Quantum Mechanics

Quantum Mechanics Shivam Prabhakaran Book Enclave Jai pur India Fir~ll.ublished : 2009 ISBN: 978-8J-8152-234-4 (D Author Published by Book Enclave Jdin Bhawan, Opp. N.E.I., Shanti Nagar, Jaipm - 302006 Tel. 0141-2221456 5110(1'1"00111 l ~- 13,55. Tower, Dhamani Street, Chama Rasta, Jaipur lei. U1-H-2311900, Fax: +91-141-2311900, E-m.lil: [email protected] • bookenc1a\'[email protected] \\'ebsitc: \\'ww.bllokenc1ave.com LII~cr T.lfP"",·lfillg ['Y Shubham Computers, Jaipur Prill led ol Roshan Offset Printers, Delhi Preface Quantum Mechanics is a theory of Mechanics, a branch of Physics that deals with the Motion of bodies and associated physical quantities such as Energy and Momentum. Quantum Mechanics has had enormous success in explaining many of the features of our world. The individual behaviour of the Microscopic Particles that make up all forms of matter can often only be satisfactorily described using Quantum Mechanics. Quantum mechanics is important for understanding how individual atoms combine to form chemicals. It provides quantitative insight into chemical bonding processes by explicitly showing which molecules are energetically favourable to which others, and by approximately how much. This book is intended to provide a comprehensive coverage of the major aspects of quantum mechanics. The most likely audience for the book consists of students and teachers of modern physics, mechanics and engineering. Shivam Prabhakaran "This page is Intentionally Left Blank" Contents Preface Introduction to Quantum Physics Max Planck's Revolutionary Hypothesis 2. Path Integrals in Quantum Mechanics 3. 4. Angular Momentum 5. Orbital Eigenfunctions: 2-D and 3-D 6. Niels Bohr and Quantum Atom 7. Time-dependent Wave Functions ~. Simple Harmonic Oscillator 9. The Hydrogen Atom 10. Electrons in One Dimension l. 1 33 60 71 81 99 117 128 143 243 "This page is Intentionally Left Blank" 1 Introduction to Quantum Physics For centuries, man has wondered on phenomena and processes happening around him. As time passed, he was successful in applying his intuition and common sense in comprehending the stars, galaxies and their behaviour, but they fail in the microscopic world of molecules, atoms and sub-atomic particles. Quantum theory provides us with the rules and regulations of the miniature world. These rules are phenomenally successful in accounting for the properties of atoms, molecules, and their constituents, and form the basis of understanding the fundamental properties of all matter. In fact, one may say that the greatest success story of the 20th-century physics is to confirm that this theory works, without a single exception, in spite of critical examination by some of the best minds spanning decades of time. The conceptual foundation of quantum theory is mysterious. It led to intense debates among scientists, and confused many. Niels Bohr, one of the most prominent scientists in this domain, once remarked, "You have not studied quantum mechanics well if you aren't confused by it." Albert Einstein, the greatest physicist of the 20th century, never approved of this theory. Bizarre though it may seem, quantum physics has led physicists step by step to a deeper view of the reality, and has answered many fundamental questions. Quantum physics is a branch of science that deals with discrete, indivisible units of energy called quanta as described by the Quantum Theory. There are five main ideas represented in Quantum Theory: 1. Energy