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List of quantum-mechanical systems with analytical solutions
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<h2 id="solvable-systems">Solvable systems</h2> <ul><li>The <a href="/facts/Two-state_quantum_system/hQrw3gR9">two-state quantum system</a> (the simplest possible quantum system)</li> <li>The <a href="/facts/Free_particle/Fjbx90yW">free particle</a></li></ul> <ul><li>The one-dimensional potentials <ul><li>The <a href="/facts/Particle_in_a_ring/MvcC9BkI">particle in a ring</a> or <a href="/facts/Ring_wave_guide/MvcC9BkI">ring wave guide</a></li> <li>The delta potential <ul><li>The <a href="/facts/Delta_potential/USFD3qxw">single delta potential</a></li> <li>The <a href="/facts/Delta_potential/USFD3qxw">double-well delta potential</a></li></ul></li> <li>The steps potentials <ul><li>The <a href="/facts/Particle_in_a_box/rezR97ky">particle in a box</a> / <a href="/facts/Infinite_potential_well/rezR97ky">infinite potential well</a></li> <li>The <a href="/facts/Finite_potential_well/85XV2JoW">finite potential well</a></li> <li>The <a href="/facts/Step_potential/Xdwx8Cku">step potential</a></li> <li>The <a href="/facts/Rectangular_potential_barrier/Cer7zMUn">rectangular potential barrier</a></li></ul></li> <li>The <a href="/facts/Airy_function/bydfB54Y">triangular potential</a></li> <li>The quadratic potentials <ul><li>The <a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">quantum harmonic oscillator</a></li> <li>The quantum harmonic oscillator with an applied uniform field<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul></li> <li>The Inverse square root potential<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li> <li>The periodic potential <ul><li>The <a href="/facts/Particle_in_a_one-dimensional_lattice_(periodic_potential)/JEof9GKh">particle in a lattice</a></li> <li>The <a href="/facts/Particle_in_a_one-dimensional_lattice/JEof9GKh">particle in a lattice of finite length</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li></ul></li> <li>The <a href="/facts/P%25C3%25B6schl%25E2%2580%2593Teller_potential/LTTgYGPn">Pöschl–Teller potential</a></li> <li>The <a href="/facts/Quantum_pendulum/57kPb5VT">quantum pendulum</a></li></ul></li> <li>The three-dimensional potentials <ul><li>The rotating system <ul><li>The <a href="/facts/Rigid_rotor/Ue8nrySx">linear rigid rotor</a></li> <li>The <a href="/facts/Rigid_rotor/Ue8nrySx">symmetric top</a></li></ul></li> <li>The <a href="/facts/Particle_in_a_spherically_symmetric_potential/9JNPLckm">particle in a spherically symmetric potential</a> <ul><li>The <a href="/facts/Hydrogen_atom/ncVpPbuE">hydrogen atom</a> or <a href="/facts/Hydrogen-like_atom/0rmuVpjT">hydrogen-like atom</a> e.g. <a href="/facts/Positronium/F35JBo9C">positronium</a></li> <li>The <a href="/facts/Hydrogen_atom/ncVpPbuE">hydrogen atom</a> in a spherical cavity with <a href="/facts/Dirichlet_boundary_condition/m9wsgG89">Dirichlet boundary conditions</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li>The Mie potential<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li>The <a href="/facts/Hooke%2527s_atom/tp7GfJwx">Hooke's atom</a></li> <li>The <a href="/facts/Morse_potential/QqMUB3Ba">Morse potential</a></li> <li>The <a href="/facts/Spherium/C8EoCjAB">Spherium</a> atom</li></ul></li></ul></li> <li>Zero range interaction in a harmonic trap<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li> <li>Multistate Landau–Zener models<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li> <li>The <a href="/facts/Luttinger_liquid/XSRzsHjH">Luttinger liquid</a> (the only exact quantum mechanical solution to a model including interparticle interactions)</li></ul> <h2 id="solutions">Solutions</h2> <table><tbody><tr><th>System</th><th>Hamiltonian</th><th>Energy</th><th>Remarks</th></tr><tr><td><a href="/facts/Two-state_quantum_system/hQrw3gR9">Two-state quantum system</a></td><td> α I + r σ ^ {\displaystyle \alpha I+\mathbf {r} {\hat {\mathbf {\sigma } }}\,} </td><td> α ± | r | {\displaystyle \alpha \pm |\mathbf {r} |\,} </td><td></td></tr><tr><td><a href="/facts/Free_particle/Fjbx90yW">Free particle</a></td><td> − ℏ 2 ∇ 2 2 m {\displaystyle -{\frac {\hbar ^{2}\nabla ^{2}}{2m}}\,} </td><td> ℏ 2 k 2 2 m , k ∈ R d {\displaystyle {\frac {\hbar ^{2}\mathbf {k} ^{2}}{2m}},\,\,\mathbf {k} \in \mathbb {R} ^{d}} </td><td>Massive quantum free particle</td></tr><tr><td><a href="/facts/Delta_potential/USFD3qxw">Delta potential</a></td><td> − 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)} </td><td> − m λ 2 2 ℏ 2 {\displaystyle -{\frac {m\lambda ^{2}}{2\hbar ^{2}}}} </td><td>Bound state</td></tr><tr><td>Symmetric <a href="/facts/Delta_potential/USFD3qxw">double-well Dirac delta potential</a></td><td> − 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)} </td><td> − 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}} </td><td> ℏ = m = 1 {\displaystyle \hbar =m=1} , <i>W</i> is <a href="/facts/Lambert_W_function/NG1wd71f">Lambert <i>W</i> function</a>, for non-symmetric potential see <a href="/facts/Delta_potential/USFD3qxw">here</a></td></tr><tr><td><a href="/facts/Particle_in_a_box/rezR97ky">Particle in a box</a></td><td> − 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}}} </td><td> 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 } </td><td>for higher dimensions see <a href="/facts/Particle_in_a_box/rezR97ky">here</a></td></tr><tr><td><a href="/facts/Particle_in_a_ring/MvcC9BkI">Particle in a ring</a></td><td> − ℏ 2 2 m R 2 d 2 d θ 2 {\displaystyle -{\frac {\hbar ^{2}}{2mR^{2}}}{\frac {d^{2}}{d\theta ^{2}}}\,} </td><td> 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 } </td><td></td></tr><tr><td><a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">Quantum harmonic oscillator</a></td><td> − 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}}\,} </td><td> ℏ ω ( n + 1 2 ) , n = 0 , 1 , 2 , … {\displaystyle \hbar \omega \left(n+{\frac {1}{2}}\right),\,\,n=0,1,2,\ldots } </td><td>for higher dimensions see <a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">here</a></td></tr><tr><td><a href="/facts/Hydrogen_atom/ncVpPbuE">Hydrogen atom</a></td><td> − 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}}} </td><td> − ( 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 } </td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/List_of_quantum-mechanical_potentials/3tIt9gSJ">List of quantum-mechanical potentials</a> – a list of physically relevant potentials without regard to analytic solubility</li> <li><a href="/facts/List_of_integrable_models/hnaKa2Gp">List of integrable models</a></li> <li><a href="/facts/WKB_approximation/XyRnYvGh">WKB approximation</a></li> <li><a href="/facts/Quasi-exactly-solvable_problems/Hf7dU0fF">Quasi-exactly-solvable problems</a></li></ul> <h2 id="reading-materials">Reading materials</h2> <ul><li>Mattis, Daniel C. (1993). <i>The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension</i>. <a href="/facts/World_Scientific/qJPTP0Yk">World Scientific</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-02-0975-9.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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) <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>