Landau / Lifshitz Quantum Mechanics

A Shorter Course of Theoretical Physics
1. Auflage 2013
ISBN: 978-1-4831-8722-8
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 382 Seiten

Quantum Mechanics deals with various aspects of quantum mechanics and covers topics ranging from the uncertainty principle and the principle of superposition to conservation laws, Schrödinger's equation, and perturbation theory. Spin, radiation, and the identity of particles are also discussed, along with the atom, the diatomic molecule, elastic and inelastic collisions, and Feynman diagrams. Comprised of 16 chapters, this volume begins with an overview of non-relativistic quantum theory and the basic concepts of quantum mechanics such as the principles of uncertainty and superposition, operators, and the density matrix. Subsequent chapters deal with conservation laws in quantum mechanics; Schrödinger's equation and general properties of its solutions; perturbations independent of time and dependent on time; spin and the spin operator; and the principle of indistinguishability of similar particles. The atom and its electron states are also examined, together with diatomic molecules; elastic and inelastic collisions; photons and electrons; Dirac's equation; and particles and antiparticles. The final chapter is devoted to Feynman diagrams, paying particular attention to the scattering matrix, radiative corrections, and radiative shift of atomic levels. This book will be of interest to physicists.

Autoren/Hrsg.

Landau, L D

Lifshitz, E. M.

Inhaltsverzeichnis

1;Front Cover;1
2;Quantum Mechanics;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;10
6;PUBLISHER'S NOTE;11
7;NOTATION;12
8;Part I: Non-relativistic theory;14
8.1;CHAPTER 1. THE BASIC CONCEPTS OF QUANTUM MECHANICS;16
8.1.1;§1. The uncertainty principle;16
8.1.2;§2. The principle of superposition;22
8.1.3;§3. Operators;25
8.1.4;§4. Addition and multiplication of operators;31
8.1.5;§5. The continuous spectrum;34
8.1.6;§6. The passage to the limiting case of classical mechanics;37
8.1.7;§7. The density matrix;39
8.2;CHAPTER 2. CONSERVATION LAWS IN QUANTUM MECHANICS;41
8.2.1;§8. The Hamiltonian operator;41
8.2.2;§9. The differentiation of operators with respect to time;42
8.2.3;§10. Stationary states;44
8.2.4;§11. Matrices of physical quantities;47
8.2.5;§12. Momentum;51
8.2.6;§13. Uncertainty relations;55
8.2.7;§14. Angular momentum;57
8.2.8;§15. Eigenvalues of the angular momentum;62
8.2.9;§16. Eigenfunctions of the angular momentum;66
8.2.10;§17. Addition of angular momenta;68
8.2.11;§18. Angular momentum selection rules;71
8.2.12;§19. Parity of a state;75
8.3;CHAPTER 3. SCHRÖDINGER'S EQUATION;80
8.3.1;§20. Schrödinger's equation;80
8.3.2;§21. The current density;82
8.3.3;§22. General properties of solutions of Schrödinger's equation;85
8.3.4;§23. Time reversal;89
8.3.5;§24. The potential well;90
8.3.6;§25. The linear oscillator;94
8.3.7;§26. The quasi-classical wave function;99
8.3.8;§27. Bohr and Sommerfeld's quantisation rule;102
8.3.9;§28. The transmission coefficient;108
8.3.10;§29. Motion in a centrally symmetric field;114
8.3.11;§30. Spherical waves;118
8.3.12;§31. Motion in a Coulomb field;123
8.4;CHAPTER 4. PERTURBATION THEORY;129
8.4.1;§32. Perturbations independent of time;129
8.4.2;§33. The secular equation;134
8.4.3;§34. Perturbations depending on time;136
8.4.4;§35. Transitions in the continuous spectrum;139
8.4.5;§36. Intermediate states;142
8.4.6;§37. The uncertainty relation for energy;143
8.4.7;§38. Quasi-stationary states;146
8.5;CHAPTER 5. SPIN;149
8.5.1;§39. Spin;149
8.5.2;§40. The spin operator;152
8.5.3;§41. Spinors;154
8.5.4;§42. Polarisation of electrons;159
8.5.5;§43. A particle in a magnetic field;162
8.5.6;§44. Motion in a uniform magnetic field;164
8.6;CHAPTER 6. IDENTITY OF PARTICLES;167
8.6.1;§45. The principle of indistinguishability of similar particles;167
8.6.2;§46. Exchange interaction;171
8.6.3;§47. Second quantisation. The case of Bose statistics;174
8.6.4;§48. Second quantisation. The case of Fermi statistics;180
8.7;CHAPTER 7. THE ATOM;182
8.7.1;§49. Atomic energy levels;182
8.7.2;§50. Electron states in the atom;184
8.7.3;§51. Fine structure of atomic levels;187
8.7.4;§52. The Mendeleev periodic system;191
8.7.5;§53. X-ray terms;197
8.7.6;§54. An atom in an electric field;200
8.7.7;§55. An atom in a magnetic field;205
8.8;CHAPTER 8. THE DIATOMIC MOLECULE;210
8.8.1;§56. Electron terms in the diatomic molecule;210
8.8.2;§57. The intersection of electron terms;212
8.8.3;§58. Valency;215
8.8.4;§59. Vibrational and rotational structures of terms in the diatomic molecule;222
8.8.5;§60. Parahydrogen and orthohydrogen;226
8.8.6;§61. Van der Waals forces;228
8.9;CHAPTER 9. ELASTIC COLLISIONS;231
8.9.1;§62. The scattering amplitude;231
8.9.2;§63. The condition for quasi-classical scattering;234
8.9.3;§64. Discrete energy levels as poles of the scattering amplitude;236
8.9.4;§65. The scattering of slow particles;238
8.9.5;§66. Resonance scattering at low energies;241
8.9.6;§67. Born's formula;243
8.9.7;§68. Rutherford's formula;250
8.9.8;§69. Collisions of like particles;252
8.9.9;§70. Elastic collisions between fast electrons and atoms;254
8.10;CHAPTER 10. INELASTIC COLLISIONS;259
8.10.1;§71. The principle of detailed balancing;259
8.10.2;§72. Elastic scattering in the presence of inelastic processes;263
8.10.3;§73. Inelastic scattering of slow particles;265
8.10.4;§74. Inelastic collisions between fast particles and atoms;266
9;Part II: Relativistic theory;270
9.1;CHAPTER 11. PHOTONS;272
9.1.1;§75. The uncertainty principle in the relativistic case;272
9.1.2;§76. Quantisation of the free electromagnetic field;277
9.1.3;§77. Photons;281
9.1.4;§78. The angular momentum and parity of the photon;284
9.2;CHAPTER 12. DIRAC'S EQUATION;288
9.2.1;§79. The Klein–Fock equation;288
9.2.2;§80. Four-dimensional spinors;290
9.2.3;§81. Inversion of spinors;294
9.2.4;§82. Dirac's equation;296
9.2.5;§83. Dirac matrices;299
9.2.6;§84. The current density in Dirac's equation;302
9.3;CHAPTER 13. PARTICLES AND ANTIPARTICLES;306
9.3.1;§85. .-operators;306
9.3.2;§86. Particles and antiparticles;309
9.3.3;§87. The relation between the spin and the statistics;313
9.3.4;§88. Strictly neutral particles;314
9.3.5;§89. Internal parity of particles;317
9.3.6;§90. The CPT theorem;320
9.3.7;§91. Neutrinos;324
9.4;CHAPTER 14. ELECTRONS IN AN EXTERNAL FIELD;327
9.4.1;§92. Dirac's equation for an electron in an external field;327
9.4.2;§93. Magnetic moment of the electron;328
9.4.3;§94. Spin–orbit interaction;332
9.5;CHAPTER 15. RADIATION;335
9.5.1;§95. The electromagnetic interaction operator;335
9.5.2;§96. Spontaneous and stimulated emission;339
9.5.3;§97. Dipole radiation;342
9.5.4;§98. Multipole radiation;344
9.5.5;§99. Radiation from atoms;346
9.5.6;§100. The infra-red catastrophe;348
9.5.7;§101. Scattering of radiation;351
9.5.8;§102. Natural width of spectral lines;356
9.6;CHAPTER 16. FEYNMAN DIAGRAMS;358
9.6.1;§103. The scattering matrix;358
9.6.2;§104. Feynman diagrams;363
9.6.3;§105. Radiative corrections;371
9.6.4;§106. Radiative shift of atomic levels;373
10;INDEX;376

Leseproben

CHAPTER 2

CONSERVATION LAWS IN QUANTUM MECHANICS

Publisher Summary

This chapter discusses conservation laws in quantum mechanics. It describes the Hamiltonian operator, the differentiation of operators with respect to time, stationary states, matrices of physical quantities, and different types of momentum. The wave function completely determines the state of a physical system in quantum mechanics. This means that, if this function is given at some instant, not only are all the properties of the system at that instant described, but its behavior at all subsequent instants is determined. The mathematical expression of this fact is that the value of the derivative of the wave function with respect to time at any given instant must be deter-mined by the value of the function itself at that instant, and, by the principle of superposition, the relation between them must be linear. Further, if the system is closed or is in a constant external field, its Hamiltonian cannot contain the time explicitly. This follows from the fact that all times are equivalent so far as the given physical system is concerned. Since any operator of course commutes with itself, a conclusion can drawn that Hamilton’s function is conserved for systems which are not in a varying external field. As is well known, a Hamilton’s function which is conserved is called the energy. The law of conservation of energy in quantum mechanics signifies that, if in a given state the energy has a definite value, this value remains constant in time. States in which the energy has definite values are called stationary states of a system.

§8. The Hamiltonian operator

The wave function ? completely determines the state of a physical system in quantum mechanics. This means that, if this function is given at some instant, not only are all the properties of the system at that instant described, but its behaviour at all subsequent instants is determined (only, of course, to the degree of completeness which is generally admissible in quantum mechanics). The mathematical expression of this fact is that the value of the derivative ??/? of the wave function with respect to time at any given instant must be determined by the value of the function itself at that instant, and, by the principle of superposition, the relation between them must be linear. In the most general form we can write

???/?t=Hˆ?, (8.1)

(8.1)

where is some linear operator; the reason for the factor will be explained later.

Since the integral is a constant independent of time, we have

dt??*?dq=??*???tdq+???* ?t?dq=0.

Substituting here from (8.1) and using in the second integral the definition of the transpose of an operator, we can write (omitting the common factor 1/)

?*Hˆ?dq-??Hˆ* ?* dq=? ?*Hˆ?dq-??* Hˆ˜* ?dq =??*( Hˆ-Hˆ+)?dq=0.

Since this equation must hold for an arbitrary function ?, it follows that we must have identically = +; the operator is therefore Hermitian. Let us find the classical quantity to which the operator corresponds. To do this, we use the limiting expression (6.1) for the wave function and write

??t=i??S?t?;

the slowly varying amplitude need not be differentiated. Comparing this equation with the definition (8.1), we see that, in the limiting case, the operator reduces to simply multiplying by ?/? This means that ?/? is the physical quantity into which the Hermitian operator passes.

The derivative ?/? is just Hamilton’s function for a mechanical system. Thus the operator is what corresponds in quantum mechanics to Hamilton’s function; this operator is called the or, more briefly, the of the system. If the form of the Hamiltonian is known, equation (8.1) determines the wave function of the physical system concerned. This fundamental equation of quantum mechanics is called the ,

§9. The differentiation of operators with respect to time

The concept of the derivative of a physical quantity with respect to time cannot be defined in quantum mechanics in the same way as in classical mechanics. For the definition of the derivative in classical mechanics involves the consideration of the values of the quantity at two neighbouring but distinct instants of time. In quantum mechanics, however, a quantity which at some instant is measured does not in general have definite values at subsequent instants; this was discussed in detail in §1.

Hence the idea of the derivative with respect to time must be differently defined in quantum mechanics. It is natural to define the of a quantity as the quantity whose mean value is equal to the derivative, with respect to time, of the mean value . Thus we have by definition

?¯=f¯?. (9.1)

(9.1)

Starting from this definition, it is easy to obtain an expression for the quantum-mechanical operator corresponding to the quantity . We can write

?¯=f¯?=dtdt??*fˆ?dq =??*?fˆ ?t?dq+???*?tfˆ ?dq+? ?*fˆ?? ?tdq.

Here is the operator obtained by differentiating the operator with respect to time; may depend on the time as a parameter. Substituting for ??/?, ??*/? their expressions according to (8.1), we obtain

?¯=??*?fˆ?t?dq+i??(Hˆ* ?*)fˆ?dq-i???*fˆ( Hˆ?)dq.

Since the operator is Hermitian, we have

(Hˆ*?*)(fˆ?)dq=??*Hˆfˆ?dq.

thus

?¯=??*(?fˆ ?t+i ?Hˆfˆ-i ?fˆHˆ)?dq.

Since, on the other hand, we must have, by the definition of mean values, , it is seen that the expression in parentheses under the integral is the required operator :

?ˆ=?fˆ?t+i?(Hˆfˆ-fˆHˆ). (9.2)

(9.2)

If the operator does not depend explicitly on time, reduces, apart from a constant factor, to the commutator of the operator and the Hamiltonian.

A very important class of physical quantities is formed by those whose operators do not depend explicitly on time, and also commute with the Hamiltonian, so that . Such quantities are said to be For these, , that is, is constant. In other words, the mean value of the quantity remains constant in time. We can also assert that, if in a given state the quantity has a definite value (i.e. the wave function is an eigenfunction of the operator ), then it will have a definite value (the same one) at subsequent instants also.

§10. Stationary states

If the system is closed or is in a external field, its Hamiltonian cannot contain the time explicitly. This follows from the fact that all times are equivalent so far as the given physical system is concerned. Since, on the other hand, any operator of course commutes with itself, we reach the conclusion that Hamilton’s function is conserved for systems which are not in a varying external field. As is well known, a Hamilton’s function which is conserved is called the (see , §6). The law of conservation of energy in quantum mechanics signifies that, if in a given state the energy has a definite value, this value remains constant in time.

States in which the energy has definite values are called of a system. They are described by wave functions ? which are the eigenfunctions of the Hamiltonian operator, i.e. which satisfy the equation ? = ?, where are the eigenvalues of the energy. Correspondingly, the wave equation (8.1) for the function ?,

???n|?t=Hˆ?n=En?n,

can be integrated at once with respect to time and gives

n=e-(i/?)Ent?n(q), (10.1)

(10.1)

where ? is a function of the coordinates only. This determines the relation between the wave functions of stationary states and the time.

We shall denote by the small letter ? the wave functions of stationary states without the time factor. These functions, and also the eigenvalues of the energy, are determined by the equation

ˆ?=E?....