Epilogue

The purpose of a physical theory is fulfilled if it provides a mathematical image of some domain of reality that allows us to relate experimental data and foresee new situations by making mathematical deductions. In this book we have restricted ourselves to this program and have never mentioned any of the further-reaching implications of quantum mechanics. Yet quantum mechanics has affected all scientific and even general human thinking. And when presented in its full generality, as done in this book, it effortlessly reveals the two facts whose lessons reach far beyond the boundaries of physics.

Classical science is based on two assumptions: (1) the deterministic nature of predictions, and (2) the atomistic nature of understanding. Quan­tum mechanics teaches us to revise both.

Classical theories are deterministic. The laws of classical physics are constructed in such a way that if the initial values of a system's dynamical variables are given, then their precise values can be calculated for any later time. These laws were not simply derived solely from experience, and then accepted as the basis of scientific philosophy and general thinking. Rather, and perhaps to a greater degree, these laws have been extricated from nature because they were in accord with the prevailing philosophical idea that nothing can be without a cause.

Probability statements in classical physics are always associated with insufficient knowledge, i.e., they are statements about the observer's know­ledge and not about the physical system, which according to the principles of classical physics can be known to unlimited accuracy. In quantum theory

503

504 Epilogue

(as described in Section 11.6 and elsewhere in the text) one can only say with what probability certain values can be expected, even if one knows the state as well as possible, i.e., even if the system is in a pure state. Thus in quantum theory statements are inherently probabilistic; the occurence of probability functions is not just a consequence of the observer's insufficient knowledge, but a property attributed to the physical systems themselves. Quantum predictions of experimental results are statements of how a micro­physical process shows up in the macrophysical domain. These traces of microphysical processes in the macrophysical domain, the only source of human knowledge about such processes, do not obey deterministic laws. Earlier traces of a microphysical process do not determine later traces uniquely, but only probabilistically. Quantum theory teaches us that there are inherent limitations to human knowledge.

The second point, that of the profoundly holistic nature (of the under­standing) of quantum physical systems, is not often emphasized, even though it is an obvious consequence of the quantum-mechanical description of physical systems. Although holism has already become rather widely ac­cepted in other disciplines (e.g., psychology), it has been resisted by the physicists, who seem to be influenced by the success of atomism in classical physics. The quantum physical system is a structured whole described by the mathematical structure of an algebra of operators. From the laws of the combination of quantum physical systems (Section III.5), it follows that there are observables-ofthe combination of the two subsystems (described by Yt'1 Q9 Yt' 2) that are incompatible with all observables of either subsystem (described by Yt'l or Yt' 2)'

Thus, in quantum physics there exist holistic properties that cannot be obtained as combinations of the properties of the subsystems. In this sense the whole is not the sum of the parts.

In the atomistic approach understanding comes from the reduction of the complex system to simpler subsystems by ever finer separations until one comes to the ultimate constituents. In quantum physics the presence of holistic properties prevents this reduction process, and the notion of ul­timate constituents loses its meaning. Atomism belongs to classical physics. A quantum physical system such as a molecule cannot be fully understood by dissecting it into nuclei and electrons, although, in the tradition of our scientific heritage, it is tempting to do this. What one arrives at in this way, however, is only the classical analogue of the quantum physical system, as in the Kepler system of proton and electron for the classical analogue of the hydrogen atom. An electron in an atom "is" something different from an electron in a linear accelerator, and the whole picture of the electron can only be displayed by giving its different aspects as they are mathematically described by the various basis systems in the space of physical states.

The visual picture that one usually requires for the process of under­standing in is quantum physics not the geometrical picture of the object, but the picture of its image in the space of physical states. The reduction from the more complex to the simpler is performed not on the physical

Epilogue 505

object, leading to simpler consituent objects, but on the space of physical states, leading to the irreducible subspaces for ever simpler structures. At every stage of this reduction one still has a whole picture describing all aspects by the various basis systems of the subspaces, but a simpler one describing a narrower domain of physics.

