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I am wondering how to evaluate the following integral of two radial hydrogenic wavefunctions: $$ \int r^{3}R^{*}_{nl} R _{10} dr $$

The whole problem is related to the calculation of the Stark effect (second-order correction to energy). Here is the answer , https://quantummechanics.ucsd.edu/ph130a/130_notes/node337.html but I would like to understand a way of calculation mentioned integral. Thank You in advance for any hints and suggestions.

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  • $\begingroup$ A few hints: Try using the general definition of $R^*_{nl}$. Remember n and l are just constants. After substituting $R^*_{nl}$ and $R^*_{10}$ in and simplifying, you will need to make use of an exponential integral identity. $\endgroup$ Apr 7, 2020 at 23:59

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The strategy is to first consider integrals of the general form $$ I_k(\lambda)=\int_0^\infty dx e^{-\lambda x} x^k \tag{1} $$ which one can solve using repeated integration by parts or by functional differentiation: $$ \frac{d I_k(\lambda)}{d\lambda}=-\lambda I_{k+1}(\lambda)\quad \Rightarrow\quad I_{k+1}=-\frac{1}{\lambda}\frac{d I_k(\lambda)}{d\lambda} $$ $I_0(\lambda)$ can be evaluated by hand and then one can generate the other values of $k$ by differentiation.

Next you can write your integral as a sum over $k$ of integrals of the type given in (1).

Note that in the link you refer to the integral is of the general form (1) but there is an additional restriction on the possible values of $\ell$ that come from the angular integration.

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