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

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

where ${\displaystyle \psi }$ is the wave function of the system, ${\displaystyle {\hat {H}}}$ is the Hamiltonian operator, and ${\displaystyle t}$ is time. Stationary states of this equation are found by solving the time-independent Schrödinger equation,

$${\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V{\left(\mathbf {r} \right)}\right]\psi {\left(\mathbf {r} \right)}=E\psi {\left(\mathbf {r} \right)},}$$

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 one-dimensional potentials
  • The particle in a ring or ring wave guide
  • The delta potential
    • The single delta potential
    • The double-well delta potential
  • The steps potentials
    • The particle in a box / infinite potential well
    • The finite potential well
    • The step potential
    • The rectangular potential barrier
  • The triangular potential
  • The quadratic potentials
    • The quantum harmonic oscillator
    • The quantum harmonic oscillator with an applied uniform field[1]
  • The Inverse square root potential[2]
  • The periodic potential
    • The particle in a lattice
    • The particle in a lattice of finite length[3]
  • The Pöschl–Teller potential
  • The quantum pendulum
• The three-dimensional potentials
  • The rotating system
    • The linear rigid rotor
    • The symmetric top
  • The particle in a spherically symmetric potential
    • The hydrogen atom or hydrogen-like atom e.g. positronium
    • The hydrogen atom in a spherical cavity with Dirichlet boundary conditions[4]
    • The Mie potential[5]
    • The Hooke's atom
    • The Morse potential
    • The Spherium atom
• Zero range interaction in a harmonic trap[6]
• Multistate Landau–Zener models[7]
• The Luttinger liquid (the only exact quantum mechanical solution to a model including interparticle interactions)

Solutions

System	Hamiltonian	Energy	Remarks
Two-state quantum system	${\displaystyle \alpha I+\mathbf {r} {\hat {\mathbf {\sigma } }}\,}$	${\displaystyle \alpha \pm |\mathbf {r} |\,}$
Free particle	${\displaystyle -{\frac {\hbar ^{2}\nabla ^{2}}{2m}}\,}$	${\displaystyle {\frac {\hbar ^{2}\mathbf {k} ^{2}}{2m}},\,\,\mathbf {k} \in \mathbb {R} ^{d}}$	Massive quantum free particle
Delta potential	${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+\lambda \delta (x)}$	${\displaystyle -{\frac {m\lambda ^{2}}{2\hbar ^{2}}}}$	Bound state
Symmetric double-well Dirac delta potential	${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+\lambda \left(\delta \left(x-{\frac {R}{2}}\right)+\delta \left(x+{\frac {R}{2}}\right)\right)}$	${\displaystyle -{\frac {1}{2R^{2}}}\left(\lambda R+W\left(\pm \lambda R\,e^{-\lambda R}\right)\right)^{2}}$	${\displaystyle \hbar =m=1}$, W is Lambert W function, for non-symmetric potential see here
Particle in a box	${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)}$ $${\displaystyle V(x)={\begin{cases}0,&0<x<L,\\\infty ,&{\text{otherwise}}\end{cases}}}$$	${\displaystyle {\frac {\pi ^{2}\hbar ^{2}n^{2}}{2mL^{2}}},\,\,n=1,2,3,\ldots }$	for higher dimensions see here
Particle in a ring	${\displaystyle -{\frac {\hbar ^{2}}{2mR^{2}}}{\frac {d^{2}}{d\theta ^{2}}}\,}$	${\displaystyle {\frac {\hbar ^{2}n^{2}}{2mR^{2}}},\,\,n=0,\pm 1,\pm 2,\ldots }$
Quantum harmonic oscillator	${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {m\omega ^{2}x^{2}}{2}}\,}$	${\displaystyle \hbar \omega \left(n+{\frac {1}{2}}\right),\,\,n=0,1,2,\ldots }$	for higher dimensions see here
Hydrogen atom	${\displaystyle -{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}-{\frac {e^{2}}{4\pi \varepsilon _{0}r}}}$	${\displaystyle -\left({\frac {\mu e^{4}}{32\pi ^{2}\epsilon _{0}^{2}\hbar ^{2}}}\right){\frac {1}{n^{2}}},\,\,n=1,2,3,\ldots }$

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

References

1. Hodgson, M.J.P. (2021). "Analytic solution to the time-dependent Schrödinger equation for the one-dimensional quantum harmonic oscillator with an applied uniform field". doi:10.13140/RG.2.2.12867.32809. {{cite journal}}: Cite journal requires |journal= (help)
2. Ishkhanyan, A. M. (2015). "Exact solution of the Schrödinger equation for the inverse square root potential ${\displaystyle V_{0}/{\sqrt {x}}}$". Europhysics Letters. 112 (1): 10006. arXiv:1509.00019. doi:10.1209/0295-5075/112/10006. S2CID 119604105.
3. Ren, S. Y. (2002). "Two Types of Electronic States in One-Dimensional Crystals of Finite Length". Annals of Physics. 301 (1): 22–30. arXiv:cond-mat/0204211. Bibcode:2002AnPhy.301...22R. doi:10.1006/aphy.2002.6298. S2CID 14490431.
4. Scott, T.C.; Zhang, Wenxing (2015). "Efficient hybrid-symbolic methods for quantum mechanical calculations". Computer Physics Communications. 191: 221–234. Bibcode:2015CoPhC.191..221S. doi:10.1016/j.cpc.2015.02.009.
5. Sever; Bucurgat; Tezcan; Yesiltas (2007). "Bound state solution of the Schrödinger equation for Mie potential". Journal of Mathematical Chemistry. 43 (2): 749–755. doi:10.1007/s10910-007-9228-8. S2CID 9887899.
6. Busch, Thomas; Englert, Berthold-Georg; Rzażewski, Kazimierz; Wilkens, Martin (1998). "Two Cold Atoms in a Harmonic Trap". Foundations of Physics. 27 (4): 549–559. Bibcode:1998FoPh...28..549B. doi:10.1023/A:1018705520999. S2CID 117745876.
7. 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. Bibcode:2017JPhA...50y5203S. doi:10.1088/1751-8121/aa6800. S2CID 119626598.

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.