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

${\displaystyle {\hat {H}}\psi \left(\mathbf {r} ,t\right)=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\left(\mathbf {r} \right)\right]\psi \left(\mathbf {r} ,t\right)=i\hbar {\frac {\partial \psi \left(\mathbf {r} ,t\right)}{\partial t}},}$

where ${\displaystyle \psi }$ is the wave function of the system, ${\displaystyle {\hat {H}}}$ is the Hamiltonian operator, and ${\displaystyle t}$ is time. Stationary states of this equation are found by solving the time-independent Schrödinger equation,

${\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\left(\mathbf {r} \right)\right]\psi \left(\mathbf {r} \right)=E\psi \left(\mathbf {r} \right),}$

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 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^[1]
• The hydrogen atom or hydrogen-like atom e.g. positronium
• The hydrogen atom in a spherical cavity with Dirichlet boundary conditions^[2]
• The particle in a one-dimensional lattice (periodic potential)
• The particle in a one-dimensional lattice of finite length^[3]
• The Morse potential
• The Mie potential^[4]
• The step potential
• The linear rigid rotor
• The symmetric top
• The Hooke's atom
• The Spherium atom
• Zero range interaction in a harmonic trap^[5]
• The quantum pendulum
• The rectangular potential barrier
• The Pöschl–Teller potential
• The Inverse square root potential^[6]
• Multistate Landau–Zener models^[7]
• The Luttinger liquid (the only exact quantum mechanical solution to a model including interparticle interactions)

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

References

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.