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. It takes the form $$\hat{H}\psi (\mathbf{r},t)=[-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V\left(\mathbf{r}\right)]\psi (\mathbf{r},t)=i\hslash \frac{\partial \psi (\mathbf{r},t)}{\partial t},$$ 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, $$[-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V\left(\mathbf{r}\right)]\psi \left(\mathbf{r}\right)=E\psi \left(\mathbf{r}\right),$$ 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
• The hydrogen atom or hydrogen-like atom e.g. positronium
• The hydrogen atom in a spherical cavity with Dirichlet boundary conditions
• The particle in a one-dimensional lattice (periodic potential)
• The particle in a one-dimensional lattice of finite length
• The Morse potential
• The Mie 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
• The quantum pendulum
• The rectangular potential barrier
• The Pöschl–Teller potential
• The Inverse square root potential
• Multistate Landau–Zener models
• The Luttinger liquid (the only exact quantum mechanical solution to a model including interparticle interactions)

Solutions

System	Hamiltonian	Energy	Remarks
Two-state quantum system	$\alpha I+\mathbf{r}\hat{\mathit{\sigma }}$	$\alpha \pm |\mathbf{r}|$
Free particle	$-\frac{{\hslash }^2{\mathrm{\nabla }}^2}{2m}$	$\frac{{\hslash }^2{\mathbf{k}}^2}{2m},\mathbf{k}\in {\mathbb{ℝ}}^d$	Massive quantum free particle
Delta potential	$-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+\lambda \delta (x)$	$-\frac{m{\lambda }^2}{2{\hslash }^2}$	Bound state
Symmetric double-well Dirac delta potential	$-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+\lambda (\delta (x-\frac{R}{2})+\delta (x+\frac{R}{2}))$	$-\frac{1}{2R^2}{(\lambda R+W(\pm \lambda R{e}^{-\lambda R}))}^2$	$\hslash =m=1$, W is Lambert W function, for non-symmetric potential see here
Particle in a box	$-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x)$ $$V(x)=\{\begin{array}{ll}0,& 0<x<L,\\ \mathrm{\infty },& \text{otherwise}\end{array}$$	$\frac{{\pi }^2{\hslash }^2{n}^2}{2m{L}^2},n=1,2,3,\dots$	for higher dimensions see here
Particle in a ring	$-\frac{{\hslash }^2}{2m{R}^2}\frac{d^2}{d{\theta }^2}$	$\frac{{\hslash }^2{n}^2}{2m{R}^2},n=0,\pm 1,\pm 2,\dots$
Quantum harmonic oscillator	$-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+\frac{m{\omega }^2{x}^2}{2}$	$\hslash \omega (n+\frac{1}{2}),n=0,1,2,\dots$	for higher dimensions see here
Hydrogen atom	$-\frac{{\hslash }^2}{2\mu }{\mathrm{\nabla }}^2-\frac{e^2}{4\pi {\epsilon }_{0}r}$	$-\left(\frac{\mu e^4}{32{\pi }^2{ϵ}_0^2{\hslash }^2}\right)\frac{1}{n^2},n=1,2,3,\dots$

See also

• List of quantum-mechanical potentials – a list of physically relevant potentials without regard to analytic solubility
• List of integrable models
• WKB approximation
• Quasi-exactly-solvable problems

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.