Dissecting a quantum physical system may destroy it. Therefore a quan­tum physical system (such as the CO molecule) cannot be understood only atomistically (as a di-atom) but is often more adequately understood holistically and functionally (as a vibrator rotator). , Atomism has been a great achievement of the past, and all technology is based on it. But quantum theory has revealed its limitations and shown that even for simple systems a holistic method is needed also.

Bibliography

The cited literature has been chosen rather arbitrarily. I have not made a systematic search for the most suitable list of books for further or sup­plementary reading, and mention just those I happen to have come across. Many of these books I have used myself.

The books and a few review articles are separated into several categories and then listed in alphabetical order; a few comments are added here and there.

1 Foundations of Quantum Mechanics

P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1958 (fourth edition). Written by one of the creators of quantum mechanics many years ago, this is still modem and one of the greatest books written on this subject.

J. M. Jauch, Foundations of Quantum Mechanics, Addison-Wesley, 1968. This text is concerned with the conceptual foundations of quantum mechanics and contains practically no applications. Furthermore, it differs from the present presentation by starting from the lattice structure of quantum mechanics. Still, it is recommended even to those who do not want to learn lattice theory.

G. Ludwig, Grundlagen der Quantenmechanik, Springer, Berlin, 1954.

J. von Neumann, M athematical Foundations of Quantum Mechanics, Springer, Berlin, 1932; Princeton University Press, 1955. This is the first book written on the Hilbert-space formulation of quantum me­chanics developed by its author.

507

508 Bibliography

2 Scattering Theory

A. I. Baz, Ya. B. Zeldovich, A. M. Peremelov, Scattering Reactions and Decay in Non­relativistic Quantum Mechanics, Israel Program for Scientific Translations, Jerusalem, 1969.

L. Fonda, G. C. Ghirardi, A. Rimini, Decay Theory of Unstable Quantum Systems, Reports on Progress in Physics, 41, 587 (1978).

M. L. Goldberger, K. M. Watson, Collision Theory, Wiley, New York, 1964.

R. G. Newton, Scattering Theory of Waves and Particles. McGraw-Hill, New York, 1966. In particular, for the formal theory of scattering this book is highly recommended.

H. M. Nussenzveig, Causality and Dispersion Relations, Academic Press, New York, 1972.

A. G. Sitenko, Lectures in Scattering Theory, Pergamon Press, 1971.

K. Smith, The Calculation of Atomic Collision Processes, Wiley-Interscience, 1971.

John R. Taylor, Scattering Theory, Wiley, New York, 1972. This is a very clearly written book, which contains more material than the second part of the present book, however, it is written in a somewhat different spirit. It is highly recommended.

3 Theory of Angular Momentum and Group Theory

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Addison­Wesley, Pub!. Co., 1979.

A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, 1957.

I. M. Gelfand, R. A. Minlos, Z. Ja. Shapiro, Representations of the Rotation Group and of the Lorentz Group, Pergamon Press, New York, 1963.

M. A. Naimark, Linear Representations of the Lorentz Group, Pergamon Press, New York,1964. The first part of this book gives an introduction to the theory of group representa­tions and discusses the rotation group in fuJI detail. This is a book written by a mathematician for physicists, very readable, and mathematically rigorous. Some of the material of Chapter I is contained in this book.

M. Hamermesh, Group Theory, Addison-Wesley, 1962.

L. Michel, Applications of Group Theory of Quantum Physics: Algebraic Aspects; and

L. O'Raifeartaigh. Unitary Representations of Lie Groups in Quantum Physics. Both published in Group Representations in Mathematics and Physics, (Ed. V. Bargmann), Springer Lecture Notes in Physics, 1970. The two reviews by Michel and O'Raifeartaigh discuss many applications of group theory to quantum physics.

