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I want to compute quickly (using maybe some scaling arguments) $\langle n^{'}l^{'}m^{'}|r^k|nlm\rangle$, where $k \in I$. $|nlm \rangle$ is the eigenfunction of the Hydrogen atom ($H$).

Example:

Quickly compute $\langle r\rangle_{nlm}$ and the likes.

Some quick help would be highly appreciated.

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    $\begingroup$ Quick help as in "my assignment is due in ten minutes and I need you to solve this for me"? The question is valid but we don't really operate on those sorts of timescales. $\endgroup$ Commented Nov 7, 2015 at 20:13
  • $\begingroup$ @EmilioPisanty Nope nothing of that sort. You can take all the time you need. "Quickly" in the sense compute faster! $\endgroup$
    – sbp
    Commented Nov 7, 2015 at 20:15
  • $\begingroup$ Closely related: Table of matrix elements of powers of r for radial functions in H atom $\endgroup$ Commented Nov 7, 2015 at 20:44
  • $\begingroup$ You can use the analytic formula for the radial matrix element in this answer, combined with the fact that the angular integration gives $\delta_{l’l}\delta_{m’m}$. $\endgroup$
    – Ghoster
    Commented Jan 30, 2023 at 5:49

3 Answers 3

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This answer gives an analytical approach for diagonal matrix elements.

First of all, since $r^k$ is spherically symmetric you can immediately integrate the angular parts: \begin{equation} \left< n' l' m' | r^k | n l m \right> = \left< n'l \right\| r^k \left\| nl \right> \delta_{l',l} \delta_{m',m}, \end{equation} where $\left< n'l \right\| r^k \left\| nl \right>$ is called the reduced matrix element.

Diagonal matrix elements: expectation values of $r^k$

Diagonal matrix elements $\langle r^k \rangle = \left< nl \right\| r^k \left\| nl \right>$ can be calculated with a combination of the Hellmann-Feynman theorem and Kramers' relation without the need to perform any integral explicitly.

The Hellmann-Feynman theorem is given by \begin{equation} \frac{dE}{d\lambda} = \left< \psi(\lambda) \right| \frac{\partial H}{\partial \lambda} \left| \psi(\lambda) \right>, \end{equation} where $\left|\psi(\lambda) \right>$ is a normalized ket for all $\lambda$. In this case you take $\lambda = l$ (note that $n=n(l)$) for $k=-2$ and $\lambda = Z$ for $k=-1$. Here $l$ is the angular momentum quantum number and $Z$ the atomic number.

Kramers' relation is a recursion relation from which all other $\langle r^k \rangle$ for integer $k$ can be found. It can be derived (see link) from the radial Schroedinger equation of the hydrogen-like atom and is given by \begin{equation} \frac{k}{4} \left[ \left( 2l + 1 \right)^2 - k^2 \right] a_0^2 \langle r^{k-2} \rangle - Z \left( 2k + 1 \right) a_0 \langle r^{k-1} \rangle + Z^2 \frac{k+1}{n^2} \langle r^k \rangle = 0, \end{equation} where $a_0$ is the Bohr radius. Note that this relation by itself gives all $\langle r^k \rangle$ for $k>-2$ since $\langle r^0 \rangle = 1$.

Off-diagonal matrix elements

For the off-diagonal matrix elements, I looked in the 8th edition of Gradshteyn and Ryzhik (page 817) for an analytical expression of the relevant integrals, but none is given there. Likely, there are no analytical expressions for the off-diagonal matrix elements. This is also confirmed by the paper linked in the answer of Emilio Pisanty.

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  • $\begingroup$ Fortunately I looked up Griffiths yesterday. The problem was given in an exercise. Either way thanks for the answer. $\endgroup$
    – sbp
    Commented Nov 8, 2015 at 9:09
  • $\begingroup$ @sbp Can you point to the exact problem you found? Does it give the expectation values only, or also the off-diagonal matrix elements? $\endgroup$ Commented Nov 11, 2015 at 12:20
  • $\begingroup$ @EmilioPisanty Kramers' relation is Problem 6.34 $\endgroup$
    – Praan
    Commented Nov 11, 2015 at 12:27
  • $\begingroup$ OK, so purely diagonal elements, so not a full answer to the question as posed. $\endgroup$ Commented Nov 11, 2015 at 12:31
  • $\begingroup$ Likely, there are no analytical expressions for the off-diagonal matrix elements. Actually, there is one. See this answer for an analytic formula for the general radial matrix element. $\endgroup$
    – Ghoster
    Commented Jan 30, 2023 at 5:56
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This looks to be a messy calculation for the most general case, and a direct attempt based on special-functions properties of the Laguerre wavefunctions is likely to simply falter and die in the not-quite-right forms of the integrals.

These matrix elements are calculated in terms of recursion relations in

Matrix-element calculations for hydrogenlike atoms. M.L. Sánchez, B. Moreno, and A. López Piñeiro. Phys. Rev. A 46 no. 11, 6908 (1992)

and the authors provide software to calculate them in

HYDMATEL: a code to calculate matrix elements for hydrogen-like atoms. M.L. Sánchez and A.López Piñeiro. Comput. Phys. Commun. 75 no. 1-2, 185 (1993)

with the software available at

HYDMATEL: a code to calculate matrix elements for hydrogen-like atoms. Computer Physics Communications Program Library, Id ACLO v1.0

under the CPC standard license.

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  • $\begingroup$ See this answer for an analytic formula for the general radial matrix element. I derived it using an integral formula I found on functions.wolfram.com. I have no idea how long that integral formula has been known. $\endgroup$
    – Ghoster
    Commented Jan 30, 2023 at 5:54
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Given that the eigenstates of the hydrogen atom can be separated into a tensor product of angular momentum eigenstates and a radial solution, the Wigner-Eckart theorem helps to calculate the matrix element of any tensor-like operator (a scalar as $r^k$ is a trivial case) between states of the form $|lm\rangle$. The rest is an integral over the radial components that can be calculated using the fact that the Laguerre polynomials (contributing to the radial component) form an orthonormal basis in $L^2$. Obviously, some integrals may be complicated to work out, but that is a different matter.

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