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Computation 2017, 5(1), 15; doi:10.3390/computation5010015

Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives

1
The Graduate School of the City University of New York, New York, NY 10016, USA
2
Department of Physics, Ningbo University, Ningbo 315211, China
*
Author to whom correspondence should be addressed.
Academic Editor: Jianmin Tao
Received: 20 January 2017 / Revised: 15 February 2017 / Accepted: 1 March 2017 / Published: 9 March 2017
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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. View Full-Text
Keywords: Schrödinger equation for electrons; self-consistency; “Quantal Newtonian” laws Schrödinger equation for electrons; self-consistency; “Quantal Newtonian” laws
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Sahni, V.; Pan, X.-Y. Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives. Computation 2017, 5, 15.

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