M. E. Rose, Elementary Theory of Angular Momentum, Wiley, New York, 1957.

Bibliography 509

4 Experimental Subjects

N. L. Alpert, W. E. Keiser, H. A. Szymanski, Theory and Practice of Infrared Spectro­scopy, Wiley, New York, 1970.

R. P. Bauman, Absorption Spectroscopy, Wiley, New York, 1962.

A. Barbaro-Galtieri, Baryon Resonances, in Advances in Particle Physics, Vol. 2 (Ed. R. L. Cook, R. E. Marshak) Interscience Publishers, 1968.

G. Herzberg, Molecular Spectra and Molecular Structure, D. van Nostrand Company, 1966 (in particular Vol. 1).

This book teaches more than just molecular physics. It is one of the most beautiful books on physics and is highly recommended to every student.

H. S. W. Massey, E. H. S. Burhop, H. B. Gilbody, Electronic and Ionic Impact Phenom­ena, Clarendon Press, Oxford, 1969 (in particular Vol. I: Collisions of Electrons with Atoms).

5 Mathematical Material

A. Bohm, The Rigged Hilbert Space and Quantum Mechanics, Springer Lecture Notes in Physics. Vol. 78 (1978).

I. M. Gel'fand and G. P. Shilov, Generalized Functions, Vols. 1,2,4, Academic Press, 1964.

A. Lichnerowicz, Linear Algebra and Analysis. Holden-Day Inc. San Francisco 1967.

K. Maurin, Hilbert Space Methods, Polish Scientific Publishers, Warsaw, 1967.

V. 1. Smirnov, A Course of Higher Mathematics, Pergamon Press, 1964 (in particular Vol. III, Part 1, on linear algebra, and Vol. III, Part 2, on functions of a complex variable).

6 History of Quantum Mechanics

Max Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hili, 1966.

F. Hund, Geschichte der Quantentheorie, BI Wissenschaftsverlag, 1975.

7 Quantum Mechanics, Textbooks and Atomic Physics

G. Baym, Lectures on Quantum Mechanics, W. A. Benjamin, New York, 1969.

D. I. Blokhintsev, Quantum Mechanics, D. Reidel Publishing Co., 1964.

H. A. Bethe, E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag, New York, 1957.

R. H. Dicke, 1. W. Wittke, Introduction to Quantum Mechanics, Addison-Wesley, 1960.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. 3,

Quantum Mechanics, Addison-Wesley, 1965.

510 Bibliography

S. Gasiorowicz, Quantum Physics, Wiley, 1974.

D. T. Gillespie, A Quantum Mechanics Primer, Intex Publisher, 1970.

K. Gottfried, Quantum Mechanics, W. A. Benjamin, New York, 1966.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics, Pergamon Press, 1958.

H. J. Lipkin, Quantum Mechanics, New Approaches to Selected Topics, North-Holland, 1973.

E. Merzbacher, Quantum Mechanics, Wiley, New York, 1961.

A. Messiah, Quantum Mechanics, Vols. 1,2, Interscience, New York, 1961.

M. Mizushima, Quantum Mechanics of Atomic Spectra and Atomic Structure, W. A. Benjamin, New York, 1970.

L. T. Schiff, Quantum Mechanics, McGraw-Hill, 1968 (third edition).

8 Prerequisites from Classical Physics

V. D. Barger, M. Olsson, Classical Mechanics: A Modern Perspective, McGraw-Hill, New York, 1973.

A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles, MacMillan, New York 1964.

H. C. Corben, Classical and Quantum Theories of Spinning Particles, Holden-Day, San Francisco. 1968.

H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass., 1950.

J. D. Jackson, Classical Electrodynamics, Wiley, New York, 1975.

F. Rohrlich, Classical Charged Particles, Addison-Wesley, Reading, Mass., 1965.

Index

A

Absorption spectrum of helium 269 Advanced Green's function 345 Algebra

of 8 (SO(4» of 8(SO(3,1» of (f (E (3» of operators generated 100

Alkali atom 189 Hamiltonian 189

Analytic, continuation 393 Analyticity 414

ofSI(P) 417 properties of the S -matrix 418

Angular momentum basis 357 intrinsic 219 operator( s) 97

commutation relations 98 orbital 220 quantum number 104 total 130,220

Angular velocity 95

Anharmonic oscillator infrared transitions 90,91 energy levels 90,91

Annihilation operator 19 Anomalous magnetic moment 230 Anticommutator 174 Antisymmetric 421 Apparatus resolution 445 Approximation, effective range 391 Argand diagram 369,435,478 Associated Legendre function 203 Associative algebra 7 Asymptotic behavior, of the lth partial

