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 modern and one of the greatest books written on this subject.

J. M. Jauch, Foundations of Quantum Mechanics, Addison-Wesley, Reading, Mass., 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, Mathematical Foundations of Quantum Mechanics, Springer, Berlin, 1932; Princeton University Press, Princeton, 1955.

574

This is the first book written on the Hilbert-space formulation of quantum mechanics developed by its author.

Bibliography 575

2 Scattering Theory

A. I. Baz, Ya. B. Zeldovich, A. M. Perelomov, 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, New York, 1971.

K. Smith, The Calculation of Atomic Collision Processes, Wiley-Interscience, New York, 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, Reading, Mass., 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 full 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, Reading, Mass., 1962.

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

L. O'Raifeartaigh, Unitary Representations of Lie Groups in Quantum Mechanics. Both published in Group Representations in Mathematics and Physics (Ed. V. Bargmann). Springer Lecture Notes in Physics, Vol. 6,1970. 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.

576 Bibliography

4 Experimental Subjects

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

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

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

G. Herzberg, Molecular Spectra and Molecular Structure, D. van Nostrand, New York, 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. 1: Collisions of Electrons with Atoms).

5 Mathematical Material

Milton Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.

A. Bohm, The Rigged Hilbert Space and Quantum Mechanics, Springer Lecture Notes in Physics, Vol. 7g, 1978.

A. Bohm, M. Gadella, Dirac Kets, Gamow Vectors and the Mathematics of Scattering and Decaying States, Springer Lecture Notes in Physics, Vol. (1986), to appear.

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

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

K. Maurin, Methods of Hilbert Spaces, Polish Scientific Publishers, Warsaw, 1967.

K. Maurin, General Eigenfunction Expansions and Unitary Representations of Topo­logical Groups, Polish Scientific Publishers, Warsaw, 1968.

V. I. Smirnov, A Course of Higher Mathematics, Pergamon Press, New York, 1964 (in particular Vol. III, Part 1, on Linear Algebra, and Vol. III, Part 2, on Functions of a Complex Variable).

G. N. Watson, Theory of Bessel Functions, Cambridge University Press, Cambridge, 1958.

6 History of Quantum Mechanics

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

Max Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill, New York, 1966.

7 Quantum Mechanics, Textbooks and Atomic Physics

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

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

D. I. Blokhintsev, Quantum Mechanics, Reidel, Dordrecht, 1964.

Bibliography 577

R. H. Dicke, J. W. Wittke, Introduction to Quantum Mechanics, Addison-Wesley, Reading, Mass., 1960.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. 3, Quantum Mechanics, Addison-Wesley, Reading, Mass., 1965.

S. Gasiorowicz, Quantum Physics, Wiley, New York, 1974.

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

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

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

H. J. Lipkin, Quantum Mechanics, New Approaches to Selected Topics, North-Holland, Amsterdam, 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, Benjamin, New York, 1970.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics, Addison-Wesley, Reading, Mass., 1974.

L. T. Schiff, Quantum Mechanics, McGraw-Hill, New York, 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., 1980 (second edition).

J. D. Jackson, Classical Electrodynamics, Wiley, New York, 1975 (second edition).

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

Index

Absolutely continuous spectrum 16 Absorption

of radiation 119 spectrum of helium 306

Addition of operators 6 of vectors 3 theorem for spherical

harmonics 238, 413 Additive semigroup 311 Adjoint

M0ller operators 403 operator 6, 37

Advanced Green's function 398 Algebra

associative 7 enveloping 132 generated 8 of 6'(E3) 182 of 6'(SO(3)) 132 of 6'(SO(3, I)) 182 of 6'(SO( 4)) 182 of 6'(SU(2)) 132 of operators

continuous 36 of the harmonic oscillator 45 of the rotator 138

operator *- 8, 161 spectrum-generating 216

of the rotator 140 sub- 7 with unit element 7 *- 8

Algebraic conjugate space 34 dual space 34

Alkali atom 223 Hamiltonian 223

Almost eigenstate 99 sharp state 99

Amplitude background 490 Breit-Wigner 483 partial-wave 415

elastic scattering 415 reaction 415 resonant 481

scattering 413 elastic 384 forward 386

Analytic continuation 446 function 444

579

580 Index

Analyticity properties of the S-matrix 467,470

Analyzer 55 Angular

momentum basis 410, 567 intrinsic 254 operator 130

commutation relations 131 total 166

orbital 255 quantum number 137 total 255

velocity 128 Anharmonic oscillator

energy levels 123, 124 infrared traOnsitions 123, 124

Annihilation operator 53 Anomalous

magnetic moment 265 Zeeman effect 270

Antiadjoint operator 508 Anticommutator 208 Antilinear

functional 34 operator 6, 508

Antisymmetric subspace 284 vectors 280

Antisymmetrical representation 279 Antiunitary 508 Anzatz (hypothesis) 29, 48, 440 Argand diagrams 422,489-491, 532,

535 Associated

Laguerre polynomials 239 Legendre polynomials 238

Associative algebra 7 Assumption

basic (see Postulate) reproducibility 348

Asymptotic behavior of the lth partial S-matrix

element 441 completeness 404

Asymptotically complete 390 Atomism 572 Attractive

in an interval 433 force 433 potential 433

Average time delay 462 value 62, 76, 345

Averaging over polarization states 381

Axial-vector 193 Axiom, basic (see Postulate)

