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.

  • $\begingroup$ Most QM books go into the spectrum of the hydrogen atom. $\endgroup$ – Dargscisyhp Aug 4 '15 at 2:17
  • $\begingroup$ @Dargscisyhp But they only go into the discrete spectrum I observe. When I say "spectrum decomposition", I mean the continuous/integral part. $\endgroup$ – tqw Aug 4 '15 at 2:45
  • $\begingroup$ Something like this 1926 paper? $\endgroup$ – Kyle Kanos Aug 4 '15 at 3:17
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    $\begingroup$ 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. $\endgroup$ – Ján Lalinský Aug 4 '15 at 17:15
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    $\begingroup$ @JánLalinský Thanks. The exact formulas are presented in §135 of the book. $\endgroup$ – tqw Aug 4 '15 at 17:27

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;


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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