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This book contains an extensive illustration of use of finite
difference method in solving the boundary value problem
numerically. A wide class of differential equations has been
numerically solved in this book. Starting with differential
equations of elementary functions like hyperbolic, sine and cosine,
we have solved those of special functions like Hermite, Laguerre
and Legendre. Those of Airy function, of stationary localised
wavepacket, of the quantum mechanical problem of a particle in a 1D
box, and the polar equation of motion under gravitational
interaction have also been solved. Mathematica 6.0 has been used to
solve the system of linear equations that we encountered and to
plot the numerical data. Comparison with known analytic solutions
showed nearly perfect agreement in every case. On reading this
book, readers will become adept in using the method.
The book contains a detailed account of numerical solutions of
differential equations of elementary problems of Physics using
Euler and 2nd order Runge-Kutta methods and Mathematica 6.0. The
problems are motion under constant force (free fall), motion under
Hooke's law force (simple harmonic motion), motion under
combination of Hooke's law force and a velocity dependent damping
force (damped harmonic motion) and radioactive decay law. Also
included are uses of Mathematica in dealing with complex numbers,
in solving system of linear equations, in carrying out
differentiation and integration, and in dealing with matrices.
The book presents in comprehensive detail numerical solutions to
boundary value problems of a number of non-linear differential
equations. Replacing derivatives by finite difference
approximations in these differential equations leads to a system of
non-linear algebraic equations which we have solved using Newton's
iterative method. In each case, we have also obtained Euler
solutions and ascertained that the iterations converge to Euler
solutions. We find that, except for the boundary values, initial
values of the 1st iteration need not be anything close to the final
convergent values of the numerical solution. Programs in
Mathematica 6.0 were written to obtain the numerical solutions.
This book is intended for undergraduate students of Mathematics,
Statistics, and Physics who know nothing about Monte Carlo Methods
but wish to know how they work. All treatments have been done as
much manually as is practicable. The treatments are deliberately
manual to let the readers get the real feel of how Monte Carlo
Methods work. Definite integrals of a total of five functions ( ),
namely Sin( ), Cos( ), e , loge( ), and 1/(1+ 2), have been
evaluated using constant, linear, Gaussian, and exponential
probability density functions ( ). It is shown that results agree
with known exact values better if ( ) is proportional to ( ).
Deviation from the proportionality results in worse agreement. This
book is on Monte Carlo Methods which are numerical methods for
Computational Physics. These are parts of a syllabus for
undergraduate students of Mathematics and Physics for the course
titled "Computational Physics." Need for the book: Besides the
three referenced books, this is the only book that teaches how
basic Monte Carlo methods work. This book is much more explicit and
easier to follow than the three referenced books. The two chapters
on the Variational Quantum Monte Carlo method are additional
contributions of the book. Pedagogical features: After a thorough
acquaintance with background knowledge in Chapter 1, five
thoroughly worked out examples on how to carry out Monte Carlo
integration is included in Chapter 2. Moreover, the book contains
two chapters on the Variational Quantum Monte Carlo method applied
to a simple harmonic oscillator and a hydrogen atom. The book is a
good read; it is intended to make readers adept at using the
method. The book is intended to aid in hands-on learning of the
Monte Carlo methods.
This book contains an extensive illustration of use of finite
difference method in solving the boundary value problem
numerically. A wide class of differential equations has been
numerically solved in this book. Starting with differential
equations of elementary functions like hyperbolic, sine and cosine,
we have solved those of special functions like Hermite, Laguerre
and Legendre. Those of Airy function, of stationary localised
wavepacket, of the quantum mechanical problem of a particle in a 1D
box, and the polar equation of motion under gravitational
interaction have also been solved. Mathematica 6.0 has been used to
solve the system of linear equations that we encountered and to
plot the numerical data. Comparison with known analytic solutions
showed nearly perfect agreement in every case. On reading this
book, readers will become adept in using the method.
The book contains a detailed account of numerical solutions of
differential equations of a number of elementary problems of
physics using Euler and second order Runge-Kutta methods using
Mathematica 6.0. The problems are motion under constant force (free
fall), motion under Hooke's law force (simple harmonic motion),
motion under combination of Hooke's law force and a velocity
dependent damping force (damped harmonic motion) and radioactive
decay law. Also included are uses of Mathematica in dealing with
complex numbers, in solving system of linear equations, in carrying
out differentiation and integration, and in dealing with matrices.
