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India is a land of enormous diversity. Cross-cultural influences are everywhere in evidence, in the food people eat, the clothes they wear, and in the places they worship. This was ever the case, and at no time more so than in the India that existed from c. 1200 to 1750, before European intervention. In this thoughtfully revised and updated second edition, readers are taken on a richly illustrated journey across the political, economic, religious, and cultural landscapes of India – from the Ghurid conquest and the Delhi Sultanate, through the rise and fall of the southern kingdom of Vijayanagara and their successors, to the peripheries of empire, to the great court of the Mughals. This was a time of conquest and consolidation, when Muslims and Hindus came together to create a literary, material, and visual culture which was uniquely their own and which still resonates today.
Partial differential equations are a vital part of any course in pure or applied mathematics. This book will be invaluable to anyone looking for a lucid but comprehensive introduction to PDEs. Designed to strike a balance between theory and practical problems, it covers all major methods as well as their historical backgrounds, theoretical rigour, and geometric significance. The book is divided into three parts. It starts with basic topics like ordinary differential equations, multivariable calculus, and geometry. This is followed by important techniques to solve certain types of partial differential equations. The last part is devoted to first, second, and higher-order PDEs. The chapters have been arranged to help students develop their knowledge gradually and systematically. Each method is discussed through theoretical descriptions in the form of theorems followed by illustrative problems to help the readers. Finally, numerous solved examples and practice problems helps the student learn to apply this knowledge.
We solve the quantum mechanical harmonic oscillator problem using an operator approach. We define the lowering and raising operators. We use the quantum mechanical harmonic oscillator to review the fundamental ideas of quantum mechanics.We study some examples of time dependence in the harmonic oscillator including the coherent state. We apply the quantum mechanical harmonic oscillator to the study of the vibrations of the nuclei of molecules.
Through the Stern-Gerlach experiment, we demonstrate several key concepts about quantum mechanics: quantum mechanics is probabilistic; spin measurements are quantized; quantum measurements disturb the system. We show how to describe the state of a quantum mechanical system mathematically using a ket, which represents all the information we can know about that state.
We learn about unbound states and find that the energies are no longer quantized. We learn about momentum eigenstates and superposing momentum eigenstates in a wave packet. We apply unbound states to the problem of scattering from potential wells and barriers in one dimension.
Following Chadwick’s discovery of the neutron in 1932, it might have seemed that all forms of matter could be explained as different combinations of fewer than 100 elements, and those elements in turn could be explained as different combinations of protons, neutrons, and electrons. Add photons to the list, and you pretty much had the universe summed up. This is the kind of model physicists like: complicated behavior arising from a few simple building blocks.
The separation of variables procedure permits us to simplify a partial differential equation by separating out the dependence on the different independent variables and creating multiple ordinary differential equations. To illustrate the method, we apply a six-step process to the classical wave equation to show how the time dependence of the wave function can be found through a separate ordinary differential equation.
Welearn the key aspect of quantum mechanics – how to predict the future with Schrödinger’s equation. We learn the general recipe for solving time-dependent problems by diagonalizing the Hamiltonian to find the energy eigenvalues and eigenvectors.
You probably learned in school that matter comes in three phases: solid, liquid, and gas. (A fourth phase called “plasma” only tends to occur in extreme environments like the center of the Sun or physics laboratories, so your teachers can be forgiven if they left it out.) Gases can flow and conform their shapes to their containers, and can also compress or expand; liquids can also flow and conform shape, but they cannot compress or expand; solids can’t really flow, conform, compress, or expand.