Local density of states near surface as operator solution of Schrodinger equation

An operator solution (OS) delta(E, H) of the Schrodinger equation is obtain ed from the Dirac delta-function mathematics. (i) In the coordinate represe ntation, it is exactly the probability density in quantum mechanics (QM). ( ii) In the representation of the eigenfunction basis, this OS is exactly th e local density of states (LDOS). (iii) In the ray space representation, th is 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 traject ory is helpful to localize the surface atom positions, say in scanning tunn eling microscopy (STM). From this trajectory picture, we find one-to-one co rrespondence between the number of trajectories and the number of eigen-equ ations. Also we try to make a compatibility between the particle and wave f unction. The dichotomy is discussed from the particle number viewpoints. Fo r 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 o pposite charges into the original charge density in the density-functional theory (DFT). This also differentiates the two mechanisms: the pure electro n 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 el ectron transfer are hole states instead of electron states. The related phy sics included are: electron and hole occupations in density of states (DOS) ; one particle representation and many-particle representation together wit h an additional 'DOS representation'; atomic stability; properties of quant um action function in 3-dimensional QM.