Containing an extensive illustration of the use of finite
difference method in solving boundary value problem numerically, a
wide class of differential equations have been numerically solved
in this book.
The book contains a detailed account of numerical solutions of
differential equations of elementary problems of Physics using
Euler and 2nd order Runge-Kutta methods and Mathematica 6.0. The
problems are motion under constant force (free fall), motion under
Hooke's law force (simple harmonic motion), motion under
combination of Hooke's law force and a velocity dependent damping
force (damped harmonic motion) and radioactive decay law. Also
included are uses of Mathematica in dealing with complex numbers,
in solving system of linear equations, in carrying out
differentiation and integration, and in dealing with matrices.
Master's Thesis from the year 2013 in the subject Physics - Quantum
Physics, grade: -, Shahjalal University of Science and Technology
(Department of Physics), course: Nanostructure Physics, language:
English, abstract: We have numerically investigated parametric
variations of transmission peaks of symmetric rectangular double
barrier in non-tunneling regime. We have compared the variations
with those for tunneling regime. One of the three variations in
non-tunneling regime is completely different from that for
tunneling regime warranting rapid dissemination. The book contains
background on Quantum Mechanics, Microelectronics and Nanostructure
Physics to enable readers assimilate the book completely.
Master's Thesis from the year 2011 in the subject Physics - Quantum
Physics, grade: -, Shahjalal University of Science and Technology
(Department of Physics), course: Nanostructure Physics, language:
English, abstract: This book contains a comprehensive account of
application of WKB method to pure Physics of nanostructures
containing single or symmetric double barrier V(x) in their band
model in presence of longitudinal magnetic field applied along x
direction. It concentrates on effects on transmission coefficient
of single and symmetric double barriers by three dimensional
electron gas (3DEG). Analytical expressions for longitudinal
magnetic field dependent transmission coefficient of single and
symmetric double barrier of general shape are obtained first. These
general expressions are then used to obtain analytical expressions
of longitudinal magnetic field dependent transmission coefficient
of single and symmetric double barriers of many different shapes we
encounter in studying nanostructure Physics. This is followed by
thorough numerical investigation to bring out effects of
longitudinal magnetic field on transmission coefficient of all
these barriers. Comparisons with standard results where available
showed excellent agreements. Results of numerical investigation
have been explained completely. The book makes well documented,
with thorough calculation and discussion, pure Physics of
semiconductor nanostructures.
We report strong, amplitude modulated, oscillatory resonant
transmission of electron through non-tunneling regime (E > V0)
of the nanostructure called symmetric double barrier. We present
thorough and complete analytical calculation and thorough numerical
investigations. Amplitude of the oscillations shows spectacular
waxing and waning as a function of energy E. We could not fathom
anything like this before the numerical investigations. We have
successfully accounted for the waxing and waning both qualitatively
and quantitatively showing nearly perfect agreement. The book
contains background on Quantum Mechanics, Microelectronics and
Nanostructure Physics so that readers having no background of
Microelectronics and Nanostructure Physics can assimilate the book
completely. Content of the book is stunning and novel: not known to
anybody or reported by anybody anywhere else, to our knowledge.
The book contains thorough and complete analytical calculation
leading to two transcendental equations satisfied by transverse
wavevector dependent confined energy levels of the semiconductor
nanostructure called isolated Quantum Well (QW). The book also
contains thorough numerical calculation to bring out effects of the
non-zero component of wavevector of electron parallel to interfaces
of QW on parametric dependence of the confined energy levels.
Complete explanations are presented of the results of numerical
investigations using a transverse wavevector dependent effective
potential. The book also deals with Physics of another
semiconductor nanostructure called single rectangular tunnel
barrier. Using a previously derived analytical expression, we have
done novel numerical investigation of transverse wavevector
dependence of transmission coefficient of single rectangular tunnel
barrier and we have provided quantitatively exact explanation of
the dependence. The book is self-contained; it contains necessary
background on Quantum Mechanics, Microelectronics and Nanostructure
Physics to enable readers assimilate the book completely.
The book deals with Physics of a Nanostructure called isolated
Quantum Well (QW) in presence of magnetic field applied
perpendicular to interfaces of the QW. It contains thorough and
complete analytical calculation leading to two transcendental
equations satisfied by the magnetic field dependent confined energy
levels of isolated GaAs-AlGaAs QWs. The book also contains thorough
numerical calculation to bring out effects of the magnetic field on
parametric dependence of the confined energy levels. Complete
explanations are presented of the results of numerical
investigations using a magnetic field dependent effective
potential. The book is self-contained; it contains necessary
background on Classical Mechanics, Quantum Mechanics,
Microelectronics and Nanostructure Physics to enable readers
assimilate the book completely.
