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I wonder if there is a nice treatment of the continuous spectrum of hydrogen atom in the physics literature--showing how the spectrum decomposition looks and how to derive it.

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tqw
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  • Most QM books go into the spectrum of the hydrogen atom. – Dargscisyhp Aug 04 '15 at 02:17
  • @Dargscisyhp But they only go into the discrete spectrum I observe. When I say "spectrum decomposition", I mean the continuous/integral part. – tqw Aug 04 '15 at 02:45
  • Something like this 1926 paper? – Kyle Kanos Aug 04 '15 at 03:17
  • @KyleKanos Thanks, but there are very few formulas in that paper unfortunately... Plus, there's no spectral expansion that I want. I guess there should be a nice writing of this somewhere. – tqw Aug 04 '15 at 03:54
  • What kind of "spectral expansion" do you seek? The continuous spectrum of the hydrogen atom is essentially made up by plane waves, and actual states are then wavepackets formed from them, but no actual eigen"states" of the continuous spectrum belong to the space of states, see e.g. this question. – ACuriousMind Aug 04 '15 at 13:32
  • Did you look at the citations to the paper? Or the references in that paper? – Kyle Kanos Aug 04 '15 at 13:32
  • @ACuriousMind The spectral expansion I'm seeking is like the one for free particle. I know the "eigenfunctions" for the continuous spectrum do not lie in $L^2$, and that's why I said "integral part". It should be certain integral transform involving special functions, and I want to know the explicit formulas. – tqw Aug 04 '15 at 14:20
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    Some discussion of eigenfunctions for continuous spectral values is in the book Landau, Lifshitz, Quantum Mechanics: Non-relativistic Theory, §37. Motion in a Coulomb field. – Ján Lalinský Aug 04 '15 at 17:15
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    @JánLalinský Thanks. The exact formulas are presented in §135 of the book. – tqw Aug 04 '15 at 17:27
  • @Z.F. Please consider answering your own question based on the above comments. – Danu Aug 15 '15 at 15:01

1 Answers1

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The term to look for is Coulomb wave. These wavefunctions are well explained in the corresponding Wikipedia article.

Depending on your mathematical background, you should be ready for a bit of a formula jolt, as these wavefunctions rely very intimately on the confluent hypergeometric function. If you want the short of it, then I can tell you that the solutions $\psi_\mathbf k^{(\pm)}(\mathbf r)$ to the continuum hydrogenic Schrödinger equation $$ \left(-\frac12\nabla^2+\frac Zr\right)\psi_\mathbf k^{(\pm)}(\mathbf r)=\frac12 k^2\psi_\mathbf k^{(\pm)}(\mathbf r) $$ with asymptotic behaviour $$ \psi_\mathbf k^{(\pm)}(\mathbf r)\approx \frac{1}{(2\pi)^{3/2}}e^{i\mathbf k·\mathbf r} \quad\text{as }\mathbf k·\mathbf r\to\mp \infty $$ are $$ \psi_\mathbf k^{(\pm)}(\mathbf r) = \frac{1}{(2\pi)^{3/2}} \Gamma(1\pm iZ/k)e^{-\pi Z/2k} e^{i\mathbf k·\mathbf r} {}_1F_1(\mp iZ/k;1;\pm i kr-i\mathbf k·\mathbf r) .$$

You can also ask for solutions with definite angular momentum (which do exist for any $m$ and $l\geq|m|$); those are detailed in the partial wave expansion section of the Wikipedia article. If you want textbooks which develop these solutions, look at

L. D. Faddeev and O. A. Yakubovskii, Lectures on quantum mechanics for mathematics students. American Mathematical Society, 2009;

and

L. A. Takhtajan, Quantum mechanics for Mathematicians, American Mathematical Society, 2008.

Hat-tip to Anatoly Kochubei for providing these references in an answer to my MathOverflow question Is zero a hydrogen eigenvalue?

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Emilio Pisanty
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