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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 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)

References

^ 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)
^ 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.
^ 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.
^ 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.
^ 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. S2CID 117745876.
^ 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.
^ 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.

[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] 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.

[Ren2002-3] 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] 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.

[5] 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. S2CID 117745876.

[6] 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.

[sinitsyn-17jpa-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.

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Informatics Educational Institutions & Programs

Solvable systems

See also

References

Reading materials