Background amplitude 490 integral 555 phase shift 480

Balmer 220 Balmer series 220 Bargmann, Vo 205 Basis

adapted to a Stern-Gerlach apparatus 343

angular momentum 410,567 components of a vector with respect

to 11 continuous 22 direct product 146, 148, 167 generalized 87

momentum eigenvector 410, 568 in an n-dimensional space 5 matrix of an operator with respect

to 20 physical 166 statement for a 60

Bell 354 Bell's inequalities 349-352, 354 Bessel function, spherical 430 Bohr

magneton 266 radius 231

Bohr, Niels 47, 81,270,353 Born approximation 380, 381, 385,

386, 405, 424, 566 Born, ~ax 47,48,80,81 Bose statistics 280 Bosons 280 Bound

state poles 440,475, 550 states 360, 390, 439

space of 211 subspace of 390

Bra 9 Branch

cut 449 line 449 points 445, 449

Branches 449 Breit - Wigner

amplitude 483, 542 approximation, generalized 522 cross section 503 curves

interfering 531 noninterfering 531

formula 483

Canonical commutation relations, three-dimensional 128

Capture state 545, 569 Cascade transition 351 Casimir operators

of t&'(SO(4» 212 of t&'(SU(2» 133

Cauchy inequality for sequences 78 integral formula 31, 444 residue theorem 446 -Riemann conditions 444 -Schwartz-Bunyakovski

inequality 5 Causality 464, 466, 467 Centrifugal

distortion 144 forces 145

Change of state 72 by the dynamical law 328 by the measuring process 328

Channel excitation 418,419 inelastic 418,419 ionization 418,419 rearrangement 419

Classical harmonic oscillator 46 Clebsch-Gordan coefficients 167

explicit expressions for 172 formula for 172 orthogonality relations 167 phase convention 170

CO molecule 55,57, 121 Coefficien t

Clebsch-Gordan 167 Fourier 25 inelasticity 421 transition 21 Wigner 167

Coherent combination 73 state vector 112

Collision cross section, total 364 rearrangement 419

Combination coherent 73 incoherent 73 linear 5 of two elementary rotators 165 of two quantum-mechanical

systems 147 interaction free 148

Commutation relation canonical 47 Heisenberg 47

Commutator 7 Commuting operators 7, 69 Compatible observables 72 Complete system

of commuting observables (c.s.c.o.) 160,388

of eigenvectors 11 Completeness

asymptotic 404

Index 581

property of a basis 360 Complex eigenvalue 12, 18,547,

562 Compliment, orthogonal 390 Component of a vector with respect to

a basis 9-11 Composite system 352 Conjugate

operator 559 space 34

Conservation of energy 379,401 Constants of the motion 211,315 Continuous

basis 22 functionals 36 operator 37 operator function 325

at a point 324 spectrum 11, 12, 16

discrete eigen val ues in 12 Contraction, Inonii-Wigner 189 Convergence

Hilbert space 35 of infinite sequences 35 radius of 445 stronger 36 weaker 36 <l>-space 36

Convolution 378, 397 Coordinate of a vector with respect to a

basis system 9-11 Copenhagen interpretation 81 Correlated photon pairs 351 Countably normed space 324 Coupling

constant 152 rule for spherical harmonics 238

Creation operator 53 Cross section

Breit-Wigner 503 classical 363 collision, total 364 differential 363-365, 367, 382, 383,

491 elastic 523

total 365 energy-averaged 498

582 Index

Cross section (continued) formula 378

for coarse-resolution experiments 379

inelastic 365 Ith partial 414,421

elastic 421 inelastic 421

observed 500 partial 364 production 363 pure resonance 482 quantum mechanical 364, 368 reaction 523 scattering 363

differential 363, 364 total 363

total inelastic 365 inelastic partial

for the 11th level 365 e.s.c.o. 160, 388 Cyclic 11

Davisson 107, 116 De Broglie, Louis 48, 107 De Broglie relation 104, 209 Decay

probability 540 for a quasistationary state 565

processes 538 rate 539, 565

initial 539, 549 partial 567 per unit energy 565 total 566

Decaying state 18, 346, 537, 538 Decomposition of a space into

orthogonal subspaces 59 Defining algebraic relations 8

of the generators of the algebra of angular momentum 166

of the harmonic oscillator 45 of the spin operators 131

Degenerate case perturbation theory 243, 252 eigenvalue 66,217 spectrum 11, 66

Degenerate case perturbation theory 243

De Haas 265 Delta-function

normalization 87 Dense subspace 38, 326

Density matrix 64, 95 Derivative of an operator 325 Detailed balance, principle of 514 Deterministic 328, 571 Diagonal form, transformation of a

matrix into 22 Diatomic molecule, vibrating, rotating,

interacting 152 Differentiable operator function at a

point in a subspace 325 Differential cross section 365 Dimension of a subspace 66, 67 Dipole

moment for the helium atom, operator for 297

resonance 487 transitions 120,216

Dirac delta-function 15 picture 321

Dirac, P. A. M. 9, 80, 160, 566 Direct

product (see Tensor product) basis 146, 148, 167 of operators 147, 157 state 331, 352, 369

