Quantum Time Evolution

given the quantum state of a quantum system at initial time, the task is to predict the quantum state at later time

List of mathematical expressions with quantities

Schrödinger Equation (Time Dependent)	$i ℏ \frac{\partial}{\partial t} \| ψ (t) ⟩ = \hat{H} \| ψ (t) ⟩$
$H$ symbol represents Quantum Hamiltonian Operator
$ℏ$ symbol represents Planck Constant
$ψ (t)$ symbol represents Quantum State Vector (Dynamic)
$t$ symbol represents Time
$i$ symbol represents Imaginary Unit

Quantum Lindblad Equation	$\frac{d}{d t} ρ = - \frac{i}{ℏ} [H, ρ] + \sum_{k = 1}^{N^{2} - 1} γ_{k} (L_{k} ρ L_{k}^{†} - \frac{1}{2} [L_{k}^{†} L_{k}, ρ]_{+})$
$H$ symbol represents Quantum Hamiltonian Operator
$L$ symbol represents Quantum Jump Operator
$γ > 0$ symbol represents Quantum Damping Rate
$ℏ$ symbol represents Planck Constant
$ρ$ symbol represents Quantum Density Operator
$t$ symbol represents Time
$i$ symbol represents Imaginary Unit

Quantum Hamiltonian (Electric Charge)	$H = H_{0} + q ϕ$
$H_{0}$ symbol represents Quantum Hamiltonian Operator
$ϕ$ symbol represents Electric Potential
$q$ symbol represents Electric Charge
$ϕ$ symbol represents Electric Potential

Expectation Value (Quantum Density)	$⟨ \hat{O} ⟩ \equiv t r (\hat{O} \hat{ρ})$
$ρ$ symbol represents Quantum Density Operator
$\hat{O}$ symbol represents Quantum Mechanical Operator
$⟨ \hat{O} ⟩$ symbol represents Expectation Value (Quantum Density)

Expectation Value (Quantum State)	$⟨ \hat{O} ⟩ \equiv ⟨ ψ \| \hat{O} \| ψ ⟩$
$\hat{O}$ symbol represents Quantum Mechanical Operator
$⟨ \hat{O} ⟩$ symbol represents Expectation Value (Quantum State)
$ψ$ symbol represents Quantum State Vector

Quantum Hamiltonian (Electric Dipole)	$H = H_{0} - μ \cdot ℰ$
$H$ symbol represents Quantum Hamiltonian Operator
$H_{0}$ symbol represents Quantum Hamiltonian Operator
$ℰ$ symbol represents Electric Field
$μ$ symbol represents Electric Dipole Moment