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Schrödinger Equation

A fundamental equation in quantum mechanics describing the time evolution of the quantum state of a physical system, crucial for quantum optics calculations.
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The statement of the theorem

Let H\mathcal{H} be a separable Hilbert space, and let ∣Ψ(t)⟩∈H|\Psi(t)\rangle \in \mathcal{H} be the state vector describing the quantum system. Define the Hamiltonian operator H^:H→H\hat{H}: \mathcal{H} \to \mathcal{H} as the generator of time evolution, such that H^=P^22m+V^(r,t)\hat{H} = \frac{\hat{P}^2}{2m} + \hat{V}(\mathbf{r}, t), where P^=−iℏ∇\hat{P} = -i\hbar\nabla is the momentum operator and V^\hat{V} is the potential energy operator. The time evolution of the state vector is governed by the equation:\niℏ∂∂t∣Ψ(t)⟩=H^∣Ψ(t)⟩\mathrm{i\hbar}\frac{\partial}{\partial t} |\Psi(t)\rangle = \hat{H} |\Psi(t)\rangle
Source: Wikipedia