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.

QUANTUM MECHANICS presents comprehensive account of the course for undergraduate students with thorough and complete calculations. The book has been written with the notion that a wave is associated with a material particle i.e. wave and particle coexist. Heisenberg's uncertainty principle has been described in light of this. A chapter is dedicated to mathematical structure of Quantum Mechanics followed by applications to one-dimensional (1D) problems. Orbital and general angular momentum are treated in two separate chapters, the latter also treats addition of angular momentum. Quantum theory of scattering, matrix formulation of Quantum Mechanics variational method and WKB approximation method have also been discussed.

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.

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 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 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.

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.

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 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.

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

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.

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.

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.