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

2. 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

3. 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

4. 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)

5. 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)

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

7. 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)

8. 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)

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

10. Quantal Source Current Density
Current density operator:

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

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

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

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

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

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

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

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

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

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

21. Ground State Wave Function

22. Ground State Wave Function

23. Ground State Wave Function

24. Relative Coordinate Component of Wave Function

25. Densities

26. Radial Probability Densities

27. Fermi-Coulomb Holes

28. Fermi-Coulomb Holes

29. Fermi-Coulomb Holes

30. Fermi-Coulomb Holes

31. Hartree Fields

32. Pauli-Coulomb Fields

33. Electron-Interaction Fields

34. Kinetic ‘Forces’

35. Ground State Divergence of Kinetic ‘Force’

36. Ground State Kinetic Energy Density (‘Quantal Decompression’)

37. Differential Density Forces

38. Ground State ‘Quantal Newtonian’ First Law

39. Excited State ‘Quantal Newtonian’ First Law

40. 1st Excited Singlet State
Hooke’s Atom
Property
Ground State
k = 0.25
1st Excited Singlet State
k = T EH
Exc
Eee
Eext E
EN=1 I

41. 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.

42. 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

43. 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:

44. 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

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