S-matrix element 388 Asymptotically complete 338 Atomistic 504

B

Background phase shift 427 Bargmann, V. 171 Basis 3

angular momentum 357 generalized 50 system, generalized 6, 127

511

512 Index

Bessel function 377 spherical 377

Bohr magneton 231 Bohr, Niels 44 Born approximation 329,334,353,

371,498 Born, Max 13,43 Bose statistics 243 Bosons 243 Bound state 310,338,386

energy eigenvalue 387 pole 386,421

Branch cut 397 line of a function 396 of a function 396 point of a function 393,396

Breit-Wigner amplitude 429

c

approximation, generalized 468 formula 429 resonances

interfering 477 noninterfering 477

Canonical commutation 12 relations, three-dimensional 95

Capture state 496, 501 Casimir

operators 100 operatorstt(SO(4» 178

Cauchy inequality 41 integral formula 391 - Schwarz - Bunyakovski

inequality 4 Causality 411 , 414 Centrifugal

distortion 110 forces 111

Channels excitation 365, 366 ionization 365, 366 inelastic 365

rearrangement 366 Clebsch - Gordan

coefficients 132 explicit expressions for 137 formulas 137 orthogonality relations 132 phase convention for the 134

CO molecule 20, 88 Coherent state vector 79 Collision

cross section, total 314 rearrangement 366

Combination of two physical systems, interaction

free 114 two quantum-mechanical

systems 113 Commutation, canonical 12 Commuting operators 34 Compatible observables 36 Complete system of commuting

observables (c.s.c.o.) 125, 126,336

Completeness property 310 Component 5 Constants of the motion 179, 279

quantum-mechanical 176 Continuous spectrum, discrete

eigenvalues in 376 Convolution 345 Coordinate 5 Copenhagen interpretation 44 Countably normed space 288 Creation operator 19 Cross section 312

differential 313,314,317,331, 332,360,437

inelastic 314 energy-averaged, for coarse

resolution experiments 444 for scattering on pure energy

state 330 formula 327

for coarse-resolution experiments 328

lth partial 361 elastic 368

inelastic 368 partial 314 quantum mechanical 313 shape of, for interference between

resonance and background 443

total inelastic partial, for the nth level 314

C.s.c.o. (complete system of commuting observables) 125, 126,336

D

De Broglie, Louis 14 De Broglie relation 72 Decay

probability 487 processes, effect of the environment

in 486 rate 486

initial 486,497 total 497, 499

Decomposition 25 Defining algebraic relations 8 Degeneracy of the energy

eigenvalues 183 Delta function 328

normalization 51 Density matrix 29,63 Detector efficiency function 444 Determination of phase shifts from

experimental data 446 Deterministic 503 Diatomic molecule

vibrating 118 rotating 118 interacting 118 parameters of 128

Dipole emission 87 moment for the helium atom 261 resonance 434

Dirac delta-function 51 formulation of quantum

mechanics 128 picture 285

Dirac, P.A.M. 44 Direct

Index 513

product (see Tensor product) sum 24

Discrete eigenvalues in the continuous spectrum 316

Dispersion of an observable 40 Distribution (see Generalized function) Double

ionization threshold of the helium atom 270

multichannel resonance 466 multichannel resonance cross

section 472 scattering 268

Doubly excited states of helium 266 Dumbbell 96,98

E

C(E3) 146 C(SO(3,1» 146 C(SO(4» 146

irreducible representation space of 178

C (SU(2», ladder representations of 100

Effective quantum number 197 Eigen-phase-shifts 464 Eigenphases 464 Eigenspaces, invariant 177 Eigenstate of observable 28, 40 Eigenvalue(s), discrete in the

