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

Question

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:

$$\langle R_{nl}|r^k|R_{n'l'}\rangle$$

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

Any help is very appreciated.

Answer 1

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

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

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.

Answer 2

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.

Answer 3

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.

Answer 4

Consider the radial matrix element between states with $n_1, l_1$ and $n_2, l_2$.

@ghoster's closed equation from The Mathematical Functions Site by Wolfram Research is incomplete. The equation cited is 05.08.21.0010.01, which only works for states with $n_{1} \neq l_{1} + 1$ and $n_{2} \neq l_{2} + 1$.

To get the complete table of radial matrix elements, one must also look to equations 01.03.21.0032.01 (for $n_1 = l_1 + 1$ and $n_2 = l_2 + 1$), and 05.08.21.0005.01 (for $n_1 = l_1 + 1$ or $n_2 = l_2 + 1$).

One may also look to equation 05.08.21.0009.01 for the simpler case of $n_1 = n_2$, $n_{1} \neq l_{1} + 1$ and $n_{2} \neq l_{2} + 1$

Further, we must note that these equations are only valid for integer $n, l$, which makes them unsuitable when applying the Rydberg-Ritz expansion for non-hydrogen hydrogenic atoms.

Also, there is the paper Matrix-element calculations for hydrogenlike atoms by Sánchez for a recurrence relation algorithm to compute all matrix elements.