List of quantum-mechanical systems with analytical solutions
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Much insight in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
can be gained from understanding the
closed-form solution In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
s to the time-dependent non-relativistic
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. It takes the form : \hat \psi\left(\mathbf, t\right) = \left - \frac \nabla^2 + V\left(\mathbf\right) \right\psi\left(\mathbf, t\right) = i\hbar \frac, where \psi is the
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
of the system, \hat is the
Hamiltonian operator Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, and t is time.
Stationary state A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ener ...
s of this equation are found by solving the time-independent Schrödinger equation, : \left - \frac \nabla^2 + V\left(\mathbf\right) \right\psi\left(\mathbf\right) = E \psi \left(\mathbf\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 In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a syst ...
(the simplest possible quantum system) *The
free particle In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
*The
delta potential In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it t ...
*The double-well Dirac delta potential *The
particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
/
infinite potential well In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
*The
finite potential well The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlike ...
*The one-dimensional triangular potential *The
particle in a ring In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle S^1) is : ...
or
ring wave guide In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle S^1) is : ...
*The
particle in a spherically symmetric potential In the quantum mechanics description of a particle in spherical coordinates, a spherically symmetric potential, is a potential that depends only on the distance between the particle and a defined centre point. One example of a spherical potential ...
*The
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最 ...
*The quantum harmonic oscillator with an applied uniform field *The
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
or
hydrogen-like atom A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such as ...
e.g.
positronium Positronium (Ps) is a system consisting of an electron and its antimatter, anti-particle, a positron, bound together into an exotic atom, specifically an onium. Unlike hydrogen, the system has no protons. The system is unstable: the two parti ...
*The
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
in a spherical cavity with
Dirichlet boundary condition In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
s *The
particle in a one-dimensional lattice (periodic potential) In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so el ...
*The
particle in a one-dimensional lattice of finite length ''L = N a'' (''N'' is a positive integer, ''a'' is the potential period) In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
*The
Morse potential The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quant ...
* The Mie potential *The
step potential Step(s) or STEP may refer to: Common meanings * Steps, making a staircase * Walking * Dance move * Military step, or march ** Marching Arts Films and television * ''Steps'' (TV series), Hong Kong * ''Step'' (film), US, 2017 Literature * '' ...
*The linear rigid rotor *The
symmetric top The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
*The
Hooke's atom Hooke's atom, also known as harmonium or hookium, refers to an artificial helium-like atom where the Coulombic electron-nucleus interaction potential is replaced by a harmonic potential. This system is of significance as it is, for certain values ...
*The
Spherium The "spherium" model consists of two electrons trapped on the surface of a sphere of radius R. It has been used by Berry and collaborators to understand both weakly and strongly correlated systems and to suggest an "alternating" version of Hund's ...
atom *Zero range interaction in a harmonic trap *The
quantum pendulum The quantum pendulum is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an ...
*The
rectangular potential barrier In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. ...
*The
Pöschl–Teller potential In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in ter ...
*The Inverse square root potential * Multistate Landau–Zener models *The
Luttinger liquid A Luttinger liquid, or Tomonaga–Luttinger liquid, is a theoretical model describing interacting electrons (or other fermions) in a one-dimensional conductor (e.g. quantum wires such as carbon nanotubes). Such a model is necessary as the commonl ...
(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 In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...
* Quasi-exactly-solvable problems


References


Reading materials

* {{cite book , last = Mattis , first = Daniel C. , authorlink = Daniel C. Mattis , title = The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension , publisher =
World Scientific World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, along with 135 journals in various f ...
, date = 1993 , isbn = 978-981-02-0975-9 Quantum models Quantum-mechanical systems with analytical solutions