# 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
${\hat {H}}\psi {\left(\mathbf {r} ,t\right)}=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V{\left(\mathbf {r} \right)}\right]\psi {\left(\mathbf {r} ,t\right)}=i\hbar {\frac {\partial \psi {\left(\mathbf {r} ,t\right)}}{\partial t}},$
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 two-state quantum system (the simplest possible quantum system)
* The free particle
* 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
** The Inverse square root potential
** The periodic potential
*** The particle in a lattice
*** The particle in a lattice of finite length
** 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
*** The Mie potential
*** The Hooke's atom
*** The Morse potential
*** The Spherium atom
* Zero range interaction in a harmonic trap
* 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 | $\alpha I+\mathbf {r} {\hat {\mathbf {\sigma } }}\,$ | $\alpha \pm |\mathbf {r} |\,$ | |
| 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 | $-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+\lambda \left(\delta \left(x-{\frac {R}{2}}\right)+\delta \left(x+{\frac {R}{2}}\right)\right)$ | $-{\frac {1}{2R^{2}}}\left(\lambda R+W\left(\pm \lambda R\,e^{-\lambda R}\right)\right)^{2}$ | $\hbar =m=1$, W is Lambert W function, for non-symmetric potential see here |
| Particle in a box | $-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)$ $V(x)={\begin{cases}0,&0<x<L,\\\infty ,&{\text{otherwise}}\end{cases}}$ | ${\frac {\pi ^{2}\hbar ^{2}n^{2}}{2mL^{2}}},\,\,n=1,2,3,\ldots$ | for higher dimensions see here |
| Particle in a ring | $-{\frac {\hbar ^{2}}{2mR^{2}}}{\frac {d^{2}}{d\theta ^{2}}}\,$ | ${\frac {\hbar ^{2}n^{2}}{2mR^{2}}},\,\,n=0,\pm 1,\pm 2,\ldots$ | |
| 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 | $-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}-{\frac {e^{2}}{4\pi \varepsilon _{0}r}}$ | $-\left({\frac {\mu e^{4}}{32\pi ^{2}\epsilon _{0}^{2}\hbar ^{2}}}\right){\frac {1}{n^{2}}},\,\,n=1,2,3,\ldots$ |