List of quantum-mechanical systems with analytical solutions

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Much insight in quantum mechanics can be gained from understanding the solutions to the time-dependent non-relativistic Schrödinger equation in an appropriate configuration space. In vector Cartesian coordinates , the equation takes the form

in which is the wavefunction of the system, H is the Hamiltonian operator, and T and V are the operators for the kinetic energy and potential energy, respectively. (Common forms of these operators appear in the square brackets.) The quantity t is the time. Stationary states of this equation are found by solving the eigenvalue-eigenfunction (time-independent) form of the Schrödinger equation,

or any equivalent formulation of this equation in a different coordinate system other than Cartesian coordinates. For example, systems with spherical symmetry are simplified when expressed with spherical coordinates. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. Fortunately, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies can be found. These quantum-mechanical systems with analytical solutions are listed below.

Solvable systems[edit]

See also[edit]


  1. ^ [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 [2].
  3. ^ Busch, Thomas (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:1509.00019Freely accessible. doi:10.1209/0295-5075/112/10006. 
  5. ^ N. A. Sinitsyn; V. Y. Chernyak (2017). "The Quest for Solvable Multistate Landau-Zener Models". arXiv:1701.01870Freely accessible [quant-ph]. 

Reading materials[edit]