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Gordan Žitković authored notes introducing stochastic processes. Conditional distributions, filtration and time evolution connect a random-variable model at one instant with a mathematically specified process.^[ref]

Homepage: gordanz.github.io/

The equation is $i\hslash {\partial }_t\psi =[-{\hslash }^2{\partial }_x^2/(2m)+V(x)]\psi$. The generator of time evolution is the sum of kinetic and potential energy operators. For a sharp stationary energy, insert $\psi (x,t)=\varphi (x)e^{-iEt/\hslash }$ to obtain $[-{\hslash }^2{\varphi }^{\prime \prime }/(2m)+V\varphi ]=E\varphi$. The probability in $dx$ is $|\psi {|}^{2}dx$, normalized to one; hence a normalized one-dimensional wavefunction has dimensions length${}^{-1/2}$. Its overall phase is unobservable, but relative phase affects interference.

Time evolution gives $e^{-i{E}_{0}t/\hslash }[{\varphi }_0+e^{-i\omega t}{\varphi }_1]/\sqrt{2}$. It becomes the opposite relative-sign superposition when $e^{-i\omega t}=-1$, first at $\boxed{t_{\min }=\pi /\omega }$. The overlap with the initial state is proportional to $1+e^{-i\omega t}$ and vanishes then; global phase is immaterial.

To match the source's order define $[a,b]={\sum }_i(a_{p_i}{b}_{q_i}-a_{q_i}{b}_{p_i})$. The chain rule and Hamilton's equations then giveAntisymmetry implies $[H,H]=0$, hence $dH/dt=H_t$, which vanishes for a time-independent Hamiltonian. With the alternative common convention $\{a,b\}={\sum }_i(a_{q_i}{b}_{p_i}-a_{p_i}{b}_{q_i})$, the same law is $dg/dt=g_t+\{g,H\}$; mixing the two orders would reverse the physical time evolution.

A system dynamicist studies feedback and time evolution using system dynamics models. Stocks describe accumulated quantities and flows describe their rates of change, while feedback links the rates to the current system state.

For a simple stock $N$ with inflow $a$ and proportional outflow $bN$, the model is $\dot{N}=a-bN$. Its equilibrium and adjustment timescale follow from the equation. A useful model also states the evidence and assumptions behind the chosen feedback relationships.

Time evolution describes the change of a physical system's state with time. A closed quantum system has unitary time evolution; dissipative dynamics need not be unitary.