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Numerical Methods in Quantum Mechanics: Analysis of Numerical Schemes on OneDimensional...
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Numerical Methods in Quantum Mechanics: Analysis of Numerical Schemes on OneDimensional Schrodinger Wave Problems
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422.pdf
Description
Title
Numerical
Methods
in
Quantum
Mechanics
:
Analysis
of
Numerical
Schemes
on
OneDimensional
Schrodinger
Wave
Problems
Subject
Schrödinger
equation
;
Finite
differences
;
Quantum
theory
;
Wave
functions
;
Blackbody
radiation
;
Photoemission
;
Harmonic
oscillators
;
Topology
;
Physics
Description
The
motion
and
behavior
of
quantum
processes
can
be
described
by the
Schrodinger
equation
using
the
wave
function
,
(x
;
t)
. The
use
of the
Schrodinger
equation
to
study
quantum
phenomena
is
known
as
Quantum
Mechanics
,
akin
to
classical
mechanics
being
the
tool
to
study
classical
physics
. This
research
focuses
on the
emphasis
of
numerical
techniques
:
FiniteDierence
,
Fast
Fourier
Transform
(spectral
method)
,
nite
dierence
schemes
such
as the
Leapfrog
method
and the
CrankNicolson
scheme
and
second
quantization
to
solve
and
analyze
the
Schrodinger
equation
for the
innite
square
well
problem
, the
free
particle
with
periodic
boundary
conditions
, the
barrier
problem
,
tightbinding
hamiltonians
and a
potential
wall
problem
.
We
discuss
these
techniques
and the
problems
created
to
test
how these
dierent
techniques
draw
both
physical
and
numerical
conclusions
in a
tabular
summary
.
We
observed
both
numerical
stability
and
quantum
stability
(conservation
of
energy
,
probability
,
momentum
,
etc.)
.
We
found
in
our
results
that the
CrankNicolson
scheme
is
an
unconditionally
stable
scheme
and
conserves
probability
(unitary)
, and
momentum
,
though
dissipative
with
energy
. The
timeindependent
problems
conserved
energy
,
momentum
and were
unitary
,
which
is
of
interest
, but
we
found
when
timedependence
was
introduced
,
quantum
stability
(i.e
.
conservation
of
mass
,
momentum
,
etc.)
was not
implied
by
numerical
stability
.
Hence
,
we
observed
schemes
that were
numerically
stable
, but not
quantum
stable
as
well
as
schemes
that were
quantum
stable
, but not
numerically
stable
for
all
of
time
,
t
.
We
also
observed
that
second
quantization
removed
the
issues
with
stability
as the
problem
was
transformed
into a
discrete
problem
.
Moreover
,
all
quantum
information
is
conserved
in
second
quantization
. This
method
,
however
,
does
not
work
universally
for
all
problems
Creator
Jones
,
Jr.
,
Marvin
Quenten
Publisher
North Carolina Agricultural and Technical State University
Date
2013
Type
Text
Format
PDF
Language
English
Major Professor
Clemence
,
Dominic
P
.
Academic Department
Mathematics
Degree
MASTER
OF
SCIENCE
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