hydrogen ground state (where $N=1$ and $L=0$), in which case there is no angular momentum. So how can there be a centrifugal force associated with it?

The formulas by the authors are in terms of a numerator and a denominator, both of which become zero for $L=0$. The quotient of numerator and denominator is tacitly assumed to yield the same (constant) value as in the more general case $L > 0$.

My questions: Is there indeed a centrifugal force acting on the electron in the Hydrogen atom, which on average balances the centripetal Coulomb force? If so, how can we understand this force in the case of $L=0$, given that the only obvious source (= angular momentum) equals zero in this particular case?

Qmechanic
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M. Wind
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2 Answers2

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The "rotating particle" picture does not work for $L=0$, unfortunately. Both the electron and the nucleus are waves who create a standing spherically symmetric wave of their relative motion.

Vladimir Kalitvianski
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For an intuitive description, let's go to a semi-classical picture of the electron movement in the atom.

I order to have an angular momentum, the electron should rotate around an axis of symmetry. Then, if in the plane perpendicular to the axis of symmetry the electron rotates, say, clockwise, the angular momentum will point along that symmetry axis. But, what to do, the electron in the atom is a non-standard creature. The level n = 1 has SPHERICAL SYMMETRY, i.e. ANY axis in the space is a symmetry axis for this level. The electron rotates on a circle around whatever axis you pick. So, in which direction can point the angular momentum?

However, with the centrifugal force the situation is better. On whichever circle rotates the electron, the centrifugal force points outwards, compensating the nuclear attraction.

I hope that this helps,

Sofia

Sofia
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