Local Density of States near Surface as Operator Solution of Schrödinger Equation

Der-Ruenn Su

《Chinese Journal of Physics》 37卷1期 (1999/02) Pp. 51-88

並列摘要

An operator solution (OS) δ(E, H) of the Schrodinger equation is obtained from the Dirac &function mathematics. (i) In the coordinate representation, it is exactly the probability density in quantum mechanics (QM). (ii) In the representation of the eigenfunction basis, this OS is exactly the local density of states (LDOS). (iii) In the ray space representation, this OS gives the quantum trajectory with a distinct classical trajectory in its classical limit. These properties, including dynamics, are added to the current probability interpretation of QM. When on surfaces, the QM trajectory is helpful to localize the surface atom positions, say in scanning tunneling microscopy (STM). From this trajectory picture, we find one-to-one correspondence between the number of trajectories and the number of eigen-equations. Also we try to make a compatibility between the particle and wave function. The dichotomy is discussed from the particle number viewpoints. For the H-atom, as a system of two particles, we find a theory in which this system is 2-dimensional in classical treatments, but it is 3-dimensional in QM. For experimental necessities in electronic devices and STM, we propose our OS for resolving the negative differential conductance (NDC) phenomena by extending this OS from standard positive to include negative values for hole states by a consideration of the neutralization of electron-hole pair in electric charges. This solves a longtime problem of incorporating the opposite charges into the original charge density in the density-functional theory (DFT). This also differentiates the two mechanisms: the pure electron LDOS process and the recombination mechanism of electron-hole pairs which emit photons like the displacement current or heat inside materials. Thus the negative nature of these NDC can be considered as if final states of electron transfer are hole states instead of electron states. The related physics included are: electron and hole occupations in density of states (DOS); one particle representation and many-particle representation together with an additional 'DOS representation'; atomic stability; properties of quantum action function in 3-dimensional QM.