Quantum Mechanics - Approximation Methods : The Variational Method

There exist systems whose Hamiltonians are known, but they can not be solved exactly or by a perturbative treatment. That is,there is no closely related Hamiltonian that can be solved exactly or approximately by perturbation theory because the first order is not sufficiently accurate. One of the approximation methods that is suitable for solving such problems is the Variational Method,which is also called the Rayleigh-Ritz method This method does not require knowledge of simple Hamiltonians that can be solved exactly. The variational method is useful for determining upper bound values for the eigeneneries of a system whose Hamiltonian is known whereas its eigenvalues and eigenfunctions are not known. It is particularly useful for determining the ground state. It becomes quite cumbersome to determine the energy levels of the excited states.