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 in an appropriate configuration space. In vector Cartesian coordinates $\mathbf {r}$ , the equation takes the form

H\psi \left(\mathbf {r} ,t\right)=\left(T+V\right)\,\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}}

in which $\psi$ 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,

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

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. However, 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

The two-state quantum system (the simplest quantum system that can exist)
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 linear 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 Morse potential
The step potential
The linear rigid rotor
The symmetric top
The Hooke's atom
The Spherium atom
Zero range interaction in a harmonic trap^[3]
The Quantum pendulum
The Rectangular potential barrier
The Pöschl-Teller potential
The Inverse square root potential^[4]
The Lambert-W step-potential
Multistate Landau–Zener Models^[5]
The Luttinger liquid (the only exact quantum mechanical solution to a model including interparticle interactions)

References

↑ Hodgson, M.J.P., 2016. Electrons in model nanostructures (Doctoral dissertation, University of York) pages 122-124.
↑ T.C. Scott and Wenxing Zhang, Efficient hybrid-symbolic methods for quantum mechanical calculations, Comput. Phys. Commun. 191, pp. 221-234, 2015 .
↑ 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.
↑ 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.
↑ N. A. Sinitsyn; V. Y. Chernyak (2017). "The Quest for Solvable Multistate Landau-Zener Models". arXiv:1701.01870 [quant-ph].

Reading materials

Mattis, Daniel C. (1993). The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension. World Scientific. ISBN 981-02-0975-4.

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[Hodgson-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 .

[3] 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.

[4] 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.

[sinitsyn-17jpa-5] N. A. Sinitsyn; V. Y. Chernyak (2017). "The Quest for Solvable Multistate Landau-Zener Models". arXiv:1701.01870 [quant-ph].

List of quantum-mechanical systems with analytical solutions

Solvable systems

See also

References

Reading materials