List of quantum-mechanical systems with analytical solutions
Much insight in quantum mechanics can be gained from understanding the closed-form solutions to the time-dependent non-relativistic Schrödinger equation. It takes the form
where is the wave function of the system, is the Hamiltonian operator, and is time. Stationary states of this equation are found by solving the time-independent Schrödinger equation,
which is an eigenvalue equation. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. However, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies, or eigenvalues, can be found. These quantum-mechanical systems with analytical solutions are listed below.Solvable systems
