Quantum Physics of Light and Matter: A Modern Introduction to Photons, Atoms and Many-Body Systems

198 Pages·2014·1.98 MB·English

by  Luca Salasnich (auth.)

UNITEXT for Physics
Luca Salasnich
Quantum Physics
of Light and
Matter
A Modern Introduction to Photons,
Atoms and Many-Body Systems
UNITEXT for Physics
Series editors
Michele Cini, Roma, Italy
A. Ferrari, Torino, Italy
Stefano Forte, Milano, Italy
Inguscio Massimo, Firenze, Italy
G. Montagna, Pavia, Italy
Oreste Nicrosini, Pavia, Italy
Luca Peliti, Napoli, Italy
Alberto Rotondi, Pavia, Italy
For further volumes:
http://www.springer.com/series/13351
Luca Salasnich
Quantum Physics of Light
and Matter
A Modern Introduction to Photons,
Atoms and Many-Body Systems
123
Luca Salasnich
Fisica e Astronomia ‘‘Galileo Galilei’’
Università di Padova
Padova
Italy
ISSN 2198-7882 ISSN 2198-7890 (electronic)
ISBN 978-3-319-05178-9 ISBN 978-3-319-05179-6 (eBook)
DOI 10.1007/978-3-319-05179-6
Springer Cham Heidelberg New York Dordrecht London
Library of Congress Control Number: 2014935224
Ó Springer International Publishing Switzerland 2014
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Preface
This book contains lecture notes prepared for the one-semester course ‘‘Structure
of Matter’’ belonging to the Master of Science in Physics at the University of
Padova. The course gives an introduction to the field quantization (second quan-
tization) of light and matter with applications to atomic physics.
Chapter 1 briefly reviews the origins of special relativity and quantum
mechanics and the basic notions of quantum information theory and quantum
statistical mechanics. Chapter 2 is devoted to the second quantization of the elec-
tromagnetic field, while Chap. 3 shows the consequences of the light field quan-
tization in the description of electromagnetic transitions. In Chap. 4, it is analyzed
the spin of the electron, and in particular its derivation from the Dirac equation,
while Chap. 5 investigates the effects of external electric and magnetic fields on the
atomic spectra (Stark and Zeeman effects). Chapter 6 describes the properties of
systems composed by many interacting identical particles. It is also discussed the
Fermi degeneracy and the Bose–Einstein condensation introducing the Har-
tree–Fock variational method, the density functional theory, and the
Born–Oppenheimer approximation. Finally, in Chap. 7, it is explained the second
quantization of the nonrelativistic matter field, i.e., the Schrödinger field, which
gives a powerful tool for the investigation of finite-temperature many-body prob-
lems and also atomic quantum optics. Moreover, in this last chapter, fermionic Fock
states and coherent states are presented and the Hamiltonians of Jaynes–Cummings
and Bose–Hubbard are introduced and investigated. Three appendices on the Dirac
delta function, the Fourier transform, and the Laplace transform complete the book.
It is important to stress that at the end of each chapter there are solved problems
which help the students to put into practice the things they learned.
Padova, January 2014 Luca Salasnich
v
Contents
1 The Origins of Modern Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Axioms of Quantum Mechanics . . . . . . . . . . . . . . . . . . 8
1.2.2 Quantum Information . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . 15
1.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Second Quantization of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 First Quantization of Light . . . . . . . . . . . . . . . . . . . . . . 23
2.1.2 Electromagnetic Potentials and Coulomb Gauge . . . . . . . 24
2.2 Second Quantization of Light . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Fock Versus Coherent States for the Light Field. . . . . . . 30
2.2.2 Linear and Angular Momentum
of the Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3 Zero-Point Energy and the Casimir Effect . . . . . . . . . . . 34
2.3 Quantum Radiation Field at Finite Temperature . . . . . . . . . . . . 36
2.4 Phase Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Electromagnetic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Classical Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Quantum Electrodynamics in the Dipole Approximation . . . . . . 53
3.2.1 Spontaneous Emission . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.2 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.3 Stimulated Emission . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Einstein Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
vii
viii Contents
3.4.1 Rate Equations for Two-Level
and Three-Level Systems . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Life-Time and Natural Line-Width . . . . . . . . . . . . . . . . . . . . . 66
3.5.1 Collisional Broadening . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5.2 Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 Minimal Coupling and Center of Mass. . . . . . . . . . . . . . . . . . . 69
3.7 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4 The Spin of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 The Pauli Equation and the Spin . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Dirac Equation with a Central Potential . . . . . . . . . . . . . . . . . . 87
4.3.1 Relativistic Hydrogen Atom and Fine Splitting. . . . . . . . 88
4.3.2 Relativistic Corrections to the Schrödinger
Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 Energy Splitting and Shift Due to External Fields . . . . . . . . . . . . . 99
