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 - \frac \nabla^2 + V \right\psi = 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 \right\psi = E \psi ,

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

Solutions

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
Exactly solvable models