Here we show an example of a multidimensional integral which appears in quantum mechanical calculations.

The ansatz for the wave function for two electrons is given by the product of two \( 1s \) wave functions as $$ \Psi({\bf r}_1,{\bf r}_2) = \exp{-(\alpha (r_1+r_2))}. $$ The integral we need to solve is the quantum mechanical expectation value of the correlation energy between two electrons, namely $$ I = \int_{-\infty}^{\infty} d{\bf r}_1d{\bf r}_2 \exp{-2(\alpha (r_1+r_2))}\frac{1}{|{\bf r}_1-{\bf r}_2|}. $$ The integral has an exact solution \( 5\pi^2/16 = 0.19277 \).