The allowed energies and states of a quantum mechanical system are the eigenvalues and eigenvectors, respectively, of the operator representing this quantity, which in the general case can be obtained by a Legendre transformation of the Lagrangian.
The time evolution of an observable is given by its commutator with this quantity according to Ehrenfest's theorem. The exponent of the product of this quantity with the time gives the propagator in quantum mechanics. The derivative of the phase space density with respect to time is equal to minus the Poisson bracket of the density with this quantity by Liouville's Theorem. The allowed energies and states of a quantum mechanical system are the eigenvalues and eigenvectors, respectively, of the operator representing this quantity, which in the general case can be obtained by a Legendre transformation of the Lagrangian. For 10 points, identify this physical quantity central to its namesake's formulation of mechanics, which represents the sum of kinetic and potential energy in a system.
The time evolution of an observable is given by its commutator with this quantity according to Ehrenfest's theorem.
Hamiltonian