Most quantum mechanical problems of interest in for example atomic, molecular, nuclear and solid state physics consist of a large number of interacting electrons and ions or nucleons.
The total number of particles \( N \) is usually sufficiently large that an exact solution cannot be found.
Typically, the expectation value for a chosen hamiltonian for a system of \( N \) particles is
$$ \langle H \rangle = \frac{\int d\boldsymbol{R}_1d\boldsymbol{R}_2\dots d\boldsymbol{R}_N \Psi^{\ast}(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N) H(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N) \Psi(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N)} {\int d\boldsymbol{R}_1d\boldsymbol{R}_2\dots d\boldsymbol{R}_N \Psi^{\ast}(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N) \Psi(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N)}, $$an in general intractable problem.
This integral is actually the starting point in a Variational Monte Carlo calculation. Gaussian quadrature: Forget it! Given 10 particles and 10 mesh points for each degree of freedom and an ideal 1 Tflops machine (all operations take the same time), how long will it take to compute the above integral? The lifetime of the universe is of the order of \( 10^{17} \) s.