Generalization of the Schrödinger Theory of Electrons

The Schrödinger theory for a system of electrons in the presence of both a static and time‐dependent electromagnetic field is generalized so as to exhibit the intrinsic self‐consistent nature of the corresponding Schrödinger equations. This is accomplished by proving that the Hamiltonian in the stationary‐state and time‐dependent cases { H ̂ ; H ̂ ( t ) } are exactly known functionals of the corresponding wave functions { Ψ ; Ψ ( t ) } , that is,H ̂ = H ̂ [ Ψ ] andH ̂ ( t ) = H ̂ [ Ψ ( t ) ] . Thus, the Schrödinger equations may be written asH ̂ [ Ψ ] Ψ = E [ Ψ ] Ψ andH ̂ [ Ψ ( t ) ] Ψ ( t ) = i ∂ Ψ ( t ) / ∂ t . As a consequence the eiegenfunctions and energy eigenvalues { Ψ , E } of the stationary‐state equation, and the wave function Ψ ( t ) of the temporal equation, can be determined self‐consistently. The proofs are based on the “Quantal Newtonian” first and second laws which are the equations of motion for the individual electron amongst the sea of electrons in the external fields. The generalization of the Schrödinger equation in this manner leads to additional new physics. The traditional description of the Schrödinger theory of electrons with the Hamiltonians { H ̂ ; H ̂ ( t ) } known constitutes a special case. © 2017 Wiley Periodicals, Inc.