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

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

where $\psi$ 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,

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

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

More information

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

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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 $V_{0}/{\sqrt {x}}$ ". 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.

List of quantum-mechanical systems with analytical solutions

Solvable systems

Solutions

See also

References

Reading materials

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