Time evolution

Time evolution operators

Consider a system with state space X for which evolution is deterministic and reversible. For concreteness let us also suppose time is a parameter that ranges over the set of real numbers R. Then time evolution is given by a family of bijective state transformations

:(\operatorname{F}{t, s} \colon X \rightarrow X){s, t \in \mathbb{R}}.

Ft, s(x) is the state of the system at time t, whose state at time s is x. The following identity holds

: \operatorname{F}{u, t} (\operatorname{F}{t, s} (x)) = \operatorname{F}_{u, s}(x).

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 propagators. In classical mechanics, the propagators are functions that operate on the phase space of a physical system. In quantum mechanics, the propagators are usually unitary operators on a Hilbert space. The propagators can be expressed as time-ordered exponentials of the integrated Hamiltonian. The asymptotic properties of time evolution are given by the scattering matrix.{{cite AV media | people = | date =October 2, 2006 | title =Lecture 1 Quantum Entanglements, Part 1 (Stanford) | medium =video | url =https://www.youtube.com/watch?v=0Eeuqh9QfNI&t=4421 | access-date =September 5, 2020 | archive-url = | archive-date = | format = | time = | location =Stanford, CA | publisher =Stanford | via=YouTube | quote =}}

A state space with a distinguished propagator is also called a dynamical system.

To say time evolution is homogeneous means that

: \operatorname{F}{u, t} = \operatorname{F}{u - t,0} for all u,t \in \mathbb{R}.

In the case of a homogeneous system, the mappings Gt = Ft,0 form a one-parameter group of transformations of X, that is

: \operatorname{G}{t+s} = \operatorname{G}{t}\operatorname{G}_{s}.

For non-reversible systems, the propagation operators Ft, s are defined whenever t ≥ s and satisfy the propagation identity

: \operatorname{F}{u, t} (\operatorname{F}{t, s} (x)) = \operatorname{F}_{u, s}(x) for any u \geq t \geq s.

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 \left| \psi (t) \right\rangle is the state of the system at time t, then

: H \left| \psi (t) \right\rangle = i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle.

This is the Schrödinger equation.

Time-independent Hamiltonian

If H is independent of time, then a state at some initial time (t = 0) can be expressed using the unitary time evolution operator U(t) is the exponential operator as

: \left| \psi (t) \right\rangle = U(t)\left| \psi (0) \right\rangle = e^{-iHt/\hbar} \left| \psi (0) \right\rangle,

or more generally, for some initial time t_0

: \left| \psi (t) \right\rangle = U(t, t_0)\left| \psi (t_0) \right\rangle = e^{-iH(t-t_0)/\hbar} \left| \psi (t_0) \right\rangle.

References

General references

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References

1. (2020). "Quantum mechanics". Wiley-VCH Verlag GmbH & Co.