In quantum mechanics we postulate that,   The time evolution operator is a [unitary operator](Unitary%20transformations%20in%20quantum%20mechanics.md#Unitary%20operators%20in%20quantum%20mechanics), that, when acting on a state vector, $|\psi(0)\rangle$ at some initial time $t=0$ gives the state vector at some later time $t$. What this looks like exactly is shown [below](time%20evolution%20operators.md#State%20vector%20transformations). Equivalently, it can also be applied to time dependent [observables](Observable.md) if we are working in the [Heisenberg picture.](time%20evolution%20operators.md#Time%20evolution%20in%20systems%20with%20time%20dependent%20Hamiltonians) For a given time independent [Hamiltonian operator](Hamiltonian%20operator.md) it takes the form of a [complex matrix exponential](Complex%20matrix%20exponentials.md)$\hat{U}(t)=e^{-i(\hat{H}/\hbar)t}$ ^65f482 where for a short time, $dt \rightarrow 0.$ $\hat{U}(dt)=1-i\hat{H}dt$ where $U^\dagger U = 1 + O(dt^2)$. This exponential form comes about from the [construction of unitary operators from observables](Unitary%20transformations%20in%20quantum%20mechanics.md#Construction%20of%20unitary%20operators%20from%20observables%20Observable%20md)  # Why unitary operators model time evolution In quantum mechanics the evolution of a [state vector](State%20vector.md) in time is modeled by a transformation to another state vector and both the initial and final state vector must be [normalized](state%20vector%20normalization.md). Note also that state vectors belong to [Hilbert spaces.](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md) The types of operator that preserves normalizations in Hilbert spaces are [unitary operators](Unitary%20transformations%20in%20quantum%20mechanics.md#Unitary%20operators%20in%20quantum%20mechanics). Thus we must use unitary operators to model time evolution. %% You may need to write about Unitary operators a bit in the Hilbert Space entry. You may need to link to an entry on Hilbert Space operators. Figure out a good way of explaining this length preserving quality. This again might have to be discussed in the section on operators in Hilbert Spaces and Hilbert spaces in quantum mechanics.%% # How quantum systems transform under time evolution operators ## State vector transformations  %%This is very simple and practically trivial - at some point you just explain that a time evolution is just U multiplied by a state vector - but next step is to explain also time order perhaps.%% %%perhaps also specify when referring to evolution in closed or open quantum systems.%% ## Density matrix transformations , thus, the time evolution of a [density matrix](density%20matrix.md) is expressed as $\hat{\rho}(t)= e^{-i(\hat{H}/\hbar)t}\hat{\rho} e^{i(\hat{H}/\hbar)t}$ under the [Schrödinger picture,](Schrödinger%20picture) where we take $\hat{H}$ to be time independent along with $\hat{\rho}$. ^9fddeb Under the [Heisenberg picture](Heisenberg%20picture) the [density matrix](density%20matrix.md) is treated as a constant and thus does not transform under a [time evolution operator.](time%20evolution%20operators.md) # Derivations of the time evolution operator from equations of motion We may derive the form of the [time evolution operators](time%20evolution%20operators.md) from any of the [quantum mechanical equations of motions](Equations%20of%20motion%20in%20quantum%20mechanics.md) in accordance with our [3rd postulate of quantum mechanics.](Postulates%20of%20Quantum%20Mechanics.md#^5b8fd6) ## Derivation of the time evolution operator from the Schrödinger equation One way of arriving at $U^\dagger(t)$ is from taking the time dependent [Schrödinger equation](Schrödinger%20Equation.md) and re-arrange it to find $i\hbar\frac{d}{dt}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle \rightarrow \frac{d}{dt}|\psi(t)\rangle = \frac{-i}{\hbar}\hat{H}|\psi(t)\rangle\rightarrow|\psi(t+dt)\rangle = (1-i\hat{H}dt)|\psi(t)\rangle$ Repeatedly applying $(1-i\hat{H}dt)$ to a state vector gives the exponential form: $\hat{U}(dt)=(1-i\hat{H}dt)^n|\psi(0)\rangle \rightarrow e^{-i\hat{H}/\hbar}|\psi(0)\rangle.$ # Time evolution in systems with time dependent Hamiltonians ([... see more](time%20evolution%20operators%20with%20time%20dependent%20Hamiltonians.md)) #QuantumMechanics/QuantumDynamics