Quantum Mechanics

561 Pages·1966·21.1 MB·English

by  A. A. et al. Sokolov

TIGHT BINDING BOOK
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CONTENTS
Removal of The Stark effect. of
degeneracy. Splitting spectral
terms. The Lorentz classical of Index of
theory dispersion.
of refraction. Polarization. Radiation
damping. Quantum theory
u
of Oscillator Ramon effect. Stokes* and
dispersion. strengths.
"anti-Stokes" lines. correction in second order
P^nergy per-
turbation A.iharmonic oscillator. Harmonic oscillator.
theory.
3
Matrix elements of x . (Perturbation for the continuous
theory
and wave conditions.
spectrum. Ingoing outgoing boundary
Phase shift. Connection between wave and three di-
partial
men sional formulations . )
v^
PART II. RELVriVISTIC MKCII \NICS 257
QUANTUM
15. The Klein-Gordon Scalar RelativiMic Wa\e 259
Equation
Relativistic invanancc of the de relations. Relativistic
Broglie
relation of a free The Klein-Gordon
energy-momentum particle.
and current Nonrclativistic limit. The
equation. Charge density.
initial data Indefiniteness of the of Inter-
problem. sign charge.
action with an external field. Relativistic
electromagnetic
levels of a in a Coulomb field. Fine
energy spinless particle
l
structure constant. The case Za > /2. and current
Charge density
in the of an field.
presence electromagnetic
16. Motion of an Electron in a Eield. Electron 268
Magnetic Spin
Classical of the Zeeman effect. Interaction of a
theory energy
Larmor moment of a
magnetic dipole. precession. Magnetic
electron. Zeeman effect in nonrelativistic
moving Schrodinger
Orbital moment. Bohr Normal and
theory. magnetic magneton.
anomalous Zeeman Emstein-de Haas
splitting. experiment.
Land factor. Stern-Gerlach Uhlenbeck-Goudsmit
g experiment.
of intrinsic momentum.
hypothesis angular Half-integral quantum
numbers for momentum. El . The Pauli
angular equation.
ecJjQn^spin
wave functions. The for intrinsic
Two-component operator mag-
netic moment. Pauli matricies.
Coupled Schro'dinger equations.
Matrix elements. Commutation relations for
Spin operators. spin
Vedtorial character of of'
operators. spin operators. Separation
and variables in a field.
spin space homogeneous magnetic
of the an direction.
Eigenvalues spin operator along arbitrary
distribution of directions.
Probability spin
17. The Uirac \\ave 285
Equation
Linearization of the Dirac matricies and their
energy operator.
relation to Pauli matricies. The and
Charge
Dirac^equation.
current External field.
density. electromagnetic Velocity oper-
ator. Statistics in second Transformation
quantization. proper-
ties of the wave function under Lorentz transformations
spinor
and rotations.
spatial
CONTENTS
18. The Dirac of the Motion of an Electron in a Central
Theory
'
'
Field of Force 293
and total momenta. Conservation laws.
Orbital, spin angular
of the total momentum
Properties angular operators. Quantization
of total momentum. Clebsch-Gordan coefficients. Spher-
angular
ical
spinors. The vector model of the addition of angular mo-
menta. Motion in a central field
including spin effects. Theory
of the rotator. Selection rules. Parity of a state. Conservation
of
parity. Solution of the Dirac equation for a free particle.
states. Nonrelativistic limit. Four-vector trans-
Negative energy
formation law of the under Lorentz
energy-momentum operators
transformations. Relativistic invariance of the scalar wsrve
Vector model.
equation. Charge conjugation.
19. The Dirac in Form 308
Equation Approximate
Two Pauli form. "Small" and
component "large" components.
2
Correction terms to order (i>/c) . Relativistic increase of mass.
Interaction of the intrinsic moment. inter-
magnetic Spin-orbit
action. Contact interaction. The and Ehren-
velocity operator
fest's theorem in the Dirac
theory.
20. The Fine Structure of the of Atoms 314
Spectra Hydrogen-like
of the method. Relativistic and
Advantages approximate spin
effects. Contact interaction. Stable motion for Z < 137. Fine
structure in the Dirac verification of the
theory. Experimental
fine structure Lamb-Rutherford Anomalous
theory. experiment.
