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

${\displaystyle {\hat {H}}\psi \left(\mathbf {r} ,t\right)=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\left(\mathbf {r} \right)\right]\psi \left(\mathbf {r} ,t\right)=i\hbar {\frac {\partial \psi \left(\mathbf {r} ,t\right)}{\partial t}},}$

where ${\displaystyle \psi }$ is the wave function of the system, ${\displaystyle {\hat {H}}}$ is the Hamiltonian operator, and ${\displaystyle t}$ is time. Stationary states of this equation are found by solving the time-independent Schrödinger equation,

${\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\left(\mathbf {r} \right)\right]\psi \left(\mathbf {r} \right)=E\psi \left(\mathbf {r} \right),}$

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.