Time evolution operator

A time evolution operator $Φ 𝑡$ on some smooth manifold $𝑀$ is a (not necessarily linear) operator describing the evolution of some point $𝜓 (𝑡) \in 𝑀$ of a dynamical system. #m/def/dynamics In the case where time evolution is governed by a differential equation

𝑑 𝑑 𝑡 𝜓 = 𝑄 (𝑡, 𝜓)

where $𝑄$ is a time-dependent vector field, then $Φ 𝑡$ is the vector flow along $𝑄$ so given any initial condition $𝜓 0 \in 𝑀$ a solution is $𝜓 (𝑡) = Φ 𝑡 (𝜓 0)$ . In general $Φ 𝑡$ has the semigroup property $Φ 𝑡 Φ 𝑠 = Φ 𝑡 + 𝑠$ and thus $(Φ 𝑡) 𝑛 = Φ 𝑛 𝑡$ .

Properties

In the case where $𝑄 (𝑡, | 𝜓 ⟩) = ˆ 𝑄 | 𝜓 ⟩$ is given by a linear operator with $𝑑 𝑑 𝑡 | 𝜓 ⟩ = ˆ 𝑄 | 𝜓 ⟩$ , time evolution is given by the matrix exponential $Φ 1 = 𝑒 ˆ 𝑄$ . See Quantum time evolution operator.

#state/develop | #lang/en | #SemBr