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Schrödinger Theory of the
Electronic Structure of
Matter from a
‘Newtonian’ Perspective
Viraht Sahni
Outline
1. Ideas from classical physics
2. Description of quantum system via
Schrödinger equation
3. ‘Newtonian’ description of quantum
system
4. Examples of ‘Newtonian’ description:
Ground and Excited states
5. Conclusions
Classical Physics
Newton’s Second 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
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
Electron-Interaction Field
Electron-interaction
‘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
Differential density
‘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
Fields
Continue self-consistent
procedure till
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
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:
(a) 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 D
Map C
(
nondegenerate ground state)
Knowledge of
uniquely determines
within a constant
to
Since
is known
and
Solution of
state
Therefore,
are known, the Hamiltonian
leads to ground and excited
.
is a basic variable of quantum mechanics
QDFT (cont’d)
Interacting system in
ground or excited state
with density
QDFT
Noninteracting fermions
with same
and
in arbitrary state
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