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



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



\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.

Solvable systems

 * The two-state quantum system (the simplest possible quantum system)
 * The free particle
 * The delta potential
 * The double-well Dirac delta potential
 * The particle in a box / infinite potential well
 * The finite potential well
 * The one-dimensional triangular potential
 * The particle in a ring or ring wave guide
 * The particle in a spherically symmetric potential
 * The quantum harmonic oscillator
 * The quantum harmonic oscillator with an applied uniform field
 * The hydrogen atom or hydrogen-like atom e.g. positronium
 * The hydrogen atom in a spherical cavity with Dirichlet boundary conditions
 * The particle in a one-dimensional lattice (periodic potential)
 * The particle in a one-dimensional lattice of finite length
 * The Morse potential
 * The Mie potential
 * The step potential
 * The linear rigid rotor
 * The symmetric top
 * The Hooke's atom
 * The Spherium atom
 * Zero range interaction in a harmonic trap
 * The quantum pendulum
 * The rectangular potential barrier
 * The Pöschl–Teller potential
 * The Inverse square root potential
 * Multistate Landau–Zener models
 * The Luttinger liquid (the only exact quantum mechanical solution to a model including interparticle interactions)