# 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

where is the wave function of the system, is the Hamiltonian operator, and is time. 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.

## 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 linear field[1]
- The hydrogen atom or hydrogen-like atom e.g. positronium
- The hydrogen atom in a spherical cavity with Dirichlet boundary conditions[2]
- The particle in a one-dimensional lattice (periodic potential)
- The Morse 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[3]
- The quantum pendulum
- The rectangular potential barrier
- The Pöschl-Teller potential
- The Inverse square root potential[4]
- Multistate Landau–Zener Models[5]
- 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

## References

- Hodgson, M.J.P., 2016. Electrons in model nanostructures (Doctoral dissertation, University of York) pages 122-124.
- T.C. Scott and Wenxing Zhang,
*Efficient hybrid-symbolic methods for quantum mechanical calculations*, Comput. Phys. Commun.**191**, pp. 221-234, 2015 . - Busch, Thomas; Englert, Berthold-Georg; Rzażewski, Kazimierz; Wilkens, Martin (1998). "Two Cold Atoms in a Harmonic Trap".
*Foundations of Physics*.**27**(4): 549–559. doi:10.1023/A:1018705520999. - Ishkhanyan, A. M. (2015). "Exact solution of the Schrödinger equation for the inverse square root potential ".
*Europhysics Letters*.**112**(1): 10006. arXiv:1509.00019. doi:10.1209/0295-5075/112/10006. - N. A. Sinitsyn; V. Y. Chernyak (2017). "The Quest for Solvable Multistate Landau-Zener Models".
*Journal of Physics A: Mathematical and Theoretical*.**50**(25): 255203. arXiv:1701.01870. doi:10.1088/1751-8121/aa6800.

## Reading materials

- Mattis, Daniel C. (1993).
*The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension*. World Scientific. ISBN 978-981-02-0975-9.