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This book presents a meticulously crafted new translation of Werner Heisenberg's ground-breaking 1925 paper, widely regarded as the birth "certificate" of quantum mechanics. The translation remains faithful to the original terminology and is accompanied by a particularly detailed, step-by-step commentary. These annotations are designed not only to clarify the meaning of this complex work for the modern reader but also to illuminate the reasoning behind Heisenberg's choices, offering deep insight into the physical, conceptual, and mathematical foundations of the theory. Great care has been taken to make the calculations as explicit as possible while keeping them concise, ensuring that every essential step is clear without unnecessary complexity. This approach allows readers to grasp the logical flow of Heisenberg's argument while appreciating the elegance of his formulation. Beyond the paper itself, the book situates Heisenberg's work within the broader intellectual landscape of the time, tracing the intense discussions and challenges in physics that led to the birth of quantum mechanics. It also provides insight into the subsequent contributions of Born, Jordan, and Heisenberg himself that culminated in the formulation of matrix mechanics. Aimed at readers with a university-level background in quantum mechanics, this book is an invaluable resource for those who wish to delve into the motivations, conceptual developments, and intricate reasoning that led Heisenberg to construct the new quantum theory in such a unique and revolutionary way.