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List of quantum-mechanical systems with analytical solutions
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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

H ^ ψ ( r , t ) = [ − ℏ 2 2 m ∇ 2 + V ( r ) ] ψ ( r , t ) = i ℏ ∂ ψ ( r , t ) ∂ t , {\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 ψ {\displaystyle \psi } is the wave function of the system, H ^ {\displaystyle {\hat {H}}} is the Hamiltonian operator, and t {\displaystyle t} is time. Stationary states of this equation are found by solving the time-independent Schrödinger equation,

[ − ℏ 2 2 m ∇ 2 + V ( r ) ] ψ ( r ) = E ψ ( r ) , {\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

Solutions

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

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.

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) /wiki/Doi_(identifier) ↩

  2. Ishkhanyan, A. M. (2015). "Exact solution of the Schrödinger equation for the inverse square root potential V 0 / x {\displaystyle V_{0}/{\sqrt {x}}} ". Europhysics Letters. 112 (1): 10006. arXiv:1509.00019. doi:10.1209/0295-5075/112/10006. S2CID 119604105. /wiki/ArXiv_(identifier) ↩

  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. /wiki/ArXiv_(identifier) ↩

  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. /wiki/Bibcode_(identifier) ↩

  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. /wiki/Doi_(identifier) ↩

  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. /wiki/Bibcode_(identifier) ↩

  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. /wiki/ArXiv_(identifier) ↩