-product space 146 sum 59

of operators 157 Discrete

eigenvalues in the continuous spectrum 12,390

spectrum 11 Dispersion

of an observable in a state 76 relations 467

Distribution 15, 27 Double

ionization threshold of the helium atom 307

multichannel resonance 520, 526 cross section of 526

resonance 520 scattering 303

Doubling, parity 195 Doubly excited states of helium 308

broad profile of 309 Dual space 34 Dumbbell 129, 131, 198

tff(E 3 ) 182 tff(SO(3» 132 tff(SO(3, 1» 182 tff(SO(4» 182,212,213

,g'(SU(2» 132 (21 + I)-dimensional irreducible

representation space of 132, 136

Effective quantum number 231 range

approximation 443 of a potential 443

Efficiency function, detector 498 Eigenfunction

expansion 23, 24 generalized 27 of an operator 23

Eigenfunctions, orthogonal 24 Eigenkets II Eigenphase 518

repulsion theorem, Wigner's 520 -shifts 518

Eigenspace of an operator 63, 65, 66, 77

invariant 211 projection operator on 68

Eigenstate of an observable 77 almost 99 physical meaning 77 pure 62

Eigenstates of the energy operator, experimental verification that the physical states are 303

Eigenvalue 6, 72 complex 18, 547 degenerate 66 generalized 38 problem in matrix notation 22 proper 38

Eigenvector expansion 10, 11

generalized 11, 18 nondegenerate 11

generalized 11,38, 162 of a linear operator 6 proper 38, 546

Eigenvectors basis system of 11 complete system of 11

Einstein, Albert 80, 107, 265, 270, 353, 354

Elastic channel 418,419 cross section 523

total 365 scattering 384

amplitude 384, 422 partial-wave amplitude for 415

Index 583

Elastically scattered 57 Elasticity 533, 536 Electromagnetic spectrum 122 Electron 253 Electronic states of a molecule 155 Electrostatic units 206 Elementary

particle, free-nonrelativistic 103 rotator 164, 165

Emission of radiation 119 Empirically 350 Energy

conservation 379,401 diagram, symmetric top 201 distribution

Gaussian 486 Lorentzian 486

-level diagram 57 of the harmonic oscillator 57 rotation-vibration band 150

with P, Q, and R branches 202 level of the hydrogen atom, n2-fold

degeneracy of 217,220 loss 57 -loss

experiments 55 spectrum of helium 305

operator 313 experimental verification that the

physical states are eigenstates of 303

for the harmonic oscillator 45, 50 spectral representation 61

for the hydrogen atom 208 kinetic (see Kinetic energy

operator) of the quantum-mechanical

symmetric top 199 resonance 482, 535 shell 379 spectrum

experimental 58 of the harmonic oscillator 54 of the rotator 140 of the symmetric top 201

Ensemble 55 state of an 58

Entire function 446 Enveloping algebra 132 Environment, measurement processes

caused by 539 EPR paradox 353 Equation of motion, Heisenberg 315 Equilibrium, thermal 381 Essential singularity 445

584 Index

Essentially self-adjoint 6, 38, 69 Euclidean

group, three-dimensional 182 space 4

Even permutation 276 Exact

eigenvector 242 equation 290 Green's functions 398 Hamiltonian 242 radial wave function 427 state vector 392 wave function 427

Excitation energies of helium 303 Excited, doubly 308 Exclusion principle 280 Expansion

eigenvector 10 Fourier 25

Expectation value 64, 70, 79, 345, 346 in a pure state 62

Experimental data 49 energy spectrum 58 verification that the physical states

are eigenstates of the energy operator 303

Experiments actual 349 conceivable 349 formation 454 gedanken (see Gedanken experiment) production 454

Exponential decay law 540 deviations from 549

Extension of an operator 25, 38 of the scalar product 37 of the unitary time development

operator 562

Factorial, double 430 Fano shape parameters 494 Fermi statistics 280 Fermions 280 Final state 366 Fine-structure 124, 151, 162, 221

constant, Sommerfeld 209, 265 interaction 269 splitting 269

Finite -dimensional space 5

spectral theorem for 10 group 276

Force attractive 433 centrifugal 145· Lorentz 263 operator 316 repulsive 433

Formation experiments 454 process 537

Forming state 538 Forward scattering amplitude 386 Fourier

coefficients 25 expansion 25 series representation 25 transform 26, 353, 396

inverse 26, 396 of the unit step function 397,461

Fraction of a resonance 533 Franck 55, 80 Franck-Hertz experiment 55 Frechet-Riesz theorem 37,87 Free

eigenvector 242 Green's functions 398 Hamiltonian 242 -nonrelativistic elementary

particle 103 radial wave function 427 state 391

vector 391 wave function 427

Frequency 121 Function

analytic 444 branch of 449 Dirac delta- 15 eigenfunction expansion of 23 entire 446 generalized 15 generating, of the Hermite

polynomials 32 meromorphic 446

in a region 445 multivalued 448 normalized 15 of a complex variable 444 of operators 69, 88 sequences of delta-type 97 square integrable 17 well behaved 17