continuous spectrum 7, 338 Eigenvector(s) 7,50

generalized 50 with complete eigenvalue 493

Elastic channel 365, 366 cross section, total 314 scattering 363

amplitude 333,368 Elasticity 479

514 Index

Electromagnetic spectrum 39 Electronic states of a molecule 121 Elementary rotator 130 Emission, dipole 87 Energy

conservation 328 diagram, symmetric top 165 eigenvalues, degeneracy of 183 -level diagram 116

band withP, Q, andR branches 165

level of hydrogen atom, n2_ fold degeneracy of 186

-loss experiments 21 spectrum of helium 266

operator, kinetic (see Kinetic energy operator)

Ensemble 20 Entire function 393 Enveloping algebra 100 Environment in decay processes,

effect of 487 Euclidean

group,3-dimensional 146 space 3

Excitation energies of helium 268 Expectation value 27 Explicit expressions for the

Clebsch -Gordan coefficients 137

Exponential decay law deviations from 496 for the nondecay amplitude 496

F

Fano shape parameter 440 Fermi

resonance, Argand diagram of 482 statistics 243

Fermions 243 Final state 315 Fine-structure 187

splitting 234

Force centrifugal III operator 280 time average in the x-direction of the

Stem -Gerlach experiment 305

Forces attractive 380 repulsive 380

Formation experiments 401 Forward-scattering amplitude 334 Fourier

transform 344 of a generalized function 344

Franck 21 Franck - Hertz experiment 21 Function

branch of a 396 multi valued 394 of a complex variable 391 sequences of delta-type 65

Functional 65 antilinear 4,50 continuous 4 generalized 51

G

Gelfand - Naimark - Segal reconstruction

theorem 8 triplet 50

Generalized basis 50 basis system 6, 127 Breit-Wigner approximation 468 eigenvector

normalized, with complex eigenvector 494

of the Hamiltonian with complex eigenvalue 493

eigenvectors 50 and rigged Hilbert space 494

function 51 optical theorem 353

Generator 8,276 of time evolution (see Hamiltonian)

Green's function 208 advanced 345,346 exact 346 free 346 retarded 346

Group, orthogonal 146 Gyromagnetic ratio 227

H

Hamiltonian 277 Hankel function, spherical 377 Heisenberg

commutation relation 12 equation of motion 279 picture 279 uncertainty relation 41

Heisenberg, Werner 13,43 Helicity 106 Helium

atom 245 doubly excited 271 excitation energies of 266 Hamiltonian 245

Hermite functions 49 polynomials 49,53

Hermitian operator 7 form 4 vector 148

Hertz 21 Highest weight 101 Hilbert, David 44 Hilbert space 5, 19, 288 Holistic 504 Hydrogen spectrum, Pauli - Bargmann

treatment 171 Hyperfine splitting 188

I

Identical quantum-mechanical particles 238

Index 515

In-state 315,318,339,341,343 Incompatible observables 36 Indistinguishable particles Inelastic

channel 365 cross section 314 reaction process 363

Inelasticity coefficient 368,479 Infinitesimal operator 276 Infrared spectrum ofHC1 90 Initial state 315 Inonu - Wigner contraction 154 Interaction

free state 318 picture (see Dirac picture)

Interfering Breit-Wigner resonances 477

Internuclear distance of HC 1 111 Intertwining relation 349 Intrinsic

angular momentum 219 parity 159

Invariant operators (see Casimir operators)

