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List of quantum-mechanical systems with analytical solutions
From Wikipedia, the free encyclopedia
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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
where is the wave function of the system, is the Hamiltonian operator, and is time. Stationary states of this equation are found by solving the time-independent Schrödinger equation,
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.
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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 steps potentials
- The triangular potential
- The quadratic potentials
- The quantum harmonic oscillator
- The quantum harmonic oscillator with an applied uniform field[1]
- The Inverse square root potential[2]
- The periodic potential
- 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[4]
- The Mie potential[5]
- The Hooke's atom
- The Morse potential
- The Spherium atom
- The rotating system
- Zero range interaction in a harmonic trap[6]
- Multistate Landau–Zener models[7]
- The Luttinger liquid (the only exact quantum mechanical solution to a model including interparticle interactions)
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Solutions
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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
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