Calculating transition matrix elements can be difficult, and I have found myself needing to use hydrogenic electric dipole transition matrix elements a fair bit.


Is there any online source that gives the nontrivial matrix elements above (for hydrogen obviously)?

  • 1
    $\begingroup$ I never worked my way through Salpeter and Bethe's book on one- and two-electron atoms, but that might may give some pointers. $\endgroup$
    – Jon Custer
    Oct 5, 2016 at 12:34
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    $\begingroup$ There is a list of spectroscopic databases from the NIST, including a database for many types of atoms that includes many (though not all) transition probabilities. $\endgroup$ Oct 6, 2016 at 4:07
  • $\begingroup$ This answer provides an analytic formula for the radial matrix elements of $r^k$. You can derive your matrix elements from this. $\endgroup$
    – Ghoster
    Jan 30 at 6:01

1 Answer 1


As a partial answer, I took the advice of @JonCuster and pulled the following equations from "Quantum Mechanics of One- & Two- Electron Atoms" by Bethe and Salpeter. All results are in C.G.S units.

The sum of dipole transition intensities for initial state $(n,l,m)$ and final states $(n',l+1,m')$, where $m'$ runs over the three allowed values ($m'=m-1,m,m+1$), is given by


Similarly, we have for final states with $l'=l-1$,


Here, the symbol $R_{n,l}^{n',l'}$ represents the radial overlap between hydrogenic states $(n,l)$ and $(n',l')$. A consequence of the above equations is that the lifetime of a state does not depend on the magnetic quantum number $m$.

The closed-form expressions for the radial overlap matrix elements in their full generality are highly complicated. Quoted once again from Bethe and Salpeter (who themselves quote W. Gordon), for $n'\neq n$,

$$\begin{align*} R_{nl}^{n',l-1}&=\frac{(-1)^{n'-1}}{4(2l-1)}!\sqrt{\frac{(n+l)!(n'+l-1)!}{(n-l-1)!(n'-1)!}}\frac{(4nn')^{l+1}(n-n')^{n+n'-2l-2}}{(n+n')^{n+n'}}\times\\ &\times\left\{F\left(-n_r,-n_r',2l,-\frac{4nn'}{(n-n')^2}\right)-\left(\frac{n-n'}{n+n'}\right)^2 F\left(-n_r,-2,-n_r',2l,-\frac{4nn'}{(n-n')^2}\right)\right\} \end{align*}$$

where $F(\alpha,\beta,\gamma,x)$ is the hypergeometric function, $n_r=n-l-1$ and $n_r'=n'-l$ are the radial quantum numbers for the two states.

EDIT: I could almost swear that I stumbled upon a random website in which you could type in whatever transition matrix element you wanted and it would spit it out (presumably from a pre-calculated table). It had an enormous amount of dipole and quadrupole transition matrix elements (for sure electric, maybe magnetic?). I really don't think I was dreaming when I came across it, but now I'm not too sure since I can't find it anywhere. I think it may have been associated with the authors of this paper, but I'm not too sure.

[1] "Quantum Mechanics of One- & Two- Electron Atoms" by Bethe and Salpeter

[2] (German) W. Gordon: Ann. d. Pys. (5) 2, 1031 (1929) - radial overlap integrals are calculated here. They are of course expressed in units of the bohr radius.

  • $\begingroup$ This answer provides an analytic formula for the radial matrix elements of $r^k$. The matrix elements in this question can be derived from this. $\endgroup$
    – Ghoster
    Jan 30 at 6:02

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