Inverse of operator 174 width 496

Ionization channel 365, 366 energy 258 -excitation channels 366 potential 258 threshold 255, 365

double, of the helium atom 270

Ionized threshold of the helium atom 270

Irreducible representation 103 space 103

oftf(SO(4» 178 Isolated systems 275,287

J

Jordan, P. 43-44

516 Index

K

Kepler problem, classical 171 Kinetic energy operator

L

for massive nonrelativistic particle 320

for photon 320

Laguerre polynomials 204 Lamb shift 187 Lande factor 227 Laurent series 392 Legendre function, associated 203 Legendre polynomial 203,438 Lenz vector 146, 173 Level shift 209 Lifetime 487

average 487 of a resonance 496

Linear combination 3 Linearly

dependent 3 independent 3

Lippman -Schwinger equation 215, 309,335,373

solution of 338 London, F. 44 Lorentz group 146 Lorentzian energy distribution

function 488 Low-lying energy levels of atomic

hydrogen 187 Ith partial

cross section 361 elastic cross section 368 inelastic cross section 368

M

Magnetic field in the Stem-Gerlach

experiment 303

moment of a neutron 230 of a spinning particle in classical

physics 226 moment operator of a spinning

particle without intrinsic magnetic moment 229

Many-particle systems 237 Mass, reduced 96 Matrix element 6 Matrix of an operator 6 Mean life 487 Measurement 27

on a state, effect of 36 Measures 128 Measuring process, change of state

by 292 Meromorphic function 393 Metastable state 401 Minimal coupling to the

electromagnetic field 230 Mixing, singlet-triplet 262 Mixture 230

of states 23 Models, quantum-mechanical 91 Moller wave operators 347, 348 Moment 358

of inertia 95 Momentum

basis 357 space wave function 55 transfer 95

Multichannel resonances 463 Multivalued function 394

N

Neumann function, spherical 377 Neutral helium 268 Newton's equation, quantum

mechanical 280 Nondecay probability 487 Nordheim, L. 44 Normalized 5 Nuclear spectral theorem 127,270

o Observable 10, 36

dispersion of 40 incompatible 36

Observed cross section for high-resolution experiment 442

Old quantum theory 13 One-parameter group of

operators 276 Operator 147

*-algebra 8 angular momentum 97 anti-adjoint 454 antilinear 454 anti unitary 454 Casimir 100

@" (SO( 4» 178 inverse of 174 isometric 349 ladder 19 level shift 209 linear 6,454 positive definite 24, 174 semilinear 454 skew-hermitian 277 square root of 174 tensor 138 unit 7 unitary 350

Optical theorem 364,369 Orbital angular momentum,

combination with spin 224 Orthogonal 5 Orthohelium 250,260,262 Orthonormal system 5

complete 5 Oscillator, harmonic 10

wave functions 53 Out-state 315,339,341,343

p

Parahelium 250,260,262 Parity 156,451

Index 517

conservation 157 doubling 159 intrinsic 159 invariance 224,452 operator for the two-electron

system 261 relative 183

Parity-doubling 161 Partial

cross section 314 for elastic scattering 362 for inelastic scattering 362

decay rate 499 -wave amplitude(s) 362,364

for elastic scattering 362 -wave expansion 356 -wave reaction amplitude 361 width 468

Particle 70,71 Pauli

exclusion principle 244 matrices 105

Pauli, W. 171 Permutation 238

even 239 odd 239

Perturbation of the continuous spectrum

213 theory 198, 207

Phase shift 384

background 427 shift analysis 435

for the Schultz resonance He-(l9.3 eV) 447

Physical sheet of S-matrix 420 system 10

conservative 282 Planck's constant 10 Pole 386

of a function 392 simple, of a function 392

Position probability 59 density 59

518 Index

Positive definite 4 definite operator 24, 174 Hermitian 4

Post-collision target 331 Potential

attractive 380 part of the phase shift 427

Precession 304 Principal quantum number 181 Principle

of detailed balance 459 of microreversibility 459

Probabilistic 504 Probability 30,43

density 43 interpretation 43

Product, direct (see Tensor product) Production

cross section 313 experiments 401

Profile index 440 of doubly excited states 272

Projectile - target system 318 Projection operator 24 Projector 24 Property 30 Proposition 30 Proton's magnetic moment 230 Pseudo-orthogonal group 146 Pure state 23, 27, 42

