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2.1 Wave–particle duality and time evolution. 2.1.1 Non-relativistic time-independent Schrödinger equation. ... The Cambridge Handbook of Physics Formulas ...
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems). In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite .
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams .
The operator ^ = ^ / is known as the time-evolution operator, and it is unitary: it preserves the inner product between vectors in the Hilbert space. [13] Unitarity is a general feature of time evolution under the Schrödinger equation.
In 1991, David Deutsch proposed a method to explain how quantum systems interact with closed timelike curves (CTCs) using time evolution equations. This method aims to address paradoxes like the grandfather paradox, [10] [11] which suggests that a time traveler who stops their own birth would create a contradiction.
Another special case of the master equation is the Fokker–Planck equation which describes the time evolution of a continuous probability distribution. [3] Complicated master equations which resist analytic treatment can be cast into this form (under various approximations), by using approximation techniques such as the system size expansion .
By definition, X nm only has the frequency E n − E m / h , so its time evolution is simple: = / = / (). This is the original form of Heisenberg's equation of motion. Given two arrays X nm and P nm describing two physical quantities, Heisenberg could form a new array of the same type by combining the terms X nk P km , which also ...
Time evolution described by a time-independent Hamiltonian is represented by a one-parameter family of unitary operators, for which the Hamiltonian is a generator: () = ^ /. In the Schrödinger picture , the unitary operators are taken to act upon the system's quantum state, whereas in the Heisenberg picture , the time dependence is ...