Zeeman effect. Weak field. Lande' factor.
magnetic g Strong
fields. Paschen-Back effect. of
magnetic "Breaking" spin-orbit
and Anomalous Zeeman
coupling. Paramagnetism diamagnetism.
effect in the vector model. (Stark effect. of meta-
Quenching
stable states. Intermediate field Paschen-Back effect.)
21. The Effect of Nuclear Structure on Atomic 334
Spectra
Reduced mass. Effect of finite nuclear size. Mesic atoms.
Ap-
harmonic oscillator for Z. of the
proximate potential large Spin
muon. of the Dirac to the neutron and
Application equation pro-
ton. Anomalous (Pauli) moment. deter-
magnetic Experimental
mination of the moments of the neutron and
magnetic proton.
Limitations on the measurement of momentum.
angular Experi-
ments of Bloch and Alvarez and of Rabi. Nuclear
magneton.
structure of the
Hyperfine hydrogen spectrum.
22. 'The Electron-Positron Vacuum and the Vacuum
347
Electromagnetic
A. Dirac of "holes." states.
theory Negative energy Discovery
of the Pair creation and annihilation.
positron. Antiparticles.
of conservation laws. Positronium. Inter-
Rigorous validity
of B. The Lamb shift of levels
convertibility particles. energy
of atomic electrons. Fluctuations of the
vacuum.
electromagnetic
Xl
CONTENTS
Virtual out" of a electron. C. Elec-
particles. "Smearing point
tron-Positron vacuum. Vacuum Anomalous
polarization. mag-
netic moments of electron,
proton, and neutron. D. Renormal-
ization. of fields.
Quantum electrodynamics. Quantum theory
Cherenkov radiation.
23. of the Helium Atom States 358
Theory Neglecting Spin
Basic of the of multielectron atoms. Indis-
principles
theory
of electrons. forces. Perturbation
tinguishability J Exchange
. solution of the helium atom. Permutation of electrons.
theory
and anti-
Exchange energy. Symmetric
degeneracy^
JSxchange
wave functions. XUoulomb interaction between elec-
symmetric
trons,"lonization 'The variational method. Derivation of
energy.
the the variational method. Hartree-
SchrOdinger equation by
self-con fields. of the
si ^tent Investigation
F]pck method__of
time.
"exchange energy. Exchange
24. of Multielectron Atoms States 378
Elementary Theory Including Spin
and states. Permutation
Symmetric antisymmetric ope
rator.
Fermi-Dirac and Bose-Einstein statistics. Pauli cntplnsinn
Tt]^
Fermion s. Bosons. Determinental wave function.
principle.
Addition of momentum. Russell-Saunders
angular coupling.
Clebsch-Gordan coefficients. LS Wave
coupling, jj coupling.
function of the helium atom and
including spins. Triplet singlet
states. Parahelium and orthohelium. of the
Energy spectrum
helium atom. Variational wave function for a Yukawa
potential.
of
Diamagnetic susceptibility parahelium.
25. of Alkali Metals 397
Optical Spectra
The structure of atoms. The Thomas-Fermi statistical
complex
method. conditions for neutral and ionized atoms.
Boundary
Solution of the Thomas-Fermi the Ritz variational
problem by
method. Total ionization distribution in
energies. Charge argon.
levels of alkali atoms. Atomic core.
Energy "Penetrating"
orbits. Polarization of the atomic core. "Effective
principle
number." of the atomic core. Fundamental
quantum Smearing
series. structure of lines. terms of
Multiplet spectral Spectral
sodium. and diffuse series.
Sharp, principle
26. Periodic of Elements 420
Mendeleyev's System
of atoms. Continuous
X-ray spectra spectra. Bremsstrahlung.
Characteristic of atoms and the structure of their inner
spectra
shells. law. structure of
Moseley's Multiplet x-ray sp
ectra.
Relativistic and effects. and doublets.
spin Regular irregular
The of law. of the
discovery Mendeleyev's periodic Filling
electron shells. of the Thomas-Fermi method. Peri-
Application
of the elements.
odicity properties
Xll CONTENTS
27. The of Molecules 137
Theory Simple
Chemical bond. molecules. Valence. Kos-
Heteropolar Affinity.
se 1 . Mo 1ecular h.vdiQg
eg_ign . foj^s^-J^aluajjoa _of
J^change
. and
Spin symmetry. Orthohydrogen
and The valence valence. Mascrs
parahydrogen. theory. Spin
and lasers.
PART III. SOME APPLICATIONS TO NL1CLFAR PHYSICS
28. Elastic of Particles 465
Scattering
Golden rule. Cross section
'Time-dependent perturbation theory.
for elastic of
scattering. Uncertainty energy.
Scattering ampli-
tude. Born a Yukawa center of
approximation. Scattering by
force. of nuclear force. Fast-electron neu-
Range scattering by
tral atoms. of Horn Partial-wave cross
Validity approximation.
sections. Phase shift. from a barrier and
Scattering spherical
spherical well. Resonant scattering. (Golden Rule #2, Density
of final states.)
29. Second t/.ut ion 480
Quant
Second of the The Ileisen-
quantization Schrodingcr equation.
of motion, numbers and c numbers. Commutation
berg equation q
relations for Boson field Creation and destruction
amplitudes.
Anti commutation relations
operators. describing particles obey-
Fermi statistics. of Maxwell's field
ing Quantization equations.
emission. Beta
Spontaneous Dipole approximation. decay.
Pauli's of the neutrino. The Fermi Weak
hypothesis theory.
and interactions. Fermi and Gamow-Teller selection
strong
rules. and Gell-Mann Non-
Feynman theory. spectrum.
/3-decay
conservation of in weak interactions. Lee and
parity Yang.
of the neutrino* Pion
Helicity decay.
APPENDIX V Hilbert and Transformation 497
Space Theory
APPENDIX I). The Statistical Assertions of Mechanics 505
Quantum
PROBLEMS 511
Preface
This textbook is based on lectures to students at the Mos-
my
cow Institute to and Moscow
Regional Pedagogical (1945 1948)
from 1945 on. In this book we set ourselves
University writing
the difficult task of in a volume the fundamentals
treating single
of atomic that nonrelativistic
theory, is, Schrodinger's theory,
Dirac's relativistic the of multielectron
theory, theory atonis,
and the basic of mechanics to solid state
applications quantum
Our aim was to combine the of the-
physics. exposition general
oretical with of the of
principles examples application quantum
mechanics to connected with atomic structure.
specific problems
To avoid this we have the treatment of
overloading book, abridged
certain but in such cases we have endeavored
specialized topics,
to references to standard works on the
supply subject.
In most textbooks the solution of with the
specific problems
of is handled in form.
help Schrodinger's equation fairly elegant
The basic mathematical tools for this are a
required purpose
of second-order differential and various
knowledge equations spe-
cial functions the and
(including Hermite, Legendre Laguerre
of Dirac's to
polynomials). However, applications theory specific
as the are on the whole handled
problems (such
hydrogen atom)
less In some cases the calculations are so
satisfactorily. long
and cumbersome that it is difficult to the mean-
perceive physical
of the solutions. In others there is no actual derivation of the
ing
results or a is In an to avoid
only rough proof given. attempt
these we have used an form of Dirac's
pitfalls, approximate equa-
tion for our treatment of the atom This
hydrogen (Chapter 19).
still enables us to obtain the formula for the fine
approximation
structure of the levels and the selection rules 18
energy (Chapter
and Our of the Lamb shift due to the electron-
20). analysis
vacuum is also somewhat
positron simplified (
Chapter 22).
Several books in mechanics are avail-
good problem quantum
and therefore we shall consider a few chosen
able,
only problems
with the aim of and the
discus-
elucidating supplementing general
sion.
The first of this book was written me and
Yu. M.
part j
ointly by
Loskutov, and the second part
jointly by me and I. M. Ternov.
Great assistance was rendered
M. M. Kolesnikova in condens-
by
notes based on lectures on mechanics and in
ing my quantum
the for the 25 was
preparing manuscript press. Chapter carefully
XlV PREFACE
read
N. N. who made a number of valuable com-
Kolesnikov,
by
ments. I would like to mention the taken S. I. Larin
great pains by
in the whole
editing manuscript.
A. A. Sokolov