Functional 33 anti linear 34, 86-88 continuous 36 linear 34 value of 37

Galilei transformation 111 Gamow

kets 542, 557 vectors 542, 557

Gaussian energy distribution 486 Gedanken experiment (too idealized

to be performed in practice, and the results of which can only be obtained by thought based on many other experiences) 78, 79, 81, 328, 329,336,463

Gelfand, I. M. 81 Gelfand

- N aimark -Segal reconstruction theorem 8

triplet 37 Generalized

basis vector expansion 556, 562 Breit -Wigner approximation 522 eigenvalue 12, 38 eigenvectors 11, 12, 38

eigenvalues corresponding to 38 complex 18

normalization 16 function 15 optical theorem 406,418

Generated algebra 8 Generator 312

of time evolution (see Hamiltonian) 313

Generators of the group SU(2) 132 set of 8

Gerlach 336 Germer 107, 116 Gibbs distribution 71, 381 Golden Rule 563, 566 Goudsmit, S. 270 Green's function

advanced 398 exact 398 free 398 for the Hamiltonian, operator for

the 243 Ith-partial-wave 434 retarded 398

Ground state 58 Group

finite 276 of operators, one-parameter 312 orthogonal, in four dimensions

182 permutation 276

antisymmetrical representation of 279

Index 585

symmetrical representation of 279 unitary representation of 276

SU(2), generators of 132 symmetric 276

Gyromagnetic ratio 262

Hif -class function 554 Hamiltonian

exact 242 free 242 Green's function for the operator

for 243 operator 313

resolvent of the 243 perturbation 242

Hankel functions, spherical 431,471 Hardy class functions 553

from above, space of 553 from below, space of 553

Hardy class vector from above 554 Harmonic oscillator

algebra of operators 45 classical 46 energy spectrum of 54 one-dimensional 44

classical mechanical 45 selection rule for 120 wave-functions 89

Heisenberg commutation relation 47 equation of motion 315 picture 315 uncertainty relation 77

Heisenberg, Werner 47,48,81,270, 287, 366

Helicity 140 Helium atom 282

absorption spectrum of 306 doubly excited 308 energy operator of 283 excitation energies 303 space of physical states 285

Hermite differential equations 30 functions, recurrence relations

for 86 polynomials 30, 86

generating function of 32 orthogonality relation for 30, 89 recurrence relation for 33 Rodrigues formula for 30

Hermitian form 4

posItIve 4, 84 positive definite 4

586 Index

Hermitian (continued) operator 6, 7

vector 181 Hertz 55 Hidden variables 348, 353

local theories 354 theories 354

Hilbert, David 80 Hilbert space 17, 54, 324

convergence 35, 325 pre- 4 realization of 17 rigged 37, 87, 99, 559

Holism 572 Huygens, C. 106 Hydrogen atom

energy levels, low-lying 221 energy operator for 208

Hyperfine interaction 272 splitting 222

Identical quantum-mechanical particles 275

Identity operator 6 spectral resolution of 13, 69

In-state 393, 395, 551 Incoherent combination of states 73 Incompatible observables 72 Independent, linearly 5 Indistinguishability 277 Indistinguishable 275

particles 274 statement that identical particles

are 277 Indivisible whole 352 Inelastic

channel 418 cross section 365 process, threshold of 418 reaction processes 417

Inelastically 57 Inelasticity 523

coefficient 421, 533 Infinite

-dimensional space 5, 10 sequence, square summable 14

Infinitesimal operator 312 Infrared spectrum of HCI 123, 126 Inhomogeneous magnetic field 340,

342 Initial

decay rate 539 state 366

Inonii-Wigner contraction 189 Integral transform 28 Interaction 148, 152

free state 369 picture 322

Interference 74 Interfering Breit - Wigner

resonances 531 Interpretation, Copenhagen

(statistical) 81 Intertwining relations 401 Intrinsic

angular momentum 49, 254 parity 195, 197

Invariance, parity 260 Invariant

operator (see Casimir operator) space 211

Inverse Fourier transform 26, 396 of an operator 7, 208

Involution 8 Ionization

channel 418 energy 296 -excitation channels 419 potential 296 threshold

nth 292,307 of the helium atom 307

Irreducible representation space of 6'(SO(4)) 212 of 6'(S U(2)) 136

Isolated physical systems 311, 318, 323, 539 singularities 445

Isometric operator 402 scalar product spaces 5

Isomorphic linear spaces 5 scalar product spaces 5

Isomorphism 5

Jordan, P. 48, 80

Kepler problem classical 206 quantum-mechanical 205,208

Kernel of an integral transform 28 Ket 9

Gamow 542, 557

Kinetic energy operator for a massive nonrelativistic

particle 370 for a photon 370

Kronecker delta 13, 15 Kronig, R. 270

Ladder operators 53 representations 133, 188

Laguerre polynomials 239 Lamb shift 269 Lande 270 Lande factor 262 Larmor

frequency 341 precession 340, 341

Laurent series 445 Legendre polynomials 238,501

orthogonality property of 414 associated 238

Lenard, P. 107 Lenz vector 207, 209

quantum-mechanical 209 Level shift 244

operator 243 Lifetime 540, 549 Line width, natural 565, 570 Linear

combination 5 functional 34 operator 6, 508

eigenvalue of 6 eigenvector of 6

space 3 Stark effect 252 topological space 324

Linearly dependent 5 independent 5

Lippmann-Schwinger equation 250, 359,387,426

Local hidden variables theories 354 London,F. 80 Lorentz

force 263 group 182

Lorentzian energy distribution 486, 543 momentum distribution 115

lth partial cross section 414,416,421 S-matrix element, symmetry relation

for 472,474

Index 587

-wave Green's function 434 Lyman series 220

Magnetic field, inhomogeneous 340, 342

Magnetic moment anomalous 265 of a neutron 265 of a proton 265 of a spinning particle in classical

