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The Hamiltonian is an operator that acts on the wave function. When we multiply the hamiltonian and the wave function, we get energy times the wave function. The wave function squared however, is the probability of electron density, or the probability of finding an electron at a certain location in an orbital. Both have something to do with finding information about an electron in a system. However, the Hamiltonian acts as an operator on the wave function in Schrodinger's equation, and wave function squared has to do with probability density.
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