The change in [closed quantum systems](Closed%20quantum%20systems.md) over time is modeled by _Unitary transformations._ This is expressed as the following postulate for describing the evolution of [closed quantum systems:](Closed%20quantum%20systems.md) ![](Postulates%20of%20Quantum%20Mechanics.md#^5b8fd6) Unitary transformations in quantum mechanics are [Linear transformations](Linear%20transformations%20in%20quantum%20mechanics.md) that are realized by [unitary operators](Unitary%20transformations%20in%20quantum%20mechanics.md#Unitary%20operators%20in%20quantum%20mechanics) where [time evolution operators](time%20evolution%20operator.md) model unitary transformations that occur within a specified time interval. ^64426c Consider a unitary operator, $\hat{U}.$ * Given a [state vector,](State%20vector.md) $|\psi\rangle,$ a [unitary transformation](Unitary%20transformations%20in%20quantum%20mechanics.md) of that state vector is expressed as $|\psi'\rangle=\hat{U}|\psi\rangle.$ * Given a [density matrix,](density%20matrix.md) $\hat{\rho},$ a [unitary transformation](Unitary%20transformations%20in%20quantum%20mechanics.md) of that density matrix is expressed as $\hat{\rho}'=\hat{U}\hat{\rho} \hat{U}^\dagger.$ # Unitary operators in quantum mechanics A [unitary operator](Unitary%20operators.md), $\hat{U}$ is a [linear operator](Linear%20transformations%20in%20quantum%20mechanics.md#Linear%20operators%20in%20quantum%20mechanics) for which [$\hat{U}^{\dagger}=\hat{U}^{-1}$](Unitary%20operators.md#^37f781) and equivalently [$\langle x|\hat{U}^{\dagger}\hat{U} |y \rangle = \langle x| y \rangle$](Unitary%20operators.md#^219d76) where here we rewrite the [definition](Unitary%20operators.md#^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.](State%20vector.md) This operator gives rise to [unitary transformations in quantum mechanics,](Unitary%20transformations%20in%20quantum%20mechanics.md) which is how evolution is modeled for [isolated quantum systems.](Closed%20quantum%20systems.md) This is also an operator that transforms between [Hilbert spaces](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md). ^7ed86c ## Construction of unitary operators from [observables](Observable.md) [Observables](Observable.md) are [hermitian operators](Hermitian%20operators.md) and thus we may express [unitary operators in quantum mechanics](Unitary%20transformations%20in%20quantum%20mechanics.md#Unitary%20operators%20in%20quantum%20mechanics) in terms of Hermitian operators. The procedure for constructing a unitary operator, $U,$ from an observable, $X,$ is as follows: ![](Unitary%20operators.md#^8e52f6) ![](Unitary%20operators.md#^932ae0) ^7f7c4e ![](Unitary%20operators.md#^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](Hamiltonian%20operator.md) governing the transformation. Thus, due to the time parameter, Unitary operators in this exponential form are referred to as [time evolution operators,](time%20evolution%20operator.md) which are expressed as ![](time%20evolution%20operator.md#^65f482) # Translation operator in quantum mechanics ![](Translation%20operator%20in%20quantum%20mechanics.md#^5ad888) ![](Translation%20operator%20in%20quantum%20mechanics.md#^9f6787) ([... see more](Translation%20operator%20in%20quantum%20mechanics.md)) # Quantum gates ![](Quantum%20gates.md#^3aa32c) ([... see more](Quantum%20gates.md)) #QuantumMechanics/FoundationsOfQuantumMechanics #QuantumMechanics/MathematicalFoundations #QuantumMechanics/QuantumDynamics