[Physics] Scattering states of Hydrogen atom in non-relativistic perturbation theory

hilbert-spacehydrogenperturbation-theoryquantum mechanicsscattering

In doing second order time-independent perturbation theory in non-relativistic quantum mechanics one has to calculate the overlap between states

$$E^{(2)}_n ~=~ \sum_{m \neq n}\frac{|\langle m | H' | n \rangle|^2}{E_n^{(0)}-E_m^{(0)}}$$

(where $E^{(k)}_n$ represents the kth order correction to the nth energy level).

For the Hydrogenic atom the spectrum of the Hamiltonian consists of a discrete set corresponding to "bound states" (the negative energy states) and a continuum of "scattering states" (the positive energy states).

Is there an example where the overlap between a bound state and the scattering states makes a measurable contribution to the energy in the perturbative regime? Should the scattering states be included in the perturbative calculations? Are there experimental results that backs this up? (For example, perhaps the electron-electron interaction in highly excited states of Helium).

As an aside, what "is" the Hilbert space of the Hydrogen atom in the position representation? I've often read the basis of eigenstates of the Hydrogen atom Hamiltonian isn't complete without the scattering states, but I've not seen any convincing argument of this. I've read that the radial bound states are dense in $L^2((0, \infty))$ (e.g. here), so including the scattering states the Hilbert space must strictly contain this.

Best Answer

The scattering states must be included in the perturbative calculations if the result is to be highly accurate. In particular, it is not justified to ignore the continuous spectrum at energies close to the dissociation threshold.

The Hilbert space in the position representation is the space of square integrable functions on $R^3\setminus\{0\}$ with respect to the inner product $$\langle\phi|\psi\rangle:=\int \frac{dx}{|x|}\phi(x)^*\psi(x).$$ The bound states alone are not dense in this space.

For thorough treatments of the hydrogen spectrum see the books
G R. Gilmore, Lie Groups, Lie Algebras & Some of Their Applications, Wiley 1974, Dover, 2002. pp. 427-430
A.O. Barut and R. Raczka, Theory of group representations and applications, 2nd. ed., Warzawa 1980. Chapter 12.2.

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