variational principle
The variational principle is the mathematical guarantee that makes the variational method trustworthy: the average energy of any normalized state is always at least the true ground-state energy. There is simply no state, however cunningly chosen, whose average energy can sink below the lowest the system actually allows. Equality holds only when your state happens to be the genuine ground state.
The reason is clean once you picture it. Any state can be written as a blend of the system's true energy states, like a chord built from pure notes. Its average energy is then a weighted mixture of the true energies, with positive weights that add to one. A weighted average of a list of numbers can never fall below the smallest number on the list — and here the smallest is the ground-state energy. The bound is exact and unavoidable.
This humble inequality is far more powerful than it looks. It turns the search for a ground state into a minimization, a problem computers handle beautifully, and it underlies an enormous range of approximation schemes. It also extends gracefully: by restricting attention to states orthogonal to the ground state, one can bound excited-state energies too, building up a ladder of estimates from a single guiding idea.
The expected energy of any state is bounded below by the ground-state energy — equality only at the true ground state.
The principle bounds the energy, not other quantities. It says nothing on its own about how accurately a trial state reproduces positions, momenta, or other observables, which can lag well behind the energy.