physics 262 of an electron 264 operator of a particle without intrinsic

magnetic moment 264 Magneton, Bohr 266 Many-particle systems 274 Mass

operator 568 reduced 46, 129

Mathematical image of a physical system 44, 100

Matrix elements 21 of a map 40 of an operator 20 orthogonal 22 symmetric 514 unitary 22

Mean life 540 Measurement 66, 76, 339

change of state by 72, 328, 338, 539 immediate repetition of 65, 66 numbers obtained 7, 39 of spin components 342 process caused by the

environment 539 Meromorphic function 446

in a region 445 Metastable states 454 Microreversability, principle of 514 Minimal coupling to the

electromagnetic field 265 Mixing, singlet-triplet 299 Mixture 57, 58, 63, 72 Meller wave operators 400

adjoint 403 Moment of inertia 128 Momentum

operator for the harmonic oscillator 45 radial 228

representation, wave functions in 90 space wave function 91 transfer 384

588 Index

Monochromator 55 Motion, Heisenberg equation of 315 Multichannel resonance 517

double 520 single 522

experimental examples 525 Multiplication

of an operator by a number 6 of two operators 6

Multipole transitions 216 Multivalued function 448 Mutually orthogonal subspaces 59

n-dimensional space 5 basis in 5

Natural line width 565, 570 Negation of a property 109 Negative-energy space 211 Neumann function, spherical 430 Newton 47, 106 Newton's equation, quantum-

mechanical 316 Nondecay probability 540 Nondegenerate spectra 159 Nonrelativistic symmetric top,

classical 199 Nonresonant background, slowly

varying 496 Nordheim, L. 80 Norm 4 Normalized 4

delta-function 15 Normed space

countably 324 nth ionization threshold Nuclear spectral theorem

163

Observable 61 dispersion of 76

292 10, 16, 162,

expectation value of 64 in a pure state 62

experimentally determined value of 61

mathematical image of 62, 71 physical 44 pure eigenstate of 62 values of 71

Observables complete system of commuting

(c.s.c.o.) 160 incompatible 73

Odd permutation 276 Off the energy shell 379 Old quantum theory 47 On the energy shell 379 One-parameter group of

operators 312,325 One-particle systems 274 Operator

addition 6 adjoint of 6, 37 angular momentum 130

total 166 antiadjoint of an 508 antilinear 6, 508 antiunitary 508 axial-vector 193 Casimir 133, 212 commuting 7, 69 conjugate 559, 562 continuous 37 derivative, at a point 325 eigenfunctions of 23 eigenspace of 66 eigenvalue of 72

generalized 38 extension of 38 for the Green's function for the

Hamiltonian 243 function 69, 324

continuous 324, 325 differentiable 325

infinitesimal 312 inverse of 7, 208 isometric 402 ladder 53 level shift 243 linear 6, 508 matrix of 20 Moller wave 400

adjoint of 403 multiplication 6

by a number 6 positive 58 positive definite 208 product 7 projection 13, 58 proper tensor 193 pseudotensor 193 regular tensor 176 scalar 176 self-adjoint

complex eigenvalue of 18, 547 spectral resolution of a 13, 16

semibounded 471 semilinear 44, 508

semiunitary 509 skew-Hermitian 313 spectral family of 69 spin 138 square root of 208 statistical 64, 65, 83, 368, 548 tensor

ofrankj 176 regular 176

trace of 59 unit 6 unitary 7, 192,402 vector 138, 176 well defined 40 spectral representation of 69 *-algebra 8, 161

Operators algebra of, of the harmonic

oscillator 45 direct product of 157 direct sum of 157 one-parameter group of 312, 325

Optical theorem 422 generalized 406, 418

Orthogonal basis functions 25 decomposition of a space into

subspaces 59 eigenfunctions 24 matrix 22 subspaces, mutually 59 vectors 4

Orthogonality property of the Legendre polynomials 414

Orthohelium 287, 296, 297 Orthonormal

basis functions 29 system 9

on an interval 32 Orthonormality of spherical

harmonics 460 Oscillator, harmonic 44

wave functions 89 Out-state 393, 395, 551

P branch 124, 149, 202 p-representation, wave function in 91 Paley-Wiener Theorem 554 Parahelium 287, 296, 297 Parity 195

conservation 300 doubling 195

intrinsic 195, 197 invariance 260, 505 -invariant 505 nonconservation 193 operation 192 operator 192

