Physics: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

[math]\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}, }[/math]

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

[math]\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), }[/math]

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)

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.