The
Dirac
Equa-on,
Zi2erbewegung
and
the
Scalar
Rela-vis-c
Approxima-on
torey
semi
October
21,
2008
The
Free‐par-cle
Dirac
Equa-on
with
and
where
‘s Ξ the Pauli matrices;
Ξ the 2x2 identity matrix;
Ξ the rest energy of free particle (electron).
Pauli
matrices
Whence
the
Dirac
Equa-on?
•  Limita-ons
of
Schrodinger
Equa-on
•  QED
(quantum
electrodynamics)
–  a
descrip-on
of
the
interac-on
of
light
and
ma2er
–  made
possible
by
Dirac’s
transforma-on
theory
–  a
rela-vis-c
quantum
field
theory
–  highly
accurate:
to
within
12
significant
digits
QED… •  Predicts
PROBABILITY
of
experimental
outcome
•  Based
on
Lagrangian
mechanics‐>path
that
minimizes
ac-on
func-onal
•  Field
equa-ons
–  Star-ng
point
:
Lagrangian
density
–  For
free
Dirac
field:
where:
adjoint
spinor
to
Expansion
gives:
which
has
features
similar
to
the
Dirac
equa-on.
Equa-ons
of
Mo-on:
we
vary
L
with
respect
to
which
leads
to
and
,
the
Schrodinger
form
of
the
Dirac
equa-on.
The
Dirac
equa-on
provides:
•  A
descrip-on
of
rela-vis-c
systems
•  An
equa-on
whose
solu-ons
lead
to
rela-vis-c
effects
–  Rela-vis-c
mass
increase
–  Electrosta-c
&
magne-c
dipole
energies
–  Spin‐orbit
energy
–  Zi2erbewegung
(Darwin
term
of
expansion)
In
1D,
we
can
write:
using the Dirac prescription to:
•  replace time-dependence with energy
•  work with an eigenvalue equation
is a 4 x 4 matrix
=>
is a four-spinor*;
*a
‘spinor’
denotes
a
wavefunc-on
with
both
linear
(physical)
momentum
and
spin
(polariza-on)
components.
In
expanded
form,
the
equa-on
becomes:
•  equa-on
is
in
block‐matrix
form
•  reduces
to
two
(mathema-cally)
equivalent
two‐spinor
equa-ons:
and
This
result
embodies
the
posi-ve
and
nega-ve
energy
solu-ons
for
which
the
Dirac
equa-on
is
legendary..
Time‐Dependent
Numerical
Solu-ons
Rever-ng
to
the
-me‐dependent
form
of
our
coupled
system
of
equa-ons,
we
can
numerically
solve
for
the
wavefunc-ons:
gives
Numerical
Solu-ons
-me
evolu-on
of
1D
Dirac
equa-on
for
gaussian
ini-al
condi-on
displays
forward
and
backward
-me
evolu-ons
(posi-ve
and
nega-ve
energies)
exhibits
strange
oscilla-ons
and
self‐interference
effects
plane
wave
solu-ons
demonstrates
phase
differences
between
components
Zi2erbewegung;
a
Rela-vis-c
Consequence
•  look
at
equa-ons
of
mo-on
for
Dirac
equa-on
in
Heisenberg
representa-on:
where
Heisenberg
picture…leads
to:
use
canonical
commuta-on
rela-ons
on
:
get
expression
for
:
Comparison
with
Classical
Space
Operator
Classically:
Quantum
Mechanically:
One‐par-cle
Picture
Zi2erbewegung
Finer
Points
•  Caused
by
interference
of
nega-ve
and
posi-ve
energies
•  Heisenberg
velocity
operator
not
a
constant
of
the
mo-on
•  Strength
propor-onal
to
ampl
of
neg
energy
waves
of
wavepacket
• 
In
presence
of
poten-al,
represented
by
Darwin
term
Zi2erbewegung
movies
(Thaller)
Dirac
Equa-on:
Connec-on
to
RDFT
•  Scalar
Rela-vis-c
Approxima-on
–  An
approxima-on
to
the
Dirac
equa-on
for
a
par-cle
in
a
(central)
poten-al
(such
as
an
electron
near
a
nucleus)
–  Applicable
to
calcula-on
of
rela-vis-c
effects
in
atoms
and
molecules
–  Approach
is
to
discard
spin‐orbit
term
Dirac
Equa-on
in
Presence
of
Central
Poten-al
mass
term
Darwin
term
non‐rel
Schrodinger
Eqn
↓
↙↘
↑
spin‐orbit
coupling
term
↙
Scalar
Rela-vis-c
Approxima-on
Spin‐orbit
term
on
order
of
α4:
Using:
Scalar
Rela-vis-c
Approxima-on
and
we
can
write:
or
Scalar
Rela-vis-c
Approxima-on
Set
note
that
for
ground
state
hydrogen,
and
we
have:
↑
negligibly
small
term
Connec-on
to
RDFT
•  Remaining
poten-al
terms
incorporated
into
Kohn‐Sham
equa-ons
of
DFT
as
effec-ve
poten-al
•  Solu-ons
sought..
References