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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.

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your reference in the last sentence doesn't mention hydrogen. –  Arnold Neumaier Nov 22 '12 at 14:30
ya...absolutely. –  Curious Nov 22 '12 at 14:44
You're right Arnold Neumaier, I made an unjustified jump. It does show explicitly the bound states are dense in $L^2(0, \infty)$. I've corrected the question. –  Edward Ross Nov 22 '12 at 22:55

1 Answer 1

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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Thanks, that makes sense to me. The bound states are complete on the subspace of square integrable functions on $R^3 \setminus \{0\}$ with respect to the inner product $\langle\phi|\psi\rangle:=\int dx\phi(x)^*\psi(x)$ and the scattering states are the orthogonal complement of this subspace in the Hilbert space you mentioned. Though I'm surprised it's $\frac{dx}{|x|}$ and not $\frac{dx}{|x|^2}$. –  Edward Ross Nov 22 '12 at 22:49
The power of the denominator is really 1 and not 2. This singular inner product is relevant to get a unitary representation of the dynamical symmetry group $SO(4,2)$ of hydrogen. –  Arnold Neumaier Nov 23 '12 at 10:57

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