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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

\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 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{{Cite journal |doi = 10.13140/RG.2.2.12867.32809|title = Analytic solution to the time-dependent Schrödinger equation for the one-dimensional quantum harmonic oscillator with an applied uniform field|year = 2021|last1 = Hodgson|first1 = M.J.P.}}
  • The Inverse square root potential{{Cite journal | last1 = Ishkhanyan | first1 = A. M.| doi = 10.1209/0295-5075/112/10006 | title = Exact solution of the Schrödinger equation for the inverse square root potential V_{0} /\sqrt{x} | journal = Europhysics Letters | volume = 112 | issue = 1 | pages = 10006 | year = 2015 | arxiv=1509.00019| s2cid = 119604105}}
  • The periodic potential
  • The particle in a lattice
  • The particle in a lattice of finite length{{cite journal | last = Ren | first = S. Y. |title = Two Types of Electronic States in One-Dimensional Crystals of Finite Length | year = 2002 | journal = Annals of Physics |volume = 301 | issue = 1 |pages = 22–30 | doi = 10.1006/aphy.2002.6298 | arxiv = cond-mat/0204211 | bibcode = 2002AnPhy.301...22R | s2cid = 14490431 }}
  • 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{{Cite journal |doi = 10.1016/j.cpc.2015.02.009|bibcode = 2015CoPhC.191..221S|title = Efficient hybrid-symbolic methods for quantum mechanical calculations|year = 2015|last1 = Scott|first1 = T.C.|last2 = Zhang|first2 = Wenxing|journal = Computer Physics Communications|volume = 191|pages = 221–234}}
  • The Mie potential{{Cite journal |doi = 10.1007/s10910-007-9228-8|title = Bound state solution of the Schrödinger equation for Mie potential | year = 2007| last1 = Sever| last2 = Bucurgat | last3 = Tezcan |last4 = Yesiltas|journal = Journal of Mathematical Chemistry |volume = 43 |issue = 2 |pages = 749–755 |s2cid = 9887899 }}
  • The Hooke's atom
  • The Morse potential
  • The Spherium atom
  • Zero range interaction in a harmonic trap{{Cite journal | last1 = Busch | first1 = Thomas| last2 = Englert | first2 = Berthold-Georg | last3 = Rzażewski | first3 = Kazimierz | last4 = Wilkens | first4 = Martin | doi = 10.1023/A:1018705520999 | title = Two Cold Atoms in a Harmonic Trap | journal = Foundations of Physics | volume = 27 | issue = 4 | pages = 549–559 | year = 1998 | bibcode = 1998FoPh...28..549B| s2cid = 117745876}}
  • Multistate Landau–Zener models{{cite journal|title=The Quest for Solvable Multistate Landau-Zener Models|journal=Journal of Physics A: Mathematical and Theoretical |volume=50 |issue=25 |pages=255203 |author1=N. A. Sinitsyn |author2=V. Y. Chernyak |arxiv=1701.01870|year=2017|doi=10.1088/1751-8121/aa6800 |bibcode=2017JPhA...50y5203S |s2cid=119626598 }}
  • The Luttinger liquid (the only exact quantum mechanical solution to a model including interparticle interactions)

Solutions

class="wikitable"

! 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}{2 R^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}{2 m L^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}{2 m R^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

{{inc-science|date=October 2024}}

See also

References

{{reflist}}

Reading materials

  • {{cite book

| last = Mattis

| first = Daniel C.

| authorlink = Daniel C. Mattis

| title = The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension

| publisher = World Scientific

| date = 1993

| isbn = 978-981-02-0975-9}}

Category:Quantum models

Quantum-mechanical systems with analytical solutions

Category:Exactly solvable models