List of quantum mechanical systems with analytical solutions
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The field of quantum mechanics is based on solutions to the Schrödinger equation, which is often represented (in its non-relativistic form) as
![H \psi\left(\mathbf{r}, t\right) = \left(T + V\right) \, \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).](../../../math/b/d/5/bd5a6f349415def6bedc744010e0746c.png)
Where ψ is the wavefunction of the system, H is the Hamiltonian operator, and T and V are the operators for the kinetic energy and potential energy, respectively. ( Common forms of these operators appear in the square brackets on the right.)
Stationary states of this equation are found by solving the eigenvalue-eigenfunction form of the Schroedinger 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).](../../../math/5/6/b/56b5196db651f69e603cc1b4fb60f642.png)
However, a given physical system may result in a potential energy function, V(r), such that a closed form solution for ψ can not be found for any (nonzero) E.
A small number of systems for which ψ and E can be found, and easily expressed, are currently known. These quantum mechanical systems with analytical solutions are quite useful for teaching and gaining intuition about quantum mechanics, and are listed below.
- The free particle
- The Delta function potential
- The particle in a box
- The finite potential well
- The particle in a ring
- The particle in a spherically symmetric potential
- The quantum harmonic oscillator
- The hydrogen atom or hydrogen-like atom
- The ring wave guide
- The particle in a one-dimensional lattice (periodic potential)
- The hydrogen-molecular ion
- The Morse potential
