separation of variables
Separation of variables is the standard trick for solving the Schrödinger equation when the Hamiltonian does not change with time. You guess that the full wavefunction, which depends on both space and time, factors into a piece that depends only on space multiplied by a piece that depends only on time. This bold assumption turns one entangled equation into two simpler ones.
When you plug the guess in, the time part solves itself: it becomes a phase that just rotates at a steady rate set by the energy. The space part is left obeying the time-independent Schrödinger equation, the eigenvalue problem for the shapes and energies. So the hard, full equation collapses into finding stationary states plus attaching a simple ticking phase to each.
The method does far more than handle time. The same factoring tames problems with several spatial dimensions: in a spherically symmetric atom, for instance, you split the wavefunction into radial and angular factors, each solved on its own. Separation of variables is one of the great workhorses of physics, the reason so many quantum problems can be solved by hand at all.
Factor space from time, and the time part becomes a simple energy-driven phase.
Separation of variables only works for special, separable cases — chiefly when the Hamiltonian is time-independent or the potential has a clean symmetry. The general solution is not a single separated product but a superposition of many such pieces.