A Note on Physical Interpretation of Quantum-Mechanical Orthogonality among Electronic Eigenstates

According to the principle in quantum mechanics, any energy eigenstate of an electron in a quantum-mechanical electron system should be orthognal to all of the other eigenstates. Quantum mechanics explains this property from the fact that the system Hamiltonian, the energy operator of the system, is mathematically self-adjoint operator. The explanation is, however, too mathematical and far from giving a physical image of the orthogonality. As shown previously by the author, the orthogonality between the valence and the core electron eigenstates in semiconductor gives many interesting physical effects in distributions of charge and momentum, in gamma-ray Compton scattering, in electron-positron pair annihilation and so on. In an aspect of theoretical treatment, these effects result from the atom-like local oscillation introduced into the valence electron wave functions through the mathematical core-orthogonalization process. However, these can be understood more physically by the fact that, in the vicinity of core region where the crystal potential field is nearly atom-like, the valence electron should behave as if it were partially an original outermost-shell electron with the nodal oscillation proper to the shell. This explanation gives us an example of physical interpretation of the abstract concept of orthogonalty and may he helpful for beginner students in quantum mechanics.