Time evolution
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Time evolution is the change of state brought about by the passage of
The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a Turing machine can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads). In this case, time is considered to be discrete steps.
Stateful systems often have dual descriptions in terms of states or in terms of observable values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in quantum mechanics where the Schrödinger picture and Heisenberg picture are (mostly)[clarification needed] equivalent descriptions of time evolution.
Time evolution operators
Consider a system with state space X for which evolution is
- .
Ft, s(x) is the state of the system at time t, whose state at time s is x. The following identity holds
To see why this is true, suppose x ∈ X is the state at time s. Then by the definition of F, Ft, s(x) is the state of the system at time t and consequently applying the definition once more, Fu, t(Ft, s(x)) is the state at time u. But this is also Fu, s(x).
In some contexts in mathematical physics, the mappings Ft, s are called propagation operators or simply
A state space with a distinguished propagator is also called a dynamical system.
To say time evolution is homogeneous means that
- for all .
In the case of a homogeneous system, the mappings Gt = Ft,0 form a one-parameter group of transformations of X, that is
For non-reversible systems, the propagation operators Ft, s are defined whenever t ≥ s and satisfy the propagation identity
- for any .
In the homogeneous case the propagators are exponentials of the Hamiltonian.
In quantum mechanics
In the Schrödinger picture, the Hamiltonian operator generates the time evolution of quantum states. If is the state of the system at time , then
This is the Schrödinger equation. Given the state at some initial time (), if is independent of time, then the unitary time evolution operator is the exponential operator as shown in the equation
See also
- Arrow of time
- Time translation symmetry
- Hamiltonian system
- Propagator
- Time evolution operator
- Hamiltonian (control theory)
References
- ^ Lecture 1 | Quantum Entanglements, Part 1 (Stanford) (video). Stanford, CA: Stanford. October 2, 2006. Retrieved September 5, 2020 – via YouTube.
General references
- Amann, H.; Arendt, W.; Neubrander, F.; Nicaise, S.; von Below, J. (2008), Amann, Herbert; Arendt, Wolfgang; Hieber, Matthias; Neubrander, Frank M; Nicaise, Serge; von Below, Joachim (eds.), Functional Analysis and Evolution Equations: The Günter Lumer Volume, Basel: Birkhäuser, MR 2402015.
- Jerome, J. W.; Polizzi, E. (2014), "Discretization of time-dependent quantum systems: real-time propagation of the evolution operator", Applicable Analysis, 93 (12): 2574–2597, S2CID 17905545.
- Lanford, O. E. (1975), "Time evolution of large classical systems", in Moser J. (ed.), Dynamical Systems, Theory and Applications, Lecture Notes in Physics, vol. 38, Berlin, Heidelberg: Springer, pp. 1–111, ISBN 978-3-540-37505-0.
- Lanford, O. E.; Lebowitz, J. L. (1975), "Time evolution and ergodic properties of harmonic systems", in Moser J. (ed.), Dynamical Systems, Theory and Applications, Lecture Notes in Physics, vol. 38, Berlin, Heidelberg: Springer, pp. 144–177, ISBN 978-3-540-37505-0.
- MR 1286099.