Index 589

for a two-electron system 298 relative 217

Partial cross section 364

lth 414 decay rate 567 -wave

amplitude 415 for elastic scattering 415 resonant 481

reaction amplitude 415 width 522

Particle 102, 103 elementary 106

free-nonrelativistic 103 extended 131

Particles identical 275 indistinguishable 274

Pauli exclusion principle 280 matrices 138

Pauli, W. 205, 270 Permutation 275

even 276 group 276

antisymmetrical representation of 279

symmetrical representation of 279 unitary representation of 276

odd 276 operation 276

Perrin, 1. 107 Perturbation 242

Hamiltonian 242 of the continuous spectrum 248 series

Rayleigh-Schrodinger 247 Wigner-Brillouin 247

theory 225, 242 degenerate case 243, 252

Phase factor 30 shift 437

analysis 489 background 480, 501 of scattering 420 potential part of 480 resonant part of 480

590 Index

Physical basis 166 observable 37, 39,44 sheet of the S-matrix 473 states 3, 138, 552

preparable 74 space of 99, 210

systems 8 conservative 318 isolated 318 mathematical image of 44, 100 theories of 124, 571

Picture Dirac 321 Heisenberg 315 interaction 322 Schrodinger 314,315

Planck 107 Planck

- Einstein relation 209 formula 107

Planck's constant 45, 48, 110, 112, 209 Plane wave 103 Podolsky 353 Polarization 368, 381 Polarized 328 Pole

bound state 440,475, 550 of order n 445 simple 445

Position operator for the harmonic oscillator 45

Positive definite

Hermitian form 4 operator 208

Hermitian form 4 operator 58

Postulates, basic I 44 II' 62 II 64,346 IlIa 67,99 IIIb 72, 100, 339 IVa 147,343 IVb 280 Va 314 Vb 315 Vc 318

Potential attractive 433

in an interval 433 effective range of 443 part of the phase shift 480 repulsive 433

Pre-Hilbert space 4 Precession, Larmor 340, 341 Preparation

apparatus 74 of a state 65, 72, 137, 346, 352

Principal quantum number 215 -value integral 249

Principle of detailed balance 514 of microreversability 514 of stability 492

Probabilistic 572 Probability 65

amplitude 105 density 80 interpretation 80 observable 65, 70 of decay 540 of nondecay 540 of transition 358

per unit time 368 Product of a vector with a number 3 Production

cross section 363 experiments 454

Profile index 494 Projection operator 13, 58, 88

on an eigenspace 68 Projector 58 Proper

eigenvalue 38 eigenvector 11, 38 tensor operator 193

Property 65 negation of 109

Proposition 65 Pseudo-orthogonal group in

(3 + I)-dimensions 182 Pseudotensor operator 193 Pure

eigenstate 62, 76 state 58, 278

mathematical image of 39, 61, 62, 64,67,72

Q-branch 202 Quantum

defect 231 electrodynamics 221 -mechanical

Lenz vector 209 Newton's equation 316

mechanical symmetric top 198 mechanics, some fundamental

properties of 63, 78, 328 number 160

angular momentum 137 principal 215 vibrational 141

Quasistationary states 454, 464, 477, 542

R-branch 124, 149,202 Rabi 265 Radial

momentum operator 228 wave function

Rate

asymptotic behavior of 435 exact 427 free 427

of decay 539 of initial decay 539 of partial decay 567 of transition 358

Ray 61 Rayleigh-Schrodinger perturbation

series 247 Reaction

amplitude, partial-wave 415 cross section 523

processes, inelastic 417 Realization 3, 93, 165

of linear spaces 18 of operators 18

Rearrangement channel 419 collision 419 -excitation channels 419

Reciprocity theorem 514 experimental check of 514

Recurrence relation for the Hermite functions 86 for the Hermite polynomials 33

Reduced mass 46, 129 matrix element 177 momentum 206

Reducible representation space 139 Reduction of a space into a sum of

irreducible total angular momentum spaces 170

Regular tensor operator 176 Relative probabilities 58 Relativistic mass effect 267

Representation 138 antisymmetrical 279 ladder 188

Index 591

momentum, wave functions in 90 Schrodinger 93 space

irreducible 136 reducible 139

symmetrical 279 unitary 182,276

Repulsive force 433 potential 433

Residue of a function 446 Resolution

of an apparatus 97, 99, 498, 499 of the identity operator with respect

to a spectral family 16, 69 Resolvant for the operator H 243 Resonance 309, 378,454,455,476,

485, 542 cross section, pure 482 double 520 energy 482, 535 fraction of 533 multichannel 517, 520, 522, 525 poles 476, 478, 483, 485 total width of 536 tripole 504 width 482

Resonant part of the phase shift 480 partial-wave amplitude 481

Rest energy 267 Retarded Green's function 398 Riccati

- Bessel function 430 -Neumann function 430

Riemann sheet 450 surface 449, 450

Rigged Hilbert space 37, 87, 99, 559 description 100

Rigid rotator 128, 140 Rodrigues formula for the Hermite

polynomials 30 Root-mean-square deviation 78 Rosen 353 Rotating-oscillator 146 Rotation 343

angle 343 axis 343 group, quantum-mechanical 138 spectra 138 -vibration band 150

592 Index

Rotational absorption spectra of HCl 143 constant of CO 152 constant of H 2 molecule 152 constants of HCl 151 invariance 257

