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 \psi\left(\mathbf, t\right) =
\left - \frac \nabla^2 + V\left(\mathbf\right) \right\psi\left(\mathbf, t\right) = i\hbar \frac,

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

:
\left - \frac \nabla^2 + V\left(\mathbf\right) \right\psi\left(\mathbf\right) = E \psi \left(\mathbf\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 ''L = N a'' (''N'' is a positive integer, ''a'' is the potential period)
*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)

See also

* List of quantum-mechanical potentials – a list of physically relevant potentials without regard to analytic solubility
* List of integrable models
* WKB approximation
* Quasi-exactly-solvable problems

References

Reading materials

* {{cite book
, last = Mattis
, first = Daniel C.
, authorlink = Daniel C. Mattis
, title = The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension
, publisher = World Scientific
, date = 1993
, isbn = 978-981-02-0975-9

Quantum models
Quantum-mechanical systems with analytical solutions