The Schrödinger equation is the basis for quantum mechanics. The solutions to the equation, known as wave functions, give complete quantum mechanical insight into the system under observation. The Hamiltonian operator, which is specific to the system's environment, acts upon the wavefunction to yield the wavefunction again, accompanied by the energy of the system (the eigenvalue). The time-dependant Schrödinger equation describes the evolution of a system’s wavefunction through time.
Thumbnail: A wave packet without dispersion (real or imaginary part). Image used with permission (Public Domain; Fffred~commonswiki).