Shaking the Atom: Electron Shells & Nucleus Resonance

[warabi]

The nucleus has positive charge, with negative electron shell(s) around it. The nucleus is located in the center of the electron shell. Electrical forces keep it there.

Now what if we shine electromagnetic radiation (linearly polarised) onto an atom? With the E-field going "up", will the nucleus have a tendency to move "up", with the electron shell wanting to move "down"? Will there be a resonant frequency in which a relatively large amount of electromagnetic energy is absorbed from the incoming radiation, and the relative movement between nucleus and shell is maximised? In other words could there be movement of the electron shell "as a whole", not e.g. excitation of individual electrons, and could it be observable?

 

[Dickfore]

A two-body problem is equivalent to a problem of a particle with reduced mass:

$$\frac{1}{m} = \frac{1}{m_{1}} + \frac{1}{m_{2}}$$

in the center-of-mass frame. For hydrogen atom, this would be the Coulomb potential. Due to conservation of angular momentum $\mathbf{L}$, the motion may be further reduced to the radial direction (one-dimensional) in an effective potential energy:

$$V_{l}(r) = \frac{L^{2}}{2 m r^{2}} - \frac{k_{0} e^{2}}{r}$$

This "effective" potential energy has a local minimum:

$$V_{l}'(r) = -\frac{L^{2}}{m r^{3}} + \frac{k_{0} e^{2}}{r^{2}} = 0$$

$$r_{0} = \frac{L^{2}}{m k_{0} e^{2}}$$

$$V_{l}''(r) = \frac{3 L^{2}}{m r^{4}} - \frac{2 k_{0} e^{2}}{r^{3}} = \frac{m^{3} (k_{0} e^{2})^{4}}{L^{6}} \equiv k > 0$$

Around this local minimum, the effective potential energy may be expanded into Taylor series:

$$V_{l}(r) = V_{l}(r_{0}) + \frac{1}{2} k (r - r_{0})^{2}$$

i.e. it looks like a harmonic oscillator. The oscillation frequency is:

$$\omega = \sqrt{\frac{k}{m}} = \frac{m (k_{0} e^{2})^{2}}{L^{3}}$$

If one uses $L \sim \hbar$, one gets that this oscillation frequency is the same as the transition frequency emitted or absorbed in the Bohr model, up to numerical factors. So, what you had described is actually the classical picture of atomic transitions. The only important feature we had neglected is that the photon carries angular momentum, so the angular momentum of the atom during such transitions is not conserved.