4 - Time-independent quantum mechanics  pp. 35-52

Time-independent quantum mechanics

By Robert K. Nesbet

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In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons.

The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N-electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.