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	$-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}-{\frac {e^{2}}{4\pi \varepsilon _{0}r}}$	$-\left({\frac {\mu e^{4}}{32\pi ^{2}\epsilon _{0}^{2}\hbar ^{2}}}\right){\frac {1}{n^{2}}},\,\,n=1,2,3,\ldots$

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.