# Table of matrix elements of powers of $r$ for radial functions in $H$ atom

Im looking for some references here. I hope it is the right place to ask.

I need to find a table of (or a formula from which to extrapolate) the matrix elements of the radial functions of the hydrogen atom evaluated in powers of $$r$$ for both diagonal and off-diagonal elements. That is, I need this:

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I found this article http://dx.doi.org/10.1088/0953-4075/28/3/007 which is great, but Im looking for something more textbook-like, say suited for undergrad. I especially need positive powers until $$k=4$$.

Any help is very appreciated.

• I do not think such a table exists. You can probably just hack the integral into Mathematica and retrieve the results. May 14, 2013 at 12:40

I haven't been able to find either tables of these matrix elements or a formula online, so I decided to provide them here myself.

The following tables give the radial matrix elements of $$r^k$$ for $$k$$ from -2 to 4, and for the six lowest-energy states (having $$n$$ from 1 to 3). The rows and columns are labeled by $$nl$$. The units are such that the Bohr radius $$a_0$$ is 1.

$$\begin{array}{c|cccccc} r^{-2} & 10 & 20 & 21 & 30 & 31 & 32 \\ \hline 10 & 2 & \frac{4 \sqrt{2}}{9} & \frac{2 \sqrt{\frac{2}{3}}}{9} & \frac{7}{12 \sqrt{3}} & \frac{1}{4 \sqrt{6}} & \frac{1}{12 \sqrt{30}} \\ 20 & \frac{4 \sqrt{2}}{9} & \frac{1}{4} & 0 & \frac{292 \sqrt{\frac{2}{3}}}{1875} & \frac{16}{625 \sqrt{3}} & -\frac{128}{1875 \sqrt{15}} \\ 21 & \frac{2 \sqrt{\frac{2}{3}}}{9} & 0 & \frac{1}{12} & \frac{2 \sqrt{2}}{625} & \frac{64}{1875} & \frac{32}{625 \sqrt{5}} \\ 30 & \frac{7}{12 \sqrt{3}} & \frac{292 \sqrt{\frac{2}{3}}}{1875} & \frac{2 \sqrt{2}}{625} & \frac{2}{27} & 0 & 0 \\ 31 & \frac{1}{4 \sqrt{6}} & \frac{16}{625 \sqrt{3}} & \frac{64}{1875} & 0 & \frac{2}{81} & 0 \\ 32 & \frac{1}{12 \sqrt{30}} & -\frac{128}{1875 \sqrt{15}} & \frac{32}{625 \sqrt{5}} & 0 & 0 & \frac{2}{135} \\ \end{array}$$

$$\begin{array}{c|cccccc} r^{-1} & 10 & 20 & 21 & 30 & 31 & 32 \\ \hline 10 & 1 & \frac{4 \sqrt{2}}{27} & \frac{8 \sqrt{\frac{2}{3}}}{27} & \frac{\sqrt{3}}{16} & \frac{5}{16 \sqrt{6}} & \frac{\sqrt{\frac{3}{10}}}{16} \\ 20 & \frac{4 \sqrt{2}}{27} & \frac{1}{4} & -\frac{1}{4 \sqrt{3}} & \frac{92 \sqrt{6}}{3125} & \frac{64}{3125 \sqrt{3}} & -\frac{448 \sqrt{\frac{3}{5}}}{3125} \\ 21 & \frac{8 \sqrt{\frac{2}{3}}}{27} & -\frac{1}{4 \sqrt{3}} & \frac{1}{4} & -\frac{72 \sqrt{2}}{3125} & \frac{192}{3125} & \frac{768}{3125 \sqrt{5}} \\ 30 & \frac{\sqrt{3}}{16} & \frac{92 \sqrt{6}}{3125} & -\frac{72 \sqrt{2}}{3125} & \frac{1}{9} & -\frac{1}{9 \sqrt{2}} & \frac{1}{9 \sqrt{10}} \\ 31 & \frac{5}{16 \sqrt{6}} & \frac{64}{3125 \sqrt{3}} & \frac{192}{3125} & -\frac{1}{9 \sqrt{2}} & \frac{1}{9} & -\frac{1}{9 \sqrt{5}} \\ 32 & \frac{\sqrt{\frac{3}{10}}}{16} & -\frac{448 \sqrt{\frac{3}{5}}}{3125} & \frac{768}{3125 \sqrt{5}} & \frac{1}{9 \sqrt{10}} & -\frac{1}{9 \sqrt{5}} & \frac{1}{9} \\ \end{array}$$

