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
ψ
ℏ
ψ
∂ ψ
ψ
is the wave function of the system,
is the Hamiltonian operator, and
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.