Problems in Quantum Mechanics and Field Theory with Mathematical Modelling

The book employs a variety of mathematical tools and techniques to solve problems in quantum mechanics and field theory. Key methods include:

1. Separation of Variables: This technique is used to simplify complex equations by breaking them down into simpler components, often in different coordinate systems like cylindrical or spherical coordinates.

2. Frobenius Method: This method is used to find solutions to differential equations with regular singular points, often resulting in series solutions.

3. Poincaré-Perron Method: This method is used to analyze the convergence of power series solutions.

4. Special Functions: Functions like confluent hypergeometric functions, Bessel functions, and Heun functions are used to solve specific differential equations that arise in the context of quantum mechanics and field theory.

5. Numerical Methods: Techniques like numerical summing of series are used to analyze and visualize solutions, particularly in the context of tunneling effects and other complex phenomena.

6. Duffin-Kemmer-Petiau Formalism: This formalism is used to describe particles with internal structure, incorporating gauge degrees of freedom and providing a more comprehensive understanding of electromagnetic fields.

These tools and techniques enable the authors to explore a wide range of problems in quantum mechanics and field theory, including those involving particles with different spins, external fields, and non-Euclidean space-time backgrounds.

The book "Problems in Quantum Mechanics and Field Theory with Mathematical Modelling" contributes significantly to our understanding of the tunneling effect in quantum mechanics by exploring various aspects and applications. It presents detailed analyses of tunneling phenomena through barriers, such as the Schwarzschild barrier for spin 1/2 particles and the Coulomb field. The book utilizes mathematical methods, including separation of variables, Frobenius solutions, and Poincaré-Perron method, to construct and analyze the solutions. It also includes numerical studies and visualizations to illustrate the tunneling effect, providing insights into the physical processes involved. Furthermore, the book discusses the tunneling effect in non-Euclidean spaces and under the influence of external fields, broadening the scope of understanding and application of tunneling in quantum mechanics.

The book "Problems in Quantum Mechanics and Field Theory with Mathematical Modelling" presents a comprehensive study of various exactly solvable problems in electrodynamics and quantum mechanics, focusing on particles with different spins in external fields and non-Euclidean space-time backgrounds. Its implications for the study of elementary particles and their interactions with electromagnetic fields are significant:

1. Enhanced Understanding of Particle Interactions: The book provides exact solutions for complex problems, offering deeper insights into the behavior of particles in electromagnetic fields, which is crucial for understanding fundamental interactions.

2. New Mathematical Tools: The use of advanced mathematical techniques, such as separation of variables, Frobenius solutions, and confluent Heun functions, provides new tools for tackling similar problems in the field.

3. Non-Euclidean Geometry: The exploration of particles in non-Euclidean spaces, like Lobachevsky and Riemannian geometries, suggests that our understanding of particle physics might benefit from considering more general geometric backgrounds.

4. Quantum Tunnelling: The study of tunneling effects through potential barriers, such as the Schwarzschild barrier, contributes to our understanding of quantum phenomena and could have implications for quantum computing and other technologies.

5. Spin Effects: The analysis of particles with different spins in various fields and geometries helps refine our understanding of spin-related effects and their role in particle interactions.

Overall, the book's findings expand the scope of quantum mechanics and field theory, offering new perspectives and tools for investigating elementary particles and their interactions with electromagnetic fields.

The book bridges the gap between theoretical research and practical applications in quantum mechanics and field theory by presenting a series of exactly solvable problems. These problems are grounded in mathematical modeling and cover various aspects of electrodynamics and quantum mechanics, including particles with different spins in external fields and non-Euclidean space-time backgrounds. By providing detailed analyses and exact solutions, the book offers researchers and students practical tools to understand and apply theoretical concepts. Additionally, the inclusion of numerical studies and visualizations of solutions helps to connect abstract theoretical models with real-world phenomena, making the book a valuable resource for both theoretical exploration and practical application.