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}}. F
t,
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, F
t,
s(
x) is the state of the system at time
t and consequently applying the definition once more, F
u,
t(F
t,
s(
x)) is the state at time
u. But this is also F
u,
s(
x). In some contexts in mathematical physics, the mappings F
t,
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. 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 G
t = F
t,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 F
t,
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. ==See also==