Q

Q-branch 166 Quantum

defect 197 electrodynamics 187 mechanics, characteristic

features 292 numbers 126

Quantum-mechanical Kepler problem 171

Quasistationary state 401,424,431

R

R andP branches 115, 164 Radial

momentum operator (conjugate to the radial operator) 194

Schrodinger equation 203 wave function

exact 374, 379 asymptotic behavior of 382 free 374, 376

Rayleight - Schrodinger perturbation series 212

Reaction process, inelastic 363 Rearrangement

channel 366 collision 366 -excitation channels 366

Reciprocity theorem 459 experimental check 460

Reduced matrix element 142 Reducible representation space 105 Reduction of the direct product into

irreproducible representations of the angular momentum 135

Regular tensor operator 139 Relations, three-dimensional canonical

commutation 95 Relativistic mass effect 231 Representations

ladder 152 unitary 147,239

Repulsive potential 380 Residue of a function 393 Resolution of the identity 34 Resolvent of an operator 208 Resonance 401,402,423,431

cross section 428 energy 428 multichannel 463

single 468 phase shift 427 pole 423,430,431 width 428

Resonant partial-wave amplitude 428

Rest energy 232 Reversible process 275 Riccati

- Bessel function 377 -Neumann function 377

Riemann sheet 397 surface 396

Rigged Hilbert space 50 description 69

Rigid rotator 95, 107 Rotating oscillator 112 Rotation

spectra 105 -vibration band 116

Rotational absorption spectrum of HC 1 109 constant of CO 118 constant of H2 molecule 118 constants ofHCl 118 invariance 221

Rotator 106 algebra of operators 95 energy spectrum 106 nonrigid 110

Runge-Lenz vector 173 Rydberg constant 184, 185

s

S-matrix 315,347,352,363,388 approach 312,328 element 420 physical sheet of 420 theory 316,354 unphysical sheet of 420

S -operator 315, 347 Scalar

operator 138 product 2,4 product space 3

Scattering amplitude, forward 334 angle 333 channels 366

cross section 313 total 313

elastic 363 experiment 312 length 390

Index 519

for a virtual state 434 operator 346 phase shift 367 potential, properties of 379 states 184,310,338 theory, formal 335

Schrodinger equation 283

for the one-electron atom 200 for the radial wave function 375 time-independent 58

picture 278,279 representation 57

Schrodinger, Erwin 13 Schrodinger's wave mechanics 43 Schultz, G. 1. 21 Schwartz space 19 Schwarz reflection principle 395 Screened coulomb field of an

atom 334 Selection

rule for dipole radiation of the rotator 108

rules for helium atom 261 rules for hydrogen bomb with

spin 235 Self-adjoint 7 Shape profile parameter 440 Siegert pole 423 Single multichannel experimental

examples 471 Singlet-triplet mixing 262

for the helium atom 261 Singularity of function, isolated

392 Sommerfeld fine-structure

constant 175 Space

antisymmetric 242 inversion 156 -inversion invariance 451 irreducible representation 103

520 Index

Space (cont.)

linear 2 of bound states 177 of physical states of the hydrogen

atom 181 of physical states of the

quantum-mechanical rotator 105

symmetric 242 Spectral

representation 26 of an operator 34,52

series of hydrogen 186 Spectrum 128

continuous 50 discrete 50 -generating algebra 106, 182 -generating group 106, 182

Spherical components of vector

operators 139 harmonics 203

addition theorem of 204 completeness relation of 204 coupling rule of 204 differential equation of 203

symmetry 356 Spin 219

and statistics, connection between 224

electron 218 combination with orbital angular

momentum 224 operators 105 -orbit interaction

interaction, matrix element of the 265

term 230 -orbital magnetic moment

interaction 264 Spinning particle, classical 22 Stable state 401 State 23

"almost eigen-" 66 exact 340 free 339

interaction free 318 nonstationary 307 preparation of a 31 quasistationary 401,424,431 stationary 284,401 of combined physical system 295 pure 66

