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The Klein–Gordon equation, + =, was the first such equation to be obtained, even before the nonrelativistic one-particle Schrödinger equation, and applies to massive spinless particles. Historically, Dirac obtained the Dirac equation by seeking a differential equation that would be first-order in both time and space, a desirable property for ...
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.
This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system. An example is a qubit used in quantum information processing.
In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known.
Hydrogen atomic orbitals of different energy levels. The more opaque areas are where one is most likely to find an electron at any given time. In quantum mechanics, a spherically symmetric potential is a system of which the potential only depends on the radial distance from the spherical center and a location in space.
The nonlinear Schrödinger equation is a simplified 1+1-dimensional form of the Ginzburg–Landau equation introduced in 1950 in their work on superconductivity, and was written down explicitly by R. Y. Chiao, E. Garmire, and C. H. Townes (1964, equation (5)) in their study of optical beams.
In some cases, the Schrödinger equation can be solved analytically on a one-dimensional lattice of finite length [5] [6] using the theory of periodic differential equations. [7] The length of the lattice is assumed to be L = N a {\displaystyle L=Na} , where a {\displaystyle a} is the potential period and the number of periods N {\displaystyle ...
In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves.The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension.