Table Of ContentTheoretical and Quantum Mechanics
Theoretical and Quantum Mechanics
Fundamentals for Chemists
by
STEFAN IVANOV
Sofia University, Bulgaria
AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress.
ISBN-10 1-4020-3365-6 (HB)
ISBN-13 978-1-4020-3365-0 (HB)
ISBN-10 1-4020-3688-4 (e-book)
ISBN-13 978-1-4020-3688-0 (e-book)
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CONTENTS
PART ONE
SELECTED TOPICS OF THEORETICAL
MECHANICS
PHYSICAL CONSTANTS AND UNITS......................................................................................xi
LIST OF SYMBOLS.....................................................................................................................xii
PREFACE.....................................................................................................................................xxi
ACKNOWLEGEMENTS..........................................................................................................xxiv
1. LAGRANGIAN FUNCTION AND LAGRANGE’S EQUATIONS................................3
1.1 TWO METHODS FOR STUDYING OF THE MECHANICAL MOTION: NEWTONIAN
AND LAGRANGIAN ........................................................................................................4
1.2 HAMILTON’S PRINCIPLE AND LAGRANGE’S EQUATIONS......................................................7
1.3 LAGRANIAN FUNCTION FOR A FREE PARTICLE..................................................................12
1.4 LAGRANIAN FUNCTION FOR A SYSTEM OF INTERACTING PARTICLES.................................15
1.5* LAGRANIAN FUNCTION FOR A CHARGED PARTICLE IN ELECTROMAGNETIC FIELD..............18
1.6 SOME MECHANICAL VALUES AND THE LAGRANGIAN........................................................20
1.7+ LAGRANIAN FUNCTION AND LAGRANGE’S EQUATIONS IN COMMONLY USED
CO-ORDINATES.................................................................................................................22
SUMMARY........................................................................................................................27
QUESTIONS......................................................................................................................29
PROBLEMS.......................................................................................................................30
2. CONSERVATION LAWS................................................................................................31
2.1 CONSERVATION OF ENERGY.............................................................................................32
2.2 CONSERVATION OF MOMENTUM.......................................................................................36
2.3 CONSERVATION OF ANGULAR MOMENTUM......................................................................38
SUMMARY........................................................................................................................42
QUESTIONS......................................................................................................................43
PROBLEMS.......................................................................................................................44
3. CENTRAL CONSERVATIVE FORCES........................................................................45
3.1 ONE-DIMENSIONAL MOTION.............................................................................................46
3.2 THE TWO-BODY PROBLEM................................................................................................48
3.3 CENTRAL CONSERVATIVE FORCES ...................................................................................52
*These sections are more difficult and can thus be omitted at the first reading.
+These sections are considered in seminars.
vi
3.4* ORBITS OF MOTION..........................................................................................................56
3.5* ELASTIC COLLISIONS........................................................................................................62
3.6* SCATTERING OF PARTICLES. FORMULA OF RUTHERFORD.................................................67
SUMMARY........................................................................................................................71
QUESTIONS......................................................................................................................73
PROBLEMS.......................................................................................................................74
4. ROTATING MOTION OF A CHARGED PARTICLE.................................................75
4.1* ROTATING FRAMES...........................................................................................................76
4.2 PARTICLE IN A UNIFORM MAGNETIC FIELD........................................................................79
4.3 LARMOR EFFECT..............................................................................................................81
4.4* INTERACTION OF MAGNETIC DIPOLE WITH A MAGNETIC FIELD..........................................84
SUMMARY........................................................................................................................88
QUESTIONS......................................................................................................................89
PROBLEMS.......................................................................................................................90
5. SMALL HARMONIC OSCILLATIONS........................................................................91
5.1 ONE-DIMENSIONAL HARMONIC OSCILLATIONS.................................................................92
5.2* OSCILLATIONS OF A SYSTEM OF PARTICLES......................................................................99
5.3 NORMAL CO-ORDINATES AND NORMAL MODES..............................................................102
5.4* COUPLED OSCILLATORS.................................................................................................104
5.5 VIBRATIONS OF MOLECULES..........................................................................................113
SUMMARY......................................................................................................................117
QUESTIONS....................................................................................................................119
PROBLEMS.....................................................................................................................120
6. HAMILTONIAN MECHANICS....................................................................................121
6.1 HAMILTONIAN FUNCTION AND HAMILTON’S EQUATIONS................................................122
6.2 CONCEPT OF CANONICAL TRANSFORMATIONS,PHASE SPACE AND LIOUVILLE’S
THEOREM.......................................................................................................................125
6.3* HAMITONIAN OF A CHARGED PARTICLE IN A ELECTROMAGNETIC FIELD..........................129
6.4 POISSON BRACKET.........................................................................................................131
6.5+ HAMILTONIAN AND HAMILTON’S EQUATIONS IN CARTESIAN,CYLINDRICAL AND
SPHERICAL CO-ORDINATES.............................................................................................134
SUMMARY......................................................................................................................143
QUESTIONS....................................................................................................................144
PROBLEMS.....................................................................................................................145
PART TWO
BASICS OF QUANTUM MECHANICS
7. THE SOURCES OF QUANTUM MECHANICS.........................................................149
7.1 EVIDENCE OF THE INADEQUACY OF CLASSICAL PHYSICS................................................151
7.1.1 Blackbody radiation..............................................................................................152
7.1.2 The photoelectric effect..........................................................................................154
7.1.3 The dependence of specific heat on temperature...................................................155
7.1.4 Atomic spectrum and structure of an atom............................................................156
7.1.5 Roentgen rays and the Compton effect..................................................................160
7.2 THE FIRST STEPS IN QUANTUM MECHANICS....................................................................162
7.2.1 Planck’s hypothesis and the blackbody radiation - quanta...................................162
7.2.2 Quanta and the photoelectric effect.......................................................................164
vii
7.2.3 Quanta and the specific heat of solids...................................................................164
7.2.4 The Bohr’s theory of the atom and atomic spectrum.............................................165
7.2.5 Photons and Compton effect..................................................................................166
7.3 WAVES VERSUS PARTICLES -DUAL NATURE OF THE RADIATION.....................................168
7.4 DEBROGLIE’S WAVES.WAVE-PARTICLE DUALITY.........................................................176
SUMMARY......................................................................................................................180
QUESTIONS....................................................................................................................181
PROBLEMS.....................................................................................................................182
8. OLD QUANTUM MECHANICS...................................................................................183
8.1 BOHR’S MODEL..............................................................................................................184
8.2 INTERPRETATION OF THE QUANTIZATION RULES............................................................190
8.3 SOMMERFELD’S MODEL.................................................................................................194
8.4 SUCCESS AND FAILURES OF THE OLD QUANTUM THEORY...............................................197
SUMMARY......................................................................................................................200
QUESTIONS....................................................................................................................201
PROBLEMS.....................................................................................................................201
9. THE WAVE FUNCTION...............................................................................................203
9.1 STATISTICAL INTERPRETATION OF THE DE BROGLIE’S WAVES........................................204
9.2 THE WAVE FUNCTION.....................................................................................................206
9.3 THE WAVE FUNCTION OF A PARTICLE SYSTEM................................................................209
9.4 MEASUREMENT,EIGENSTATES AND EIGENVALUES OF THE OBSERVABLES.....................211
9.5 THE SUPERPOSITION PRINCIPLE......................................................................................215
9.6* SUPERPOSITION OF STATES WITH A CONTINUOUS SPECTRUM..........................................221
SUMMARY......................................................................................................................225
QUESTIONS....................................................................................................................226
PROBLEMS.....................................................................................................................227
10. OPERATORS..................................................................................................................229
10.1 MEAN VALUES AND OPERATORS....................................................................................230
10.2 OPERATORS IN QUANTUM MECHANICS...........................................................................234
10.3 EIGENVALUES AND EIGENFUNCTIONS OF OPERATORS.....................................................236
10.4 OPERATORS OF THE CO-ORDINATES AND THE MOMENTUM AND THEIR EIGENFUNCTIOS AND
EIGENVALUES................................................................................................................240
10.5+ ANGULAR-MOMENTUM OPERATOR................................................................................243
10.6 EIGENVALUES AND EIGENFUNCTIONS OF THE OPERATORS Lˆ AND Lˆ2...........................248
z
10.7 VECTOR INTERPRETATION OF THE ANGULAR MOMENTUM..............................................251
10.8 ENERGY OPERATORS......................................................................................................254
10.9 THE HEISENBERG UNCERTAINTY PRINCIPLE...................................................................256
SUMMARY......................................................................................................................260
QUESTIONS....................................................................................................................261
PROBLEMS.....................................................................................................................262
11. THE SCHROEDINGER EQUATION AND TIME DEPENDENCE OF THE
OBSERVABLES............................................................................................................265
11.1 THE SCHROEDINGER WAVE EQUATION...........................................................................266
11.2 STATIONARY STATE........................................................................................................269
11.3* DERIVATIVE OF AN OPERATOR WITH RESPECT TO TIME...................................................272
11.4 CONSTANTS OF THE MOTION..........................................................................................275
SUMMARY......................................................................................................................278
QUESTIONS....................................................................................................................279
PROBLEMS.....................................................................................................................280
viii
12. MOTION IN ONE-DIMENTIONAL POTENTIAL....................................................281
12.1 CONSTANT POTENTIAL FIELD..........................................................................................282
12.2 THE INFINITE SQUARE WELL POTENTIAL.........................................................................285
12.3 THE STEP POTENTIAL......................................................................................................290
12.4* THE FINITE SQUARE WELL POTENTIAL.............................................................................294
12.5+ THE BARRIER POTENTIAL...............................................................................................301
12.6 THE TUNNEL EFFECT AND (cid:505)LPHA-DECAY.......................................................................303
SUMMARY......................................................................................................................309
QUESTIONS....................................................................................................................310
PROBLEMS.....................................................................................................................311
13. THE HARMONIC QUANTUM OSCILLATOR.........................................................313
13.1* THE SOLUTION OF THE OSCILLATOR EQUATION..............................................................314
13.2 GROUND STATE AND ZERO-POINT ENERGY.....................................................................320
13.3 EIGENSTATES OF THE HARMONIC OSCILLATOR...............................................................322
13.4* COUPLED QUANTUM-MECHANICAL OSCILLATORS AND VAN DER WAALS FORCES..........326
SUMMARY......................................................................................................................333
QUESTIONS....................................................................................................................334
PROBLEMS.....................................................................................................................335
14. THE TWO-BODY PROBLEM – THE HYDROGEN ATOM AND A DIATOMIC
MOLECULE.....................................................................................................................337
14.1* PARTICLE MOTION IN A CENTRAL SYMMETRIC FIELD......................................................338
14.2* ELECTRON IN A COULOMB FIELD....................................................................................341
14.3 SPECTRUM OF THE ENERGY OF THE HYDROGEN ATOM AND HYDROGEN-LIKE IONS.........347
14.4 PROBABILITY DISTRIBUTION OF THE ELECTRON..............................................................354
14.5 HYDROGEN-LIKE ATOMS................................................................................................360
14.6 POTENTIAL ENERGY OF A DIATOMIC MOLECULE.............................................................364
14.7* VIBRATIONAL AND ROTATIONAL LEVELS OF A DIATOMIC MOLECULE.............................367
14.8 SPECTRA OF A DIATOMIC MOLECULE..............................................................................373
SUMMARY......................................................................................................................378
QUESTIONS....................................................................................................................380
PROBLEMS.....................................................................................................................382
15. PERTURBATION THEORY AND STARK EFFECT................................................385
15.1* ELEMENTS OF MATRIX MECHANICS................................................................................386
15.2* PERTURBATION THEORY................................................................................................392
15.3* TIME-INDEPENDENT DEGENERATE PERTURBATION THEORY...........................................397
15.4* THE STARK EFFECT........................................................................................................400
SUMMARY......................................................................................................................404
QUESTIONS....................................................................................................................406
PROBLEMS.....................................................................................................................406
16. SPIN..................................................................................................................................409
16.1 SPIN,SPIN MATRICES AND SPIN FUNCTIONS....................................................................410
16.2 ADDITION OF ANGULAR MOMENTA IN QUANTUM MECHANICS........................................416
16.3 TOTAL ANGULAR MOMENTUM.......................................................................................421
16.4 LS AND JJ COUPLING.....................................................................................................424
SUMMARY......................................................................................................................427
QUESTIONS....................................................................................................................429
PROBLEMS.....................................................................................................................430
ix
17. INTERACTION OF AN ELECTRON IN A ATOM WITH A MAGNETIC FIELD431
17.1 A MAGNETIC FIELD AND AN ELECTRON IN AN ATOM.......................................................432
17.2 THE SPIN-ORBIT INTERACTION........................................................................................438
17.3 NORMAL ZEEMAN EFFECT..............................................................................................444
17.4* COUPLING TO THE ELECTROMAGNETIC FIELD ................................................................448
17.5 ANOMALOUS ZEEMAN EFFECT.......................................................................................451
SUMMARY......................................................................................................................455
QUESTIONS....................................................................................................................456
PROBLEMS.....................................................................................................................457
18. THE HELIUM ATOM....................................................................................................459
18.1 IDENTICAL PARTICLES....................................................................................................460
18.2 SYMMETRIC AND ANTISYMMETRIC STATES....................................................................463
18.3 SPIN FUNCTIONS OF HELIUM...........................................................................................465
18.4 VECTOR INTERPRETATION OF THE SPIN STATES. PARAHELIUM AND ORTHOHELIUM.......468
SUMMARY......................................................................................................................472
QUESTIONS....................................................................................................................473
PROBLEMS.....................................................................................................................474
19. PERIODIC TABLE.........................................................................................................475
19.1 MULTIELECTRON ATOMS – THE HARTREE THEORY........................................................476
19.2 SHELL STRUCTURE OF THE ATOM...................................................................................481
19.3 ELECTRON CONFIGURATION AND THE PERIODIC TABLE..................................................488
19.4 HUND’S RULE.................................................................................................................492
19.5 FINE STRUCTURE OF THE MULTIELECTRON ATOMS.........................................................497
SUMMARY......................................................................................................................500
QUESTIONS....................................................................................................................501
PROBLEMS.....................................................................................................................502
REFERENCES (WITH VERY SHORT COMMENTS FOR EACH BOOK).................503
INDEX..........................................................................................................................................507
PHYSICAL CONSTANTS AND UNITS
Name Notation Value
Angstrom Å 10−10 m
Atomic mass unit amu 1,660.10-27 kg
Avogadro's number N 6,022.1023 mol-1
A
Boltzmann constant k 1,381.1023 J.K-1
Bohr magneton μ 9,274.10-24 J.T-1
B
Bohr radius a 0,529.10−10 m
0
Calorie cal 4,19 J
Coulomb constant k =1/4πε 8,99.109 N.m2/C2
0 0
Electron charge e 1,602.10-19 C
Electron mass m 9,109.10−31 kg
e
Electronvolt eV 1,602.10-19 J
Fine-structure constant α=e2k (cid:61)c 1137,036=7,297.10−3
0
Gauss Gs 10−4 T
Gravitational constant G 6,67.10−11 N.m2.kg-2
0
Permeability of vacuum μ 4π.10−7 N.s2/C2
0
Permittivity of vacuum ε 8,854.10−12 C2/N.m2
0
Planck's constant h 6,626.10−34 J.s
(cid:61)=h 2π 1,055.10−34 J.s
Rydberg constant R 1,097.107m-1
ℜ=ch 2,17.10−18J=13,6 eV
R=Rc 3,27.1015 s-1
Speed of light c 2,998.108 m/s
Universal gas constant R 8,31 J/mol.K
LIST OF SYMBOLS
Notation Quantity Page
Latin
A amplitude of the displacement 94
A vector potential 19
Al signed minor 101
nk
~
A complex amplitude 97
a semimajor axis of an ellipse 82
amplitude of oscillation 95
width of a potential well 285
a acceleration 5
a radius of Bohr 82, 185, 355
0
B magnetic induction 18
~
B complex amplitude 97
b semiminor axis of an ellipse 82
C constant quantity 14, 52
C superposition coefficients 218
i
C~i complex coefficients (amplitude of the oscillation of q with ω) 101
k k l
c constant 94
d constant 94
distance between two point sources 170
d,d electrical dipole moment 401
dW probability of finding object in an element of volume dV 207
dr element of the displacement (change of all co-ordinates) 53
ds magnitude of the element of the displacement dr 22
E energy 21
E energy of the state with quantum number n 186, 287, 319, 347
n
E rotational energy 341
l
E intensity of the electrical field 18
E
normalised (dimensionless) energy of a particle 297, 315, 343
e charge of the electron 18
eccentricity 60
F,F force 5
Fτ tangential component of the force 37
F normal component of the force 37
n
F magnetic force 79
m
F force acting of the particle α 16
α
G gravitational force 37
G gravitational constant 52
0
g Lande factor (gyromagnetic ratio) 85, 434
