If Ψ is the wavefunction for a physical system at an initial time and the system is free of external interactions, then the evolution in time of the wavefunction is given by
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where H is the Hamiltonian operator formed from the classical Hamiltonian by substituting for the classical observables their corresponding quantum mechanical operators. For a mechanical system, the classical Hamiltonian would be just the kinetic energy plus the potential energy, i.e., the expression for energy. The role of the Hamiltonian in both space and time is contained in the Schrodinger equation.
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