Rotator 139 algebra of operators 138 elementary 164, 165 energy spectrum 140 nonrigid 144 space of physical states of 138 spectrum generating algebra of 140

Runge-Lenz vector 207 Rydberg constant 218,219

S-matrix 366,399,404,416 pair of second sheet poles of 542 theory 366, 406

S-operator 366, 399, 417 Scalar

operator 176 product 4

continuous case 14 extension of 37

product space 4 Scattered elastically 57 Scattering

amplitude 413 elastic 384, 422 forward 386

angle 384 channels 419 cross section 363

differential 363, 364 total 363

elastic 384 partial-wave amplitude for 415

experiment 363 lengths 443 operator 399, 401 phase shift 420 potentials 432 states 218,360,390

space of 389, 390 theory, formal 387

Schrodinger equation 318, 319

for the one-electron atom 235 for the radial wave function 428 time-independent 93

picture 214,215 representation 93

Schrodinger, Erwin 47, 48

Schrodinger's wave mechanics 80 Schulz resonance 454, 456 Schwartz, L. 81 Schwartz space 99, 439 Schwarz reflection principle 447 Schwinger 265. Selection

of a subensemble 72, 100 rules 120, 270

for dipole radiation of the rotator 141

for the harmonic oscillator 120 for the helium atom 298 for the hydrogen atom with

spin 270 Self-adjoint operator 6

complex eigenvalues of 547 essentially 38

Semibounded operator 471 Semigroup, additive 311 Semilinear operator 44, 508 Semi unitary operator 509 Set of generators 8 Shape

parameter, Fano 494 profile parameter 494

Sheet physical 473 Riemann 450 unphysical 473

Siegert poles 476,481, 542 Simple

pole 445 spectra 159

Simultaneous eigenstates 82 eigenvectors 133, 344 measurements 82, 344, 345

Singlet states 287, 296, 346 -triplet mixing 299

Singularities 445 essential 445 isolated 445

Skew-Hermitian operator 313 vector operator 181

Slowly varying nonresonant background 496

Solid angle delta-function 385 Sommerfeld fine-structure

constant 209, 265 Space

antisymmetric 284 conjugate 34

countably normed 324 decomposition of 59 direct-product 146 dual 34 Euclidean 4

three-dimensional 59 finite dimensional 5 Hilbert 17, 54 inversIOn 192 irreducible representation 136 linear 3 linear topological 37, 324 n-dimensional 5

basis in 5 negative-energy 211 of bound states 211 of continuous antilinear

functionals 11 of continuous functionals 36 of Hardy class functions from

above 553 of Hardy class functions from

below 553 of physical states 9, 99, 147, 210,

325, 554 of the hydrogen atom 215 of the quantum-mechanical

rotator 138 of scattering states 390 of square integrable functions 17,

557 pre-Hilbert 4, 17 reducible representation 139 rigged Hilbert 37, 554 Schwartz 17,40, 439 self-dual 35 spanned by a vector 59 symmetric 284

Spectral decomposition of a vector 10 family of an operator 69 representation of an operator with

discrete spectrum 61,69 resolution

of a self-adjoint operator 13, 16 of the identity operator 13, 69

series of hydrogen 220 theorem 10- 12, 68, 162

Spectrum 10, 11, 54 absolutely continuous 16 continuous 11, 87

discrete eigenvalues in 12 degenerate 66 discrete 11, 87 electromagnetic 122

generating-algebra 216 of the rotator 140

generating-group 216 nondegenerate 159 simple 159

Spherical

Index 593

Bessel function 430 components of a vector

operator 176 coordinates 236,371

delta-function normalization in 432

Hankel functions 431,471 harmonics 237

addition theorem for 238,413 completeness relation of 238 coupling rule for 238 orthonormality of 460

Neumann function 430 symmetry 416

Spherically symmetric interaction 384 Spin 255

and statistics, connection between 280

correlation observable 344 correlations in a singlet state 342,

343 -down 332 electron 253 operators 131, 138 -orbit interaction 265, 300, 302 -up 332

Splitting fine-structure 269 hyperfine 222

Square integrable function 17 summable infinite sequence 14

Stable states 454 Standard deviation 78 Stark effect, linear 252 State

almost eigen- 99 almost sharp 99 capture 545, 569 change of 72 controlled 560 decaying 18,537,538 discrete physical 58 dispersion of an observable in 76 exact 392 forming 538 free 391 ground 58 in- 393, 395

594 Index

State (continued) interaction free 369 mathematical image of 62, 64, 67 metastable 454 momentum space wave function

for 91 nonstationary 357 of a combination of physical

systems 331,369 of a system after a measurement 72 of an ensemble 58 of the target 369 out- 393, 395 parity of 195 physically preparable 74, 99 preparation of a 65 pure 58, 62, 64, 67, 278 quasistationary 454,464,477, 542 sharp 97 singlet 287, 296 stable 454 stationary 320, 454 time-reversed 509 transition probability into 358 value of an observable in 61 vector 61

exact 392 free 391

vectors, coherent combination of 73 velocity reversed 509 virtual 475, 550

Statement for a basis 60 that identical particles are

indistinguishable 277 States

incoherent combination of 73 mixture of 58 space of bound 211 space of physical 99, 210 space of scattering 390 subspace of bound 390 subspace of scattering 389 triplet of 287, 296

