List of quantum-mechanical systems with analytical solutions explained

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{H} \psi\left( r , t\right) = \left[ - (\hbar ^{2})/(2 m) \nabla ^{2} + V\left( r \right) \right] \psi\left( r , t\right) = i\hbar (\partial\psi\left( r , t\right))/(\partial t) ,

where

\psi is the wave function of the system, \hat{H} is the Hamiltonian operator, and t is time. Stationary states of this equation are found by solving the time-independent Schrödinger equation,\left[ - (\hbar ^{2})/(2 m) \nabla ^{2} + V\left( r \right) \right] \psi\left( r \right) = E \psi \left( r \right),

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 uniform 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 particle in a one-dimensional lattice of finite length^[3]
• The Morse potential
• The Mie potential^[4]
• The step potential
  • The linear rigid rotor The symmetric top
• The Hooke's atom
• The Spherium atom
• Zero range interaction in a harmonic trap^[5]
• The quantum pendulum
• The rectangular potential barrier
• The Pöschl–Teller potential
• The Inverse square root potential^[6]
• Multistate Landau–Zener models^[7]
• 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
• Quasi-exactly-solvable problems

Reading materials

• Book: Mattis , Daniel C. . Daniel C. Mattis

. Daniel C. Mattis . The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension . . 1993 . 978-981-02-0975-9.

Notes and References

1. 10.13140/RG.2.2.12867.32809. Analytic solution to the time-dependent Schrödinger equation for the one-dimensional quantum harmonic oscillator with an applied uniform field. 2021. Hodgson. M.J.P..
2. 10.1016/j.cpc.2015.02.009. 2015CoPhC.191..221S. Efficient hybrid-symbolic methods for quantum mechanical calculations. 2015. Scott. T.C.. Zhang. Wenxing. Computer Physics Communications. 191. 221–234.
3. Ren . S. Y. . Two Types of Electronic States in One-Dimensional Crystals of Finite Length . 2002 . Annals of Physics . 301 . 1 . 22–30 . 10.1006/aphy.2002.6298 . cond-mat/0204211 . 2002AnPhy.301...22R . 14490431 .
4. 10.1007/s10910-007-9228-8. Bound state solution of the Schrödinger equation for Mie potential . 2007. Sever. Bucurgat . Tezcan . Yesiltas. Journal of Mathematical Chemistry . 43 . 2 . 749–755 . 9887899 .
5. Busch . Thomas. Englert . Berthold-Georg . Rzażewski . Kazimierz . Wilkens . Martin . 10.1023/A:1018705520999 . Two Cold Atoms in a Harmonic Trap . Foundations of Physics . 27 . 4 . 549–559 . 1998 . 117745876.
6. Ishkhanyan . A. M.. 10.1209/0295-5075/112/10006 . Exact solution of the Schrödinger equation for the inverse square root potential V _{0} /\sqrt{x} . Europhysics Letters . 112 . 1 . 10006 . 2015 . 1509.00019. 119604105.
7. The Quest for Solvable Multistate Landau-Zener Models. Journal of Physics A: Mathematical and Theoretical . 50 . 25 . 255203 . N. A. Sinitsyn . V. Y. Chernyak . 1701.01870. 2017. 10.1088/1751-8121/aa6800 . 2017JPhA...50y5203S . 119626598 .