$$\begin{array}{c|cccccc} r^0 & 10 & 20 & 21 & 30 & 31 & 32 \\ \hline 10 & 1 & 0 & \frac{16 \sqrt{\frac{2}{3}}}{27} & 0 & \frac{3 \sqrt{\frac{3}{2}}}{16} & \frac{3 \sqrt{\frac{3}{10}}}{16} \\ 20 & 0 & 1 & -\frac{\sqrt{3}}{2} & 0 & \frac{384 \sqrt{3}}{3125} & -\frac{3072 \sqrt{\frac{3}{5}}}{3125} \\ 21 & \frac{16 \sqrt{\frac{2}{3}}}{27} & -\frac{\sqrt{3}}{2} & 1 & -\frac{144 \sqrt{2}}{3125} & 0 & \frac{4608}{3125 \sqrt{5}} \\ 30 & 0 & 0 & -\frac{144 \sqrt{2}}{3125} & 1 & -\frac{2 \sqrt{2}}{3} & \sqrt{\frac{2}{5}} \\ 31 & \frac{3 \sqrt{\frac{3}{2}}}{16} & \frac{384 \sqrt{3}}{3125} & 0 & -\frac{2 \sqrt{2}}{3} & 1 & -\frac{\sqrt{5}}{3} \\ 32 & \frac{3 \sqrt{\frac{3}{10}}}{16} & -\frac{3072 \sqrt{\frac{3}{5}}}{3125} & \frac{4608}{3125 \sqrt{5}} & \sqrt{\frac{2}{5}} & -\frac{\sqrt{5}}{3} & 1 \\ \end{array}$$

$$\begin{array}{c|cccccc} r & 10 & 20 & 21 & 30 & 31 & 32 \\ \hline 10 & \frac{3}{2} & -\frac{32 \sqrt{2}}{81} & \frac{128 \sqrt{\frac{2}{3}}}{81} & -\frac{9 \sqrt{3}}{64} & \frac{27 \sqrt{\frac{3}{2}}}{64} & \frac{9 \sqrt{\frac{15}{2}}}{64} \\ 20 & -\frac{32 \sqrt{2}}{81} & 6 & -3 \sqrt{3} & -\frac{11808 \sqrt{6}}{15625} & \frac{27648 \sqrt{3}}{15625} & -\frac{119808 \sqrt{\frac{3}{5}}}{15625} \\ 21 & \frac{128 \sqrt{\frac{2}{3}}}{81} & -3 \sqrt{3} & 5 & \frac{10368 \sqrt{2}}{15625} & -\frac{27648}{15625} & \frac{165888}{15625 \sqrt{5}} \\ 30 & -\frac{9 \sqrt{3}}{64} & -\frac{11808 \sqrt{6}}{15625} & \frac{10368 \sqrt{2}}{15625} & \frac{27}{2} & -9 \sqrt{2} & 3 \sqrt{10} \\ 31 & \frac{27 \sqrt{\frac{3}{2}}}{64} & \frac{27648 \sqrt{3}}{15625} & -\frac{27648}{15625} & -9 \sqrt{2} & \frac{25}{2} & -\frac{9 \sqrt{5}}{2} \\ 32 & \frac{9 \sqrt{\frac{15}{2}}}{64} & -\frac{119808 \sqrt{\frac{3}{5}}}{15625} & \frac{165888}{15625 \sqrt{5}} & 3 \sqrt{10} & -\frac{9 \sqrt{5}}{2} & \frac{21}{2} \\ \end{array}$$

