the variational method
When a Hamiltonian is too hard to solve and no small parameter is available to perturb around, the variational method offers a different kind of leverage: guess a trial wavefunction with adjustable parameters, and let a rigorous inequality guarantee that your best guess never underestimates the ground-state energy. It turns solving a differential equation into minimizing a number -- a problem a computer, or a clever hand, can attack directly.
The principle is an exact theorem. For any normalized trial state |psi>, the expectation value of the Hamiltonian is an upper bound on the true ground-state energy: <psi| H |psi> is greater than or equal to E_ground, with equality only when |psi> is the exact ground state. The proof is one line: expand |psi> in the true eigenstates and note that mixing in any excited state, which has higher energy, can only raise the average. So you write a trial wavefunction with free parameters, compute <H> as a function of those parameters, and minimize -- the lowest value you can reach is your best estimate of (and a guaranteed ceiling on) the ground-state energy. The more flexible the trial function, the tighter the bound.
The method is a mainstay precisely where perturbation theory fails -- strongly interacting systems, with no natural small parameter. It gives the ground-state energy of helium to a fraction of a percent from a simple trial function, underlies the Hartree-Fock and density-functional methods that power quantum chemistry, and provides the variational Monte Carlo approach used for many-body systems. Two honest caveats: the bound applies rigorously only to the ground state (excited states need extra orthogonality constraints), and a good energy does not guarantee a good wavefunction -- the energy is stationary, so it is insensitive to errors in psi, meaning your estimate can be excellent even when the trial state's shape is mediocre.
For the helium atom, a trial wavefunction that is a product of two hydrogen-like 1s orbitals with an adjustable effective nuclear charge Z_eff gives a ground-state energy within about 2 percent of experiment when you minimize over Z_eff, landing near Z_eff = 1.69 -- less than the true nuclear charge of 2, a quantitative measure of how each electron screens the nucleus from the other.
Minimizing over an effective charge gives helium's energy to ~2 percent and quantifies electron screening (Z_eff below 2).
The variational energy is a rigorous upper bound only for the ground state, and only if the trial state is properly normalized; the bound says nothing about whether the wavefunction itself is accurate. Because <H> is stationary at the true state, a mediocre trial shape can still give an excellent energy -- so never judge the wavefunction's quality by the energy alone.