The work was done by Polash Ranjan Dey as M.S. thesis in the
Department of Physics, Shahjalal University of Science and
Technology, Sylhet 3114, Bangladesh. The work was supervised by Dr.
Sujaul Chowdhury, [email protected]. The work gives a
thorough illustration of use of wave mechanics to semiconductor
nanostructure in presence of magnetic field. The work covers
complete analytical calculation and thorough numerical
investigation with complete explanation of the results obtained.
We have studied Physics of a Nanostructure called Quantum Well
(QW). We have numerically calculated confined energy levels of
electrons, light holes and heavy holes in isolated GaAs-AlGaAs QWs
of various widths and depths. We have presented both graphically as
well as in the form of data tables variation of the energy levels
as width and depth of QW are varied. We find that the values and
variations of energy are exactly the same as the values and
variations of energy of resonant transmission peaks of symmetric
rectangular double barrier. Surprisingly, this is exact numerical
agreement. The agreement remains exact even if we take effects of
effective mass inequality into account. The book is self-contained;
it contains necessary Nanostructure Physics, Microelectronics,
Quantum Mechanics, as well as results for symmetric rectangular
double barrier for ready reference. Interested reader will be able
to assimilate the book completely.
We have demonstrated Physics for which we need to modify boundary
condition to be used in matching solutions of Schroedinger equation
at finite potential discontinuities i.e. at interfaces between
barrier and quantum well materials, because of difference of
effective mass of electron in their conduction bands. Using the
modified boundary condition, we have presented thorough and
complete analytical calculation of transmission coefficient of
nanostructures called single rectangular tunnel barrier and
symmetric rectangular double barrier. We have presented numerical
investigations for the two nanostructures. We find that parametric
dependences of energy of resonant transmission peaks of the double
barrier do not change qualitatively because of taking effective
mass inequality into account; but we find changes in energy and in
full width at half maxima (FWHM) of the resonant peaks. The book is
self contained; it contains necessary Nanostructure Physics,
Microelectronics and Quantum Mechanics to enable interested reader
assimilate the book completely.
The work provides a thorough illustration of use of WKB method to
barrier transmission problem. On reading the book, the reader will
be adept in using the method. Single and symmetric double barriers
of general shape are treated first. Then single and symmetric
double barriers of many different shapes we encounter in studying
Nanostructure Physics are covered. Complete analytical calculation
and results of extensive numerical investigation are reported along
with comparison with standard results where available showing
excellent agreement.
The book contains thorough and complete analytical calculation of
transverse wavevector dependent transmission coefficient of the
semiconductor nanostructure called single rectangular tunnel
barrier. Thorough numerical investigations are presented to bring
out dependence of transmission coefficient on angle of incidence,
transverse wavevector, total energy and longitudinal part of total
energy of incident electron. Complete explanations are presented of
results of numerical investigations. Physics of the problem has
been solved completely in this book. Interested reader will be able
to enjoy the Physics with complete satisfaction. The book is
self-contained; it contains background on Nanostructure Physics,
Microelectronics and Quantum Mechanics to enable the reader
assimilate the book completely.
Experiments at the turn of the 21st century have made revolutionary
advancements in the research area of two-dimensional (2D)
superlattices. In conjunction with contemporary and subsequent
theories, there have been ground breaking advancements in
understanding what electrons do in two-dimensional periodic
potential and a perpendicular magnetic field. The work is expected
to get recognition by Nobel Prize in Physics. The book contains a
marvelous account by a graduate student of the breakthroughs in
fundamental nanoelectronics research on 2D superlattices in both
experimental and theoretical fronts.
The book deals with a semiconductor nanostructure called symmetric
rectangular double barrier. Complete and thorough analytical
calculation of transmission coefficient of GaAs-AlGaAs symmetric
rectangular double barrier (for E
This book presents in comprehensive detail numerical solutions to
boundary value problems of a number of differential equations using
the so-called Shooting Method. 4th order Runge-Kutta method,
Newton's forward difference interpolation method and bisection
method for root finding have been employed in this regard. Programs
in Mathematica 6.0 were written to obtain the numerical solutions.
This monograph on Shooting Method is the only available detailed
resource of the topic.
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