5.1 Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Zeeman Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.1 Strong-Field Zeeman Effect . . . . . . . . . . . . . . . . . . . . . 102
5.2.2 Weak-Field Zeeman Effect. . . . . . . . . . . . . . . . . . . . . . 103
5.3 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Many-Body Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1 Identical Quantum Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Non-interacting Identical Particles . . . . . . . . . . . . . . . . . . . . . . 117
6.2.1 Uniform Gas of Non-interacting Fermions . . . . . . . . . . . 119
6.2.2 Atomic Shell Structure and the Periodic
Table of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3 Interacting Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3.1 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3.2 Hartree for Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3.3 Hartree-Fock for Fermions . . . . . . . . . . . . . . . . . . . . . . 126
6.3.4 Mean-Field Approximation . . . . . . . . . . . . . . . . . . . . . 129
6.4 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5 Molecules and the Born-Oppenheimer Approximation . . . . . . . . 137
6.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Contents ix
7 Second Quantization of Matter. . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1 Schrödinger Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Second Quantization of the Schrödinger Field. . . . . . . . . . . . . . 147
7.2.1 Bosonic and Fermionic Matter Field . . . . . . . . . . . . . . . 150
7.3 Connection Between First and Second Quantization . . . . . . . . . 153
7.4 Coherent States for Bosonic and Fermionic Matter Fields . . . . . 156
7.5 Quantum Matter Field at Finite Temperature . . . . . . . . . . . . . . 160
7.6 Matter-Radiation Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.6.1 Cavity Quantum Electrodynamics . . . . . . . . . . . . . . . . . 163
7.7 Bosons in a Double-Well Potential . . . . . . . . . . . . . . . . . . . . . 167
7.7.1 Analytical Results with N ¼ 1 and N ¼ 2 . . . . . . . . . . . 170
7.8 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Appendix A: Dirac Delta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Appendix B: Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Appendix C: Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Chapter 1
The Origins of Modern Physics
In this chapter we review the main results of early modern physics (which we suppose
the reader learned in previous introductory courses), namely the special relativity of
Einstein and the old quantum mechanics due to Planck, Bohr, Schrödinger, and
others. The chapter gives also a brief description of the relevant axioms of quan-
tum mechanics, historically introduced by Dirac and Von Neumann, and elementary
notions of quantum information. The chapter ends with elements of quantum statis-
tical mechanics.
1.1 Special Relativity
In 1887 Albert Michelson and Edward Morley made a break-thought experiment of
optical interferometry showing that the speed of light in the vacuum is
8
c = 3 × 10 m/s, (1.1)
independently on the relative motion of the observer (here we have reported an
approximated value of c which is correct within three digits). Two years later, Henry
Poincaré suggested that the speed of light is the maximum possible value for any
kind of velocity. On the basis of previous ideas of George Francis FitzGerald, in
1904 Hendrik Lorentz found that the Maxwell equations of electromagnetism are
invariant with respect to this kind of space-time transformations
∧ x − vt
x = √ (1.2)
v2
1 −
c2
∧
y = y (1.3)
∧
z = z (1.4)
L. Salasnich, Quantum Physics of Light and Matter, UNITEXT for Physics, 1
DOI: 10.1007/978-3-319-05179-6_1, © Springer International Publishing Switzerland 2014
2 1 The Origins of Modern Physics
2
t − vx/c
∧
t = √ , (1.5)
2
v
1 −
2
c
which are called Lorentz (or Lorentz-FitzGerald) transformations. This research
activity on light and invariant transformations was summarized in 1905 by Albert
Einstein, who decided to adopt two striking postulates:
(i) the law of physics are the same for all inertial frames;
(ii) the speed of light in the vacuum is the same in all inertial frames.
From these two postulates Einstein deduced that the laws of physics are invariant with
respect to Lorentz transformations but the law of Newtonian mechanics (which are
not) must be modified. In this way Einstein developed a new mechanics, the special
relativistic mechanics, which reduces to the Newtonian mechanics when the involved
velocity v is much smaller than the speed of light c. One of the amazing results of
relativistic kinematics is the length contraction: the length L of a rod measured by
an observer which moves at velocity v with respect to the rod is given by
√
2
v
L = L0 1 − , (1.6)
2
c
where L0 is the proper length of the rod. Another astonishing result is the time
dilatation: the time interval T of a clock measured by an observer which moves at
velocity v with respect to the clock is given by
T0
T = √ , (1.7)
2
v
1 −
2
c
where T0 is the proper time interval of the clock.
We conclude this section by observing that, according to the relativistic mechanics
of Einstein, the energy E of a particle of rest mass m and linear momentum p = |p|
is given by
√
2 2 2 4
E = p c + m c . (1.8)
If the particle has zero linear momentum p, i.e. p = 0, then
2
E = mc , (1.9)
which is the rest energy of the particle. Instead, if the linear momentum p is finite
2
but the condition pc/(mc ) ≪ 1 holds one can expand the square root finding
2
p
2 4
E = mc + + O(p ), (1.10)
2m