Stationary state 284,401 systems 284

Statistical operator 29,318 of a decaying state 495

Stem -Gerlach experiment 219, 292

Structureless particles 20 Subspace 3

invariant 105 orthogonal 25

Sum, direct 24 S-wave scattering length 390 Symmetric 241

(permutation) group 239 top

classical nonrelativistic 163 energy diagram 165 quantum-mechanical 162

Symmetry breaking 197 group 314 relation 420

T

relations of the Clebsch -Gordan coefficients 137

spherical 356 transformation 282

T-matrix 316,363 elements

off the energy shell 328 on the energy shell 328

for a spherically symmetric problem 360

Taylor series 392 Tensor

operator 138 proper 157 pseudo- 157

product 112 Tenn

diagram for energy levels of helium 259

value 119 Thomas

factor 231 precession 231

Threshold inelastic 365 ionization 255,365

Time delay 405, 410

average 409 -dependent external forces 287 development, continuous

unitary 35 evolution 274

continuity of 275 reversal 453 reversal invariance 455

Titchmarsh theorem 493 Topological space, linear 5,288 Torque 227, 304 Total width 499 Trace of the operator 25 Transition

coefficients between momentum and angular momentum bases 359

probabilities 307 probability 306, 308 rate 308,309,312

Transpositions 239 Two-electron excitation 268

in neutral helium 269

u Uncertainty 41 Unit

ray 26

Index 521

step function 341 Unitarity of the S-operator 364 Unitary 157

circle 369,436 representation

of the group SO(4) 147 of the pennutation group 239 of the group SO(3,1) 147

Units 175 Unphysical sheet of S-matrix 420

v

Vector operator 139 skew-hennitian 148 spherical components of 139

Vectors 3 Velocity reversal 453 Vibrating rotator 112, 113 Vibration-rotation band of carbon

monoxide 92 Vibrational levels of CO 22, 88, 91 Virtual-state pole 422,434 Voight profile 448 Von Kampen causality condition 411 Von Neumann, John 44

w

Wave 71,72 function 43, 73

momentum space 55 exact 373 free 373 in the momentum

representation 55 in the P-representation 55

packet 72 -particle duality

Weight 100 diagram 103 vector 100

Wigner - Brillouin perturbation series 212

522 Index

Wigner ( cont.)

coefficients 131 -Eckart theorem 138, 139 theorem 458 3-jsymbols 137,140,141

Wigner's causality inequality 413 eigenphase repulsion theorem

466

y

Yukawa interaction 334

z Zero-point energy 62

*-algebra 8

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Zeitschrift fiir Physik appears in three parts- A: Atoms and Nuclei; B: Condensed Matter and Quanta; C: Particles and Fields. Each part may be ordered separately. Coordinating editor for Zeitschrift flir Physik, Parts A, B and C, is O. Haxel, Heidelberg.

Zeitschrift fiir Physik C-Particles and Fields is devoted to the experimental and theoretical investigation of elementary particles. In view of the steadily growing interplay of theory and experiment in this field, particular emphasis is given to a clear and complete presentation of research.

The topics covered include: strong, electromagnetic, and weak interactions of elementary particles, interaction and classification of constituents, and symmetry and unification schemes of different interactions.

Lecture Notes in Physics Managing Editor w. Beiglbock

This series reports on new developments in physical research and teaching­quickly, informally, and at a high level. The type of material considered for pUblication includes preliminary drafts of original papers and monographs, lectures on a new field or lectures that present a new angle on a classical field, collections of seminar papers, and reports of meetings.

VoL 81 M.H. MacGregor, The Nature of the Elementary Particle. 1978. xxii, 482p.

Vol. 83 Experimental Methods in Heavy Ion Physics. Edited by K. Bethge. 1978. v, 251p.

Vol. 84 Stochastic Processes in Nonequilibrium Systems. Edited by L. Garrido, P. Seglar, and PJ. Shepard. 1978. xi, 355p

Vol. 88 K. Hutter and A.A.F. Van De Ven, Field Matter Interactions in Thermoelastic Solids. 1979. viii, 231p.

Vol. 89 Microscopic Optical Potentials. Edited by H. von Geramb. 1979. xi, 481p.

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