$$\begin{array}{c|cccccc} r^2 & 10 & 20 & 21 & 30 & 31 & 32 \\ \hline 10 & 3 & -\frac{512 \sqrt{2}}{243} & \frac{1280 \sqrt{\frac{2}{3}}}{243} & -\frac{81 \sqrt{3}}{128} & \frac{135 \sqrt{\frac{3}{2}}}{128} & \frac{81 \sqrt{\frac{15}{2}}}{128} \\ 20 & -\frac{512 \sqrt{2}}{243} & 42 & -20 \sqrt{3} & -\frac{953856 \sqrt{6}}{78125} & \frac{1714176 \sqrt{3}}{78125} & -\frac{5308416 \sqrt{\frac{3}{5}}}{78125} \\ 21 & \frac{1280 \sqrt{\frac{2}{3}}}{243} & -20 \sqrt{3} & 30 & \frac{1057536 \sqrt{2}}{78125} & -\frac{1990656}{78125} & \frac{6967296}{78125 \sqrt{5}} \\ 30 & -\frac{81 \sqrt{3}}{128} & -\frac{953856 \sqrt{6}}{78125} & \frac{1057536 \sqrt{2}}{78125} & 207 & -135 \sqrt{2} & 45 \sqrt{10} \\ 31 & \frac{135 \sqrt{\frac{3}{2}}}{128} & \frac{1714176 \sqrt{3}}{78125} & -\frac{1990656}{78125} & -135 \sqrt{2} & 180 & -63 \sqrt{5} \\ 32 & \frac{81 \sqrt{\frac{15}{2}}}{128} & -\frac{5308416 \sqrt{\frac{3}{5}}}{78125} & \frac{6967296}{78125 \sqrt{5}} & 45 \sqrt{10} & -63 \sqrt{5} & 126 \\ \end{array}$$

$$\begin{array}{c|cccccc} r^3 & 10 & 20 & 21 & 30 & 31 & 32 \\ \hline 10 & \frac{15}{2} & -\frac{2560 \sqrt{2}}{243} & \frac{5120 \sqrt{\frac{2}{3}}}{243} & -\frac{1215 \sqrt{3}}{512} & \frac{1215 \sqrt{\frac{3}{2}}}{512} & \frac{1701 \sqrt{\frac{15}{2}}}{512} \\ 20 & -\frac{2560 \sqrt{2}}{243} & 330 & -150 \sqrt{3} & -\frac{66313728 \sqrt{6}}{390625} & \frac{105504768 \sqrt{3}}{390625} & -\frac{264757248 \sqrt{\frac{3}{5}}}{390625} \\ 21 & \frac{5120 \sqrt{\frac{2}{3}}}{243} & -150 \sqrt{3} & 210 & \frac{76889088 \sqrt{2}}{390625} & -\frac{125411328}{390625} & \frac{334430208}{390625 \sqrt{5}} \\ 30 & -\frac{1215 \sqrt{3}}{512} & -\frac{66313728 \sqrt{6}}{390625} & \frac{76889088 \sqrt{2}}{390625} & \frac{6885}{2} & -\frac{8775}{2 \sqrt{2}} & \frac{2835 \sqrt{\frac{5}{2}}}{2} \\ 31 & \frac{1215 \sqrt{\frac{3}{2}}}{512} & \frac{105504768 \sqrt{3}}{390625} & -\frac{125411328}{390625} & -\frac{8775}{2 \sqrt{2}} & 2835 & -945 \sqrt{5} \\ 32 & \frac{1701 \sqrt{\frac{15}{2}}}{512} & -\frac{264757248 \sqrt{\frac{3}{5}}}{390625} & \frac{334430208}{390625 \sqrt{5}} & \frac{2835 \sqrt{\frac{5}{2}}}{2} & -945 \sqrt{5} & 1701 \\ \end{array}$$

$$\begin{array}{c|cccccc} r^4 & 10 & 20 & 21 & 30 & 31 & 32 \\ \hline 10 & \frac{45}{2} & -\frac{40960 \sqrt{2}}{729} & \frac{71680 \sqrt{\frac{2}{3}}}{729} & -\frac{3645 \sqrt{3}}{512} & 0 & \frac{5103 \sqrt{\frac{15}{2}}}{256} \\ 20 & -\frac{40960 \sqrt{2}}{729} & 2880 & -1260 \sqrt{3} & -\frac{4592443392 \sqrt{6}}{1953125} & \frac{6772211712 \sqrt{3}}{1953125} & -\frac{14714929152 \sqrt{\frac{3}{5}}}{1953125} \\ 21 & \frac{71680 \sqrt{\frac{2}{3}}}{729} & -1260 \sqrt{3} & 1680 & \frac{5361334272 \sqrt{2}}{1953125} & -\frac{8026324992}{1953125} & \frac{18059231232}{1953125 \sqrt{5}} \\ 30 & -\frac{3645 \sqrt{3}}{512} & -\frac{4592443392 \sqrt{6}}{1953125} & \frac{5361334272 \sqrt{2}}{1953125} & \frac{122715}{2} & -\frac{76545}{\sqrt{2}} & 11907 \sqrt{10} \\ 31 & 0 & \frac{6772211712 \sqrt{3}}{1953125} & -\frac{8026324992}{1953125} & -\frac{76545}{\sqrt{2}} & 48195 & -15309 \sqrt{5} \\ 32 & \frac{5103 \sqrt{\frac{15}{2}}}{256} & -\frac{14714929152 \sqrt{\frac{3}{5}}}{1953125} & \frac{18059231232}{1953125 \sqrt{5}} & 11907 \sqrt{10} & -15309 \sqrt{5} & 25515 \\ \end{array}$$

