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

See also


  1. Hodgson, M.J.P., 2016. Electrons in model nanostructures (Doctoral dissertation, University of York) pages 122-124.
  2. T.C. Scott and Wenxing Zhang, Efficient hybrid-symbolic methods for quantum mechanical calculations, Comput. Phys. Commun. 191, pp. 221-234, 2015 .
  3. 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.
  4. 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.
  5. 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.
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