E*Phi(r)==H*Phi(r). E is the allowed energy levels that a particle may have; H is the Hamiltonian operator for the system; and Phi(r) is the wave function of the particle at position r. This can only be used when time is held constant (for determining an initial condition, for example)

The Hamiltonian is the operator which tells you how to move a quantum mechanical state ahead in time. The "Time-independent Schrödinger Equation" is really a search for the eigenvalues and eigenvectors of the Hamiltonian since these tell you how any given state vector evolves in time.

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