Stationary state 320,454 system 320

Statistical operator 64, 83, 95, 368 choice of 65 of a decaying state 548 of a pure state 95

Statistics Bose 280 Fermi 280

Stern 336

Stern-Gerlach experiment 254, 329, 342

Stronger convergence 36 Structureless particles 55 SU(2)

enveloping algebra of 132 generators of the group 132

Subalgebra 7 Subensemble, selection of 72, 100 Subspace 5

antisymmetric 284 dense 326 invariant 139 of bound states 390 of scattering states 389 symmetric 284

Subspaces, orthogonal 59 Sum, direct 59 Superposition 73

principle 74 Symmetric

(permutation) group 276 interaction, spherically 384 operator 6 subspace 284 top

classical nonrelativistic 199 energy diagram 202 quantum mechanical 198

energy operator of the 199 energy spectrum of the 201

vectors 280 Symmetrical representation 279 Symmetry

breaking 231 relation for the lth partial S-matrix

element 472,474 relations of the Clebsch-Gordan

coefficien ts 172 spherical 416 transformation 318

System energy operator of 313 Hamiltonian operator of 313 isolated physical 318 many-particle 274 mathematical image of 62 one-particle 274 stationary 320

T-matrix 367,379,416 off-the-energy-shell 379 on-the-energy-shell 379, 413

Target 369 Taylor series 444 Tensor

operator 176 of rankj 176 proper 193 pseudo- 193 regular 176

product 146 Term

diagram for energy levels of helium 295

value 153 Thermal equilibrium 381 Thomas

factor 266 precession 266

Thomas, L. H. 270 Thomson, G. P. 107 Thomson, J. J. 106 Threshold

nth ionization 292 of an inelastic process 418

Time 310 delay 458, 462

average 462 dependence, explicit 323 evolution 310

generator of 313 -reversal 507

in variance 509 operator 507

-reversed state 509 Titchmarch Theorem 554 Top (see Symmetric top) Topological

space 35 linear 324

structures 19, 35 Topology 35, 36, 99, 325

coarser 36 finer 36

Torque 262,263 Total

-angular momentum operators 166 collision cross section 364 decay rate 566 elastic cross section 365 inelastic cross section 365 inelastic partial cross section for the

11th level 365 scattering cross section 363 width of a resonance 536

Trace of an operator 59 Transformation matrix 21

Transition amplitude 523 cascade 351 coefficients 21, 23, 27 dipole 216 matrix 21 muItipole 216 probability 358

per unit time 368 rate 358 two-step 351

Transpositions 276 Triplet states 287, 296

Index 595

Tripole resonance 504 Two-electron excitation states in

neutral helium 306 Two-step transition 351

Uhlenbeck, G. E. 270 Uncertainty

of classical physics 354 quantum-mechanical 354 relation, Heisenberg 77, 82

Unit operator 6 step function 393

Fourier transform of 397,461 ray 61

Unitarity of the S-operator 417 Unitary

circle 422, 490 deficiency 403 matrix 22 operator 7,192,402 representation

of E3 192 of SO(3, 1) 192 of SO(4) 192 of the permutation group 276

Units 109-112,209 Unphysical sheet of the S-matrix 473

Value of an observable 71, 76, 344 average 62, 76, 345 expectation 64, 345

in a pure state 62 experimentally

determined 61, 72, 76, 86, 344 produced 347, 353 uncovered 347, 353

Vector 2 addition 3 antisymmetric 280

596 Index

Vector (continued) axial- 193 components of II exact state 392 free state 391 Gamow 542, 557 Hardy class

from above 554 operators 138, 176

Hermitian 181 skew-Hermitian 181 spherical components of 176

product with a number 3 space spanned by a 59 spectral decomposition of 10 state 61 symmetric 280 value of a functional at a 37 weight 133

Velocity-reversed state 509 Vibrating rotator 146, 148 Vibration-rotation band of carbon

monoxide 124, 125 Virtual

state 475, 550 -state poles 475, 487

Voigt integral 502 V oigt profile 502 Von Kampen causality condition 464 Von Neumann, John 80,324

Wave 103 function 80, 105

exact 427 exact radial 427 free 427 free radial 427 in the momentum

representation 90 momentum space 91

number 121 operator, Moller 400

packet 103, 550 particle-duality 81 plane 103

Wave functions in the momentum representation 90

Weaker convergence 36 Weight 133

diagram 136 function 16, 568 highest 134 vectors 133

Well-behaved functions 15, 17, 27, 40 Well-defined operator 40 Wheeler, J. A. 351, 366 Width

natural line 565, 570 of a resonance 482

total 536 Wigner

-Brillouin perturbation series 247 coefficients 167 - Eckart theorem 177 theorem 132,509,512 3-j symbols 173-175

Wigner, E. P. 138 Wigner's

causality inequality 466 eigenphase repulsion theorem 520

Young 106 Yukawa interaction 385

Zeeman effect 273 anomalous 270

Zero operator 6 Zero vector 3

*-algebra 8 *-operation 8 =2 214