Abstract
The Schrödinger theory of electrons in an external electromagnetic field is described from the new perspective of the individual electron. The perspective is arrived at via the time-dependent “Quantal Newtonian” law (or differential virial theorem). (The time-independent law, a special case, provides a similar description of stationary-state theory). These laws are in terms of “classical” fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. The laws reveal the following physics: (a) in addition to the external field, each electron experiences an internal field whose components are representative of a specific property of the system such as the correlations due to the Pauli exclusion principle and Coulomb repulsion, the electron density, kinetic effects, and an internal magnetic field component. The response of the electron is described by the current density field; (b) the scalar potential energy of an electron is the work done in a conservative field. It is thus path-independent. The conservative field is the sum of the internal and Lorentz fields. Hence, the potential is inherently related to the properties of the system, and its constituent property-related components known. As the sources of the fields are functionals of the wave function, so are the respective fields, and, therefore, the scalar potential is a known functional of the wave function; (c) as such, the system Hamiltonian is a known functional of the wave function. This reveals the intrinsic self-consistent nature of the Schrödinger equation, thereby providing a path for the determination of the exact wave functions and energies of the system; (d) with the Schrödinger equation written in self-consistent form, the Hamiltonian now admits via the Lorentz field a new term that explicitly involves the external magnetic field. The new understandings are explicated for the stationary state case by application to two quantum dots in a magnetostatic field, one in a ground state and the other in an excited state. For the time-dependent case, the evolution of the same states of the quantum dots in both a magnetostatic and a time-dependent electric field is described. In each case, the satisfaction of the corresponding “Quantal Newtonian” law is demonstrated.
Highlights
In this paper, we explain new understandings [1] of Schrödinger theory of the electronic structure of matter, and of the interaction of matter with external static and time-dependent electromagnetic fields
As the sources of the fields are functionals of the wave function, so are the respective fields, and, the scalar potential is a known functional of the wave function; (c) as such, the system Hamiltonian is a known functional of the wave function
This reveals the intrinsic self-consistent nature of the Schrödinger equation, thereby providing a path for the determination of the exact wave functions and energies of the system; (d) with the Schrödinger equation written in self-consistent form, the Hamiltonian admits via the Lorentz field a new term that explicitly involves the external magnetic field
Summary
We explain new understandings [1] of Schrödinger theory of the electronic structure of matter, and of the interaction of matter with external static and time-dependent electromagnetic fields. The eigenfunctions Ψ and eigenenergies E of the Schrödinger equation can be obtained self-consistently This form of eigenvalue equation is mathematically akin to that of Hartree–Fock and Hartree theories in which the corresponding Hamiltonian Ĥ HF is a functional of the single particle orbitals φi of the Slater determinant wave function. In the time-dependent case, it is shown via the “Quantal Newtonian” second law that the Hamiltonian Ĥ (t) = Ĥ [Ψ(t)], so that the self-consistent form of the Schrödinger equation is Ĥ [Ψ(t)]Ψ(t) = i∂Ψ(t)/∂t. In this instance, it is the evolution of the many-electron time-dependent wave function Ψ(t) that is obtained self-consistently. Constrained-search variational methods [24,25,26] for the determination of the wave function
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