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Hamiltonian Operator

The Hamiltonian operator, H = \sum_{i} \hat{x}_i \frac{\partial}{\partial x_i} + V(\vec{x}), represents the total energy of the system and governs its time evolution.
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The statement of the theorem

Let H\mathcal{H} be a Hilbert space defined over the configuration space x∈Rn\mathbf{x} \in \mathbb{R}^n. Define the Hamiltonian operator H^:D→H\hat{H}: \mathcal{D} \rightarrow \mathcal{H} acting on a state vector ∣ψ⟩∈D⊂H|\psi\rangle \in \mathcal{D} \subset \mathcal{H} as: H^=12m∑i=1np^i2+V(x)⋅I \hat{H} = \frac{1}{2m} \sum_{i=1}^{n} \hat{p}_i^2 + V(\mathbf{x}) \cdot \mathbf{I} where p^i=−iℏ∂∂xi\hat{p}_i = -i\hbar \frac{\partial}{\partial x_i} is the canonical momentum operator, and V(x)V(\mathbf{x}) is the potential energy function. The time evolution of the state is governed by the Schrödinger equation: iℏ∂∂t∣ψ(t)⟩=H^∣ψ(t)⟩ i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle Furthermore, H^\hat{H} is self-adjoint, ensuring that the expectation value of the energy, ⟨H^⟩=⟨ψ∣H^∣ψ⟩\langle\hat{H}\rangle = \langle\psi|\hat{H}|\psi\rangle, is real.
Source: Wikipedia