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## Meet the editors

Dr. Muhammad Bilal Tahir is a distinguished physicist and academic leader who currently serves as the director of the Institute of Physics. With a career dedicated to the advancement of physics and education, he has made significant contributions to the field. He holds a Ph.D. in Physics, earned through rigorous research and a commitment to excellence. Dr. Tahir's expertise spans several areas of physics, including quantum mechanics,

condensed matter physics, and theoretical physics, making him a versatile and wellrounded scientist. As the director of the Institute of Physics, Dr. Tahir has played a pivotal role in shaping the institution's academic and research programs. His visionary leadership has driven the development of cutting-edge research initiatives and cultivated a vibrant academic community. Under his guidance, the institute has flourished as a center of excellence for both undergraduate and graduate students, fostering a spirit of innovation and inquiry. Beyond his administrative role, Dr. Tahir remains an active researcher, with numerous publications in reputable scientific journals. His work reflects a deep commitment to advancing our understanding of the fundamental principles that govern the physical world. Dr. Tahir's dedication to physics and education has left an indelible mark on the academic community and the broader scientific landscape. His passion for research, teaching, and leadership continues to inspire the next generation of physicists and scholars.

Professor Dr. Muhammad Sagir is a highly accomplished academic and professional in chemical and environmental engineering. Currently serving as the director of the Institute of Chemical and Environmental Engineering, Khawaja Fareed University of Engineering and Information Technology (KFUEIT), Pakistan, his career has been marked by exceptional contributions to the field. His passion for teaching, research,

and academic administration continues to inspire students, researchers, and professionals alike.

Dr. Muhammad Isa Khan is an esteemed academic and scholar who is currently an Associate Professor in Physics at The Islamia University of Bahawalpur, Pakistan. With a career dedicated to education and research, Dr. Khan has made significant contributions to his field. Dr. Khan holds a Ph.D. in Physics, a testament to his dedication to academic excellence and a profound understanding of his subject matter. His expertise spans

several domains, reflecting his versatility and comprehensive knowledge in his area of expertise. Dr. Khan is an active researcher with numerous publications in prestigious academic journals. His work reflects a deep commitment to advancing knowledge in his field, and his research has made a tangible impact on the academic and professional community.

Dr. Muhamamd Rafique holds a Ph.D. in Physics, which serves as a testament to his dedication to academic rigor and the depth of his subject knowledge. His expertise spans various domains, and he has established himself as a versatile and well-rounded academic. In his role as an assistant professor at the University of Sahiwal, Pakistan, Dr. Rafique has made substantial contributions to the institution's academic and research programs. His commitment to teaching and

mentoring students has helped foster a vibrant academic community, and his guidance has proven invaluable in shaping the academic journeys of many aspiring scholars.

### Contents



Preface

The 20th century bore witness to two monumental scientific revolutions that forever altered the landscape of our understanding of the universe. First came the theory of relativity, which fundamentally transformed our conceptions of space and time, weaving a tapestry of spacetime that reshaped the way we perceive the fabric of the cosmos. Concurrently, quantum mechanics emerged, a groundbreaking framework that revolutionized our comprehension of the states of matter and energy, delving into the depths of the microcosmic world. Quantum mechanics, with its enigmatic principles and mathematical intricacies, has emerged as a captivating and challenging subject in the realms of science, engineering, and technology. It serves as the key to unlocking a wide array of phenomena, unveiling the beauty of mathematical techniques, Schrödinger's equation, and the ethereal realm of wave functions. It has enabled us to probe the mysteries of matter and energy at

It is with great enthusiasm that we present this book, *Schrödinger Equation –* 

*Fundamental Aspects and Potential Applications*. Within its pages, we explore a myriad of topics encompassing the essence of quantum mechanics. Our journey commences with an introduction to the spectral theory of the one and multi-dimensional Schrödinger equation, offering a foundation for understanding its mathematical underpinnings. As we delve deeper, we venture into the realm of scattering theory for the multi-dimensional non-relativistic Schrödinger equation, shedding light on the intricacies of quantum interactions. The quantization process and the profound concept of Feynman path integrals also find their place within our exploration as we provide insight into the mechanics of quantum systems and their behavior. Moreover, we examine the innovative application of relative entropy methods and transformation theory, unveiling their significance in constructing singular diffusion processes that bear an astonishing resemblance to Schrödinger equations. These methodologies provide fresh perspectives on the underlying principles governing

We have endeavored to make this book accessible, well-organized, and self-contained, ensuring that it caters to a broad audience, including undergraduates, graduates, researchers, and professionals in the fields of functional analysis, probability theory, and quantum dynamics. Our aim is to facilitate a comprehensive understanding of the Schrödinger equation and its multifaceted applications. We trust that readers will find this book to be a valuable resource, brimming with profound insights into the Schrödinger equation and its far-reaching implications. This book is a comprehensive guide that both educates and inspires, fostering a deeper appreciation for the pro-

their most fundamental levels.

quantum dynamics.

found wonders of quantum mechanics.

*by Masaru Hamano and Masahiro Ikeda*
