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Hamiltonian Operator

Represents the total energy of a quantum mechanical system, including kinetic and potential energy terms.
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The statement of the theorem

Define the Hamiltonian operator H^\hat{H} for a system of particles with mass mkm_k and position rk\mathbf{r}_k as the sum of the kinetic energy operator T^\hat{T} and the potential energy operator V^\hat{V}: \nH^=T^+V^\hat{H} = \hat{T} + \hat{V} \nwhere T^=∑kpk22mk\hat{T} = \sum_{k} \frac{\mathbf{p}_k^2}{2m_k} and V^=∑k<le24πϵ0rkl+Vext(r1,…,rN)\hat{V} = \sum_{k<l} \frac{e^2}{4\pi\epsilon_0\mathbf{r}_{kl}} + V_{ext}(\mathbf{r}_1, \dots, \mathbf{r}_N). Here, pk\mathbf{p}_k is the momentum operator, pk=−iℏ∇k\mathbf{p}_k = -i\hbar\nabla_k.
Source: Wikipedia