Schrödinger Theory of the Electronic Structure of Matter from a ‘Newtonian’ Perspective Viraht Sahni

Outline 1. Ideas from classical physics Description of quantum system via Schrödinger equation 3. ‘Newtonian’ description of quantum system Examples of ‘Newtonian’ description: Ground and Excited states 5. Conclusions

Classical Physics Newton’s Second Law Newton’s First Law N particles interacting via Newton’s 3rd Law forces (Newton’s 3rd Law) Newton’s First Law

Classical Physics ‘A new concept appeared in physics, the most important invention since Newton's time: the field. It needed great scientific imagination to realize that it is not the charges nor the particles but the field in the space between the charges and particles that is essential for the description of the physical phenomenon’. Einstein and Infeld (The Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta, Simon and Schuster, New York, 1938)

Classical Physics Electric Field and Potential Energy Coulomb’s Law z′ Force: r′ - r r’ q r x′ or y′ Potential energy of test charge Provided is conservative: smooth in a simply connected region) Work done is path-independent Total Potential Energy (in Integral Virial Form)

Time-Dependent Schrödinger Theory N electrons is path-independent) (non-conserved)

Quantal Sources Electron Density ( is a local or static quantal source Density operator: Sum Rule: ( is a local or static quantal source distribution for each t)

Quantal Source Pair-Correlation Density Pair function: Pair correlation operator: Sum Rule: (for each electron position r) Fermi-Coulomb hole Sum Rule: (for each electron position r) ( and are nonlocal or dynamic quantal source distributions for each t)

Quantal Source Spinless Single-Particle ``` Density Matrix Density Matrix operator: Sum Rule: (non-idempotent)

Quantal Source Current Density Current density operator:

‘Classical’ Fields Electron-Interaction Field ‘force’ (Coulomb’s Law) or Since Hartree Field Pauli-Coulomb Field In general:

‘Classical’ Field Kinetic Field Kinetic ‘force’ Kinetic energy density tensor In general:

‘Classical’ Field Differential Density Field ‘force’ In general:

‘Classical’ Field Current Density Field Current density ‘force’: In general:

‘Quantal Newtonian’ Second Law Physical Interpretation of External Potential (conservative field) Work done is path-independent

Self-Consistent Nature of Schrödinger Equation Quantal sources Continue self-consistent procedure till Fields

Energy Components Electron-interaction Hartree Pauli-Coulomb Kinetic External (All expressions independent of whether the Fields are conservative or not)

Ehrenfest’s Theorem ‘QN’ 2nd Law Operate with

Time-Independent Schrödinger Theory N electrons ‘Quantal Newtonian’ First Law Since , work done is path-independent. Ehrenfest’s Theorem:

Examples of the ‘Newtonian’ Perspective Hooke’s Atom Ground State First Excited Singlet State all known.

Ground State Wave Function

Ground State Wave Function

Ground State Wave Function

Relative Coordinate Component of Wave Function

Densities

Radial Probability Densities

Fermi-Coulomb Holes

Fermi-Coulomb Holes

Fermi-Coulomb Holes

Fermi-Coulomb Holes

Hartree Fields

Pauli-Coulomb Fields

Electron-Interaction Fields

Kinetic ‘Forces’

Ground State Divergence of Kinetic ‘Force’

Ground State Kinetic Energy Density (‘Quantal Decompression’)

Differential Density Forces

Ground State ‘Quantal Newtonian’ First Law

Excited State ‘Quantal Newtonian’ First Law

1st Excited Singlet State Hooke’s Atom Property Ground State k = 0.25 1st Excited Singlet State k = 0.14498 T 0.664418 0.876262 EH 1.030250 0.722217 Exc -0.582807 -0.370075 Eee 0.447448 0.352142 Eext 0.888141 1.052372 E 2.000000 2.280775 EN=1 0.750000 0.570194 I -1.250000 -1.710581

Conclusion It is possible to describe Schrödinger theory of the electronic structure of matter from a ‘Newtonian’ perspective of ‘classical’ fields and quantal sources. The fields are representative of the system density, kinetic effects, and electron correlations due to the Pauli Exclusion Principle and Coulomb repulsion. The ‘Newtonian’ description is: tangible, (b) leads to further insights into the electronic structure, (c) knowledge of classical physics can be made to bear on this understanding.

Quantal Density Functional Theory N electrons Conservative external field ( path-independent) Hohenberg-Kohn Theorem Map C Map D ( nondegenerate ground state) Knowledge of uniquely determines to within a constant Since and are known, the Hamiltonian is known Solution of leads to ground and excited state Therefore, . is a basic variable of quantum mechanics

QDFT (cont’d) Interacting system in ground or excited state with density Noninteracting fermions with same and in arbitrary state QDFT Existence of noninteracting system is an assumption ( : correlations due to the Pauli exclusion principle, Coulomb repulsion, and Correlation-Kinetic effects.) Wave function: Slater determinant Density:

QDFT (cont’d) ‘QN’ First Law for Model System Dirac density matrix ‘QN’ First Law for Interacting System Local Electron-interaction Potential Energy is path-independent

QDFT (cont’d) Correlation-Kinetic Field Total Energy