Separating space and time variables

The time-dependent Schrddinger equation involves differentiation with respect to both time and position, the latter contained in the kinetic energy of the Hamiltonian operator. 

For (bound) systems where the potential energy operator is time-independent (V(r,t) = V(r)), the Hamiltonian operator becomes time-independent and yields the total energy when acting on the wave function. The energy is a constant, independent of time, but depends on the space variables. 

Inserting this in the time-dependent Schrodinger equation shows that the time and space variables of the wave function can be separated. 

The latter follows from solving the first-order differential equation with respect to time, and shows that the time dependence can be written as a simple phase factor multiplied with the spatial wave function. For time-independent problems, this phase factor is normally neglected, and the starting point is taken as the time-independent Schrodinger equation. 

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