xContents2855 APPROXIMATIONMETHODS5.1Time-Independent Perturbation Theory:285Nondegenerate Case5.2Time-Independent Perturbation Theory:298The Degenerate Case5.3304Hydrogenlike Atoms: Fine Structure and the Zeeman Effect5.4313Variational Methods5.5316Time-Dependent Potentials:The Interaction Picture3255.6Time-Dependent Perturbation Theory5.7Applications to Interactions with the Classical335Radiation Field3415.8Energy Shift and Decay Width345Problems3576 IDENTICALPARTICLES3576.1Permutation Symmetry3616.2SymmetrizationPostulate3636.3Two-Electron System3666.4The Helium Atom3706.5Permutation Symmetry and Young Tableaux377Problems3797SCATTERING THEORY7.1379The Lippmann-Schwinger Equation7.2386TheBorn Approximation7.3390Optical Theorem7.4392Eikonal Approximation7.5Free-Particle States: Plane Waves Versus Spherical Waves3953997.6Method of Partial Waves7.7410Low-Energy Scatteringand Bound States4187.8Resonance Scattering4217.9Identical Particles and Scattering4227.10 Symmetry Considerations in Scattering4247.11 Time-Dependent Formulation of Scattering4297.12InelasticElectron-AtomScattering4347.13 Coulomb Scattering441Problems446Appendix A456AppendixB458Appendix C464Supplement I Adiabatic Change and Geometrical Phase481Supplement II Non-Exponential Decays487Bibliography491Index
Modern Quantum Mechanics
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CHAPTER 1Fundamental ConceptsThe revolutionary change in our understanding of microscopic phenomenathat took place during the first 27 years of the twentieth century isunprecedented in the history of natural sciences. Not only did we witnesssevere limitations in the validity of classical physics, but we found thealternative theory that replaced the classical physical theories to be farricher in scope and far richer in its range of applicability.The most traditional way to begin a study of quantum mechanics isto follow the historical developments-—Planck's radiation law, the Einstein-Debye theory of specific heats, the Bohr atom, de Broglie's matter waves,and so forth-together with careful analyses of some key experiments suchas the Compton effect, the Franck-Hertz experiment, and the Davisson-Germer-Thompson experiment. In that way we may come to appreciate howthe physicists in the first quarter of the twentieth century were forced toabandon,little by little,the cherished concepts of classical physics and how,despite earlier false starts and wrong turns, the great masters--Heisenberg,Schrodinger, and Dirac, among others-finally succeeded in formulatingquantum mechanics as we know it today.However, we do not follow the historical approach in this book.Instead, we start with an example that illustrates, perhaps more than anyother example, the inadequacy of classical concepts in a fundamental way.We hope that by exposing the reader to a "shock treatment" at the onset, he1
2Fundamental Conceptsor she may be attuned to what we might call the "quantum-mechanical wayof thinking at a very early stage.1.1. THE STERN-GERLACH EXPERIMENTThe example we concentrate on in this section is the Stern-Gerlach experi-ment, originally conceived by O. Stern in 1921 and carried out in Frankfurtby him in collaboration with W.Gerlach in 1922.This experirnent illustratesin a dramatic manner the necessity for a radical departure from theconcepts of classical mechanics. In the subsequent sections the basic for-malism of quantum mechanics is presented in a somewhat axiomatic mannerbut always with the example of the Stern-Gerlach experiment in the back ofour minds. In a certain sense, a two-state system of the Stern-Gerlach typeis the least classical, most quantum-mechanical system. A solid understand-ing of problems involving two-state systems will turn out to be rewarding toany serious student of quantum mechanics. It is for this reason that we referrepeatedly to two-state problems throughout this book.Description of the ExperimentWe now present a brief discussion of the Stern-Gerlach experiment,which is discussed in almost any book on modern physics.* First, silver (Ag)atoms are heated in an oven. The oven has a small hole through which someof the silver atoms escape. As shown in Figure 1.1, the beam goes through acollimator and is then subjected to an inhomogeneous magnetic fieldproduced by a pair of pole pieces, one of which has a very sharp edge.We must now work out the effect of the magnetic field on the silveratoms.For our purpose the following oversimplified model of the silveratom suffices. The silver atom is made up of a nucleus and 47 electrons,where 46 out of the 47 electrons can be visualized as forming a sphericallysymmetrical electron cloud with no net angular momentum.If we ignore thenuclear spin, which is irrelevant to our discussion, we see that the atom as awhole does have an angular momentum, which is due solely to the spin-intrinsic as opposed to orbital-angular momentum of the single 47th (5s)electron.The 47 electrons are attached to the nucleus, which is ~ 2× 105times heavier than the electron; as a result, the heavy atom as a wholepossesses a magnetic moment equal to the spin magnetic moment of the47th electron.In other words, the magnetic moment μ of the atom is* For an elemcntary but cnlightcning discussion of thc Stern-Gerlach experiment, see Frenchand Taylor (1978, 432-38)