Physlib.QuantumMechanics.FiniteTarget

Given a quantum mechanical system $A$ with a finite-dimensional Hilbert space $H$ and a Hamiltonian operator $\hat{H}$, the unitary time evolution is the one-parameter group of unitary operators $U(t)$ defined by $U(t) = \exp\left(-\frac{i}{\hbar} t \hat{H}\right)$, where $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant.

For a quantum mechanical system $A$ with a finite-dimensional Hilbert space and Hamiltonian operator $\hat{H}$, the generator $G$ of the unitary time evolution one-parameter group $U(t)$ is given by: $$G = \frac{1}{\hbar} \hat{H}$$ where $\hbar$ is the reduced Planck constant. This generator $G$ is the self-adjoint operator such that the time evolution is expressed as $U(t) = \exp(-itG)$.

For a quantum mechanical system $A$ with a finite-dimensional Hilbert space and Hamiltonian operator $\hat{H}$, the time evolution operator $U(t)$ at time $t \in \mathbb{R}$ is given by the exponential of the operator $-\frac{it}{\hbar}\hat{H}$, where $i$ is the imaginary unit and $\hbar$ is the reduced Planck's constant. That is: $$U(t) = \exp\left(-\frac{it}{\hbar} \hat{H}\right)$$