In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?
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The Hamiltonian in a conservative system describes the total internal energy of the system. The formula $H=T+V$ with the traditional form of the kinetic energy is valid for a frictionless nonrelativistic system in Cartesian coordinates, possibly with external forces. |
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The Hamiltonian is a constant of the motion as long as it is independent of time. More deeply that means that the Lagrangian it comes from must be independent of time. A constant Hamiltonian is the total energy if the potential is velocity independent. |
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By definition, the Hamiltonian is always related to the total energy of the system via $\langle E \rangle = \mathrm{Tr}\{ H \rho \}$. But depending on what do you mean by $T$ and $V$ you equation may or may not be general. |
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