Physlib.QuantumMechanics.FiniteTarget.Basic

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

For a finite quantum mechanical system with an $n$-dimensional Hilbert space $H$ and a time evolution operator $U(t) = \exp\left(-\frac{i t}{\hbar} \hat{H}\right)$, given a time $t \in \mathbb{R}$ and a basis $b = \{v_1, \dots, v_n\}$ of $H$ over $\mathbb{C}$, the time evolution matrix is the $n \times n$ complex matrix representing the linear operator $U(t)$ with respect to the basis $b$.

For a finite-dimensional quantum mechanical system with an $n$-dimensional Hilbert space $H$, the time evolution matrix in the standard basis at time $t \in \mathbb{R}$ is the $n \times n$ complex matrix representing the time evolution operator $U(t) = \exp\left(-\frac{i t}{\hbar} \hat{H}\right)$ with respect to the basis $b$ canonically derived from the isomorphism $H \cong \mathbb{C}^n$.