The change in [closed quantum systems]( over time is modeled by _Unitary transformations._ This is expressed as the following postulate for describing the evolution of [closed quantum systems:]( ![](^5b8fd6) Unitary transformations in quantum mechanics are [Linear transformations]( that are realized by [unitary operators]( where [time evolution operators]( model unitary transformations that occur within a specified time interval. ^64426c Consider a unitary operator, $\hat{U}.$ * Given a [state vector,]( $|\psi\rangle,$ a [unitary transformation]( of that state vector is expressed as $|\psi'\rangle=\hat{U}|\psi\rangle.$ * Given a [density matrix,]( $\hat{\rho},$ a [unitary transformation]( of that density matrix is expressed as $\hat{\rho}'=\hat{U}\hat{\rho} \hat{U}^\dagger.$ # Unitary operators in quantum mechanics A [unitary operator](, $\hat{U}$ is a [linear operator]( for which [$\hat{U}^{\dagger}=\hat{U}^{-1}$](^37f781) and equivalently [$\langle x|\hat{U}^{\dagger}\hat{U} |y \rangle = \langle x| y \rangle$](^219d76) where here we rewrite the [definition](^219d76) in [Bra-ket notation.](Quantum%20Mechanics%20(index).md#Bra-ket%20notation) This notation also makes it clear that $\hat{U}^{\dagger}\hat{U}=1$ and $|x\rangle$ and $|y\rangle$ are some arbitrary pair of [state vectors.]( This operator gives rise to [unitary transformations in quantum mechanics,]( which is how evolution is modeled for [isolated quantum systems.]( This is also an operator that transforms between [Hilbert spaces]( ^7ed86c ## Construction of unitary operators from [observables]( [Observables]( are [hermitian operators]( and thus we may express [unitary operators in quantum mechanics]( in terms of Hermitian operators. The procedure for constructing a unitary operator, $U,$ from an observable, $X,$ is as follows: ![](^8e52f6) ![](^932ae0) ^7f7c4e ![](^7dab13) In physical realizations of unitary transformations the parameter $t$ is some elapsed time, $t_2-t_1,$ where $t_1$ and $t_2$ are start and end points for that unitary transformation. In addition, $-\hbar X$ is a [Hamiltonian operator]( governing the transformation. Thus, due to the time parameter, Unitary operators in this exponential form are referred to as [time evolution operators,]( which are expressed as ![](^65f482) # Translation operator in quantum mechanics ![](^5ad888) ![](^9f6787) ([... see more]( # Quantum gates ![](^3aa32c)([... see more]( #QuantumMechanics/FoundationsOfQuantumMechanics #QuantumMechanics/MathematicalFoundations #QuantumMechanics/QuantumDynamics