List of quantum-mechanical systems with analytical solutions
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.
- 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
- 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 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
- 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)
- 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.