To clarify, these radial matrix elements are

$$\langle n' l' | \, r^k \, | n l \rangle \equiv \int_0^\infty r^2 dr \, R_{n'l'}(r) \, r^k \, R_{nl}(r)$$

where

$$R_{nl}(r) = N_{nl} \, e^{-\rho/2} \rho^l L_{n-l+1}^{2l+1}(\rho)$$

is the radial part of the hydrogen wavefunction

$$\psi_{nlm}(r, \theta, \phi) = R_{nl}(r) Y_l^m(\theta, \phi),$$

$$\rho$$ is related to $$r$$ by

$$\rho \equiv \frac{2r}{n},$$

and the normalization factor $$N_{nl}$$ is

$$N_{nl} \equiv \sqrt{\sqrt{\left(\frac{2}{n}\right)^3} \frac{(n-l-1)!}{2n(n+l)!}}.$$

Remember, I am using units where $$a_0=1$$.

The integral for a particular matrix element can be done in any computer algebra system; it’s just a polynomial times an exponential. The TeX output for the tables above was generated by Mathematica.

A general formula for an arbitrary matrix element can be obtained by using the following integral formula from functions.wolfram.com:

$$I(\alpha, p; m, \lambda, a; n, \beta, b) \equiv \int_0^\infty t^{\alpha-1} e^{-pt} L_m^\lambda(at) L_n^\beta(bt)\,dt \\ =\frac{\Gamma(\alpha)}{p^\alpha} \frac{(\lambda+1)_m}{m!} \frac{(\beta+1)_n}{n!} \sum_{j=0}^m \frac{(-m)_j(\alpha)_j}{(\lambda+1)_j j!} \left(\frac{a}{p}\right)^j \sum_{k=0}^n \frac{(-n)_k(j+\alpha)_k}{(\beta+1)_k k!} \left(\frac{b}{p}\right)^k$$

where $$(x)_n$$ is the Pochhammer symbol

$$(x)_n \equiv x(x+1)\dots(x+n-1).$$

The result for the radial matrix element is

$$\langle n' l' | \, r^k \, | n l \rangle = N_{n'l'} N_{nl} \left(\frac{2}{n'}\right)^{l'} \left(\frac{2}{n}\right)^l \\ \times I\left(3+k+l'+l, \frac{1}{n'}+\frac{1}{n}; n'-l'-1, 2l'+1, \frac{2}{n’}; n-l-1, 2l+1,\frac{2}{n}\right).$$

Note that the double-sum $$I$$ has $$(n'-l')(n-l)$$ terms, each of which is rational if the $$k$$ in $$r^k$$ is. The square roots in the matrix elements come from the normalization factors.

This formula is considerably quicker (at least in Mathematica) than evaluating the individual integrals, especially when $$n$$ or $$n’$$ get large.

This formula works for non-integral $$k$$ as well.

In the closed form solution the radial component is given by one of the Laguerre polynomials the coefficients and properties of which are tabulated in many places such as Abramowitz and Stegun.

From this you can compute the values you want easily.

• Yea i know, i was just looking for a table. Anyway, thanks for that. May 15, 2013 at 14:22
1. Bockaste.K(1974), MEAN-VALUES OF POWERS OF RADIUS FOR HYDROGENIC ELECTRON ORBITS. Physical Review A. 9(3): p. 1087-1089, DOI: 10.1103/PhysRevA.9.1087. This is the standard ref and has a table.
• Hello Steve, and welcome to Physics.SE! It would be very helpful to everyone involved if you stated and/or explained the content of your source. Mar 27, 2015 at 19:51
• This paper does not deal with off-diagonal matrix elements. Jan 30 at 0:59