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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 .
For the sake of pedagogy, the Heisenberg picture is introduced here from the subsequent, but more familiar, Schrödinger picture. According to Schrödinger's equation, the quantum state at time is | = | , where () = is the time-evolution operator induced by a Hamiltonian () that could depend on time, and | is the initial state.
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 ...
If the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with ,, leaving , time-independent. We proceed assuming that this is the case.
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.
Within the field of social evolution, Hamiltonian spite is a term for spiteful behaviors occurring among conspecifics that have a cost for the actor and a negative impact upon the recipient. Theories on altruism and spitefulness
Thus, the time evolution of a function on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian.
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory.