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- Herausgeber: Wiley-VCH Verlag GmbH & Co. KGaA
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This modern textbook offers an introduction to Quantum Mechanics as a theory that underlies the world around us, from atoms and molecules to materials, lasers, and other applications. The main features of the book are:
- Emphasis on the key principles with minimal mathematical formalism
- Demystifying discussions of the basic features of quantum systems, using dimensional analysis and order-of-magnitude estimates to develop intuition
- Comprehensive overview of the key concepts of quantum chemistry and the electronic structure of solids
- Extensive discussion of the basic processes and applications of light-matter interactions
- Online supplement with advanced theory, multiple-choice quizzes, etc.
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Seitenzahl: 1028
Veröffentlichungsjahr: 2017
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Table of Contents
Cover
Title Page
Copyright
Dedication
Foreword
Preface
Editors' Note
Part I: Fundamental Principles
Chapter 1: The Principle of Wave–Particle Duality: An Overview
1.1 Introduction
1.2 The Principle of Wave–Particle Duality of Light
1.3 The Principle of Wave–Particle Duality of Matter
1.4 Dimensional Analysis and Quantum Physics
Chapter 2: The Schrödinger Equation and Its Statistical Interpretation
2.1 Introduction
2.2 The Schrödinger Equation
2.3 Statistical Interpretation of Quantum Mechanics
2.4 Further Development of the Statistical Interpretation: The Mean-Value Formula
2.5 Time Evolution of Wavefunctions and Superposition States
2.6 Self-Consistency of the Statistical Interpretation and the Mathematical Structure of Quantum Mechanics
2.7 Summary: Quantum Mechanics in a Nutshell
Chapter 3: The Uncertainty Principle
3.1 Introduction
3.2 The Position–Momentum Uncertainty Principle
3.3 The Time–Energy Uncertainty Principle
3.4 The Uncertainty Principle in the Classical Limit
3.5 General Investigation of the Uncertainty Principle
Part II: Simple Quantum Systems
Chapter 4: Square Potentials. I: Discrete Spectrum—Bound States
4.1 Introduction
4.2 Particle in a One-Dimensional Box: The Infinite Potential Well
4.3 The Square Potential Well
Chapter 5: Square Potentials. II: Continuous Spectrum—Scattering States
5.1 Introduction
5.2 The Square Potential Step: Reflection and Transmission
5.3 Rectangular Potential Barrier: Tunneling Effect
Chapter 6: The Harmonic Oscillator
6.1 Introduction
6.2 Solution of the Schrödinger Equation
6.3 Discussion of the Results
6.4 A Plausible Question: Can We Use the Polynomial Method to Solve Potentials Other than the Harmonic Oscillator?
Chapter 7: The Polynomial Method1: Systematic Theory and Applications
7.1 Introduction: The Power-Series Method
7.2 Sufficient Conditions for the Existence of Polynomial Solutions: Bidimensional Equations
7.3 The Polynomial Method in Action: Exact Solution of the Kratzer and Morse Potentials
7.4 Mathematical Afterword
Chapter 8: The Hydrogen Atom. I: Spherically Symmetric Solutions
8.1 Introduction
8.2 Solving the Schrödinger Equation for the Spherically Symmetric Eigenfunctions
8.3 Discussion of the Results
8.4 What Is the Electron Doing in the Hydrogen Atom after All? A First Discussion on the Basic Questions of Quantum Mechanics
7
Chapter 9: The Hydrogen Atom. II: Solutions with Angular Dependence
9.1 Introduction
9.2 The Schrödinger Equation in an Arbitrary Central Potential: Separation of Variables
9.3 The Hydrogen Atom
Chapter 10: Atoms in a Magnetic Field and the Emergence of Spin
10.1 Introduction
10.2 Atomic Electrons as Microscopic Magnets: Magnetic Moment and Angular Momentum
10.3 The Zeeman Effect and the Evidence for the Existence of Spin
10.4 The Stern–Gerlach Experiment: Unequivocal Experimental Confirmation of the Existence of Spin
10.5 What is Spin?
10.6 Time Evolution of Spin in a Magnetic Field
10.7 Total Angular Momentum of Atoms: Addition of Angular Momenta
Chapter 11: Identical Particles and the Pauli Principle
11.1 Introduction
11.2 The Principle of Indistinguishability of Identical Particles in Quantum Mechanics
11.3 Indistinguishability of Identical Particles and the Pauli Principle
11.4 The Role of Spin: Complete Formulation of the Pauli Principle
11.5 The Pauli Exclusion Principle
11.6 Which Particles Are Fermions and Which Are Bosons
11.7 Exchange Degeneracy: The Problem and Its Solution
Part III: Quantum Mechanics in Action: The Structure of Matter
Chapter 12: Atoms: The Periodic Table of the Elements
12.1 Introduction
12.2 Arrangement of Energy Levels in Many-Electron Atoms: The Screening Effect
12.3 Quantum Mechanical Explanation of the Periodic Table: The “Small Periodic Table”
Chapter 13: Molecules. I: Elementary Theory of the Chemical Bond
13.1 Introduction
13.2 The Double-Well Model of Chemical Bonding
13.3 Examples of Simple Molecules
13.4 Molecular Spectra
Chapter 14: Molecules. II: The Chemistry of Carbon
14.1 Introduction
14.2 Hybridization: The First Basic Deviation from the Elementary Theory of the Chemical Bond
14.3 Delocalization: The Second Basic Deviation from the Elementary Theory of the Chemical Bond
Chapter 15: Solids: Conductors, Semiconductors, Insulators
15.1 Introduction
15.2 Periodicity and Band Structure
15.3 Band Structure and the “Mystery of Conductivity.” Conductors, Semiconductors, Insulators
15.4 Crystal Momentum, Effective Mass, and Electron Mobility
15.5 Fermi Energy and Density of States
Chapter 16: Matter and Light: The Interaction of Atoms with Electromagnetic Radiation
16.1 Introduction
16.2 The Four Fundamental Processes: Resonance, Scattering, Ionization, and Spontaneous Emission
16.3 Quantitative Description of the Fundamental Processes: Transition Rate, Effective Cross Section, Mean Free Path
16.4 Matter and Light in Resonance. I: Theory
16.5 Matter and Light in Resonance. II: The Laser
16.6 Spontaneous Emission
16.7 Theory of Time-dependent Perturbations: Fermi's Rule
16.8 The Light Itself: Polarized Photons and Their Quantum Mechanical Description
Appendix
Bibliography
Index
End User License Agreement
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Guide
Cover
Table of Contents
Foreword
Preface
Part I: Fundamental Principles
Begin Reading
List of Illustrations
Chapter 1: The Principle of Wave–Particle Duality: An Overview
Figure 1.1 The kinetic energy of electrons as a function of photon frequency . The experimental curve is a straight line whose slope is equal to Planck's constant.
Figure 1.2 The standard experimental setup for studying the photoelectric effect. The photoelectric current occurs only when and vanishes when gets smaller than the threshold frequency . The kinetic energy of the extracted electrons is measured by reversing the polarity of the source up to a value —known as the
cutoff potential
—for which the photoelectric current vanishes and we get .
Figure 1.3 A photon colliding with a stationary electron. The photon is scattered at an angle with a wavelength that is greater than its initial wavelength . The electron recoils at an angle with energy and momentum .
Figure 1.4 Standing classical waves on a string. A standing wave of this kind can only be formed when an integer number of half-waves fit on the string. That is, .
Figure 1.5 A standing matter wave of spherical shape. A particle trapped inside a bounded region—a spherical volume in our case—of linear dimension , is described (in the state of lowest energy) by a spherical standing wave that vanishes only at the boundary of this region. For its wavelength we thus have .
Figure 1.6 Total energy of the hydrogen atom as a function of its size, . The real radius of the atom is the one that minimizes its total energy.
Figure 1.7 Typical one-dimensional wavefunctions. (a) An extended wavefunction: The position of the particle is known with very low precision. There is a significant probability of locating the particle in regions away from the “most frequented” location at . (b) A localized wavefunction: The position of the particle is known with very high precision. In the vast majority of the measurements, we would detect the particle in the immediate vicinity of .
Figure 1.8 The energy-level diagram for the hydrogen atom and the two basic quantities associated with it. The ionization energy is the minimum energy needed to remove the electron from the atom. The minimum excitation energy is the energy required to affect the smallest possible change to the atom in its ground state.
Figure 1.9 The correct quantum picture for the ground state of the hydrogen atom. The wave nature of the electron is incompatible with motion along some classical orbit. Instead, we are forced to think of the electron (in its ground state) as a spherically symmetric
probability cloud
about the nucleus.
Figure 1.10 A false picture that should be discarded. Here the electron supposedly forms something like sinusoidal standing matter waves along a circle of radius . But this picture is a flawed projection to three-dimensional space of the classical picture for a wave on a string. Three-dimensional waves—quantum or classical—typically fill the space and surely do not look like standing sound waves in a circular tube.
Figure 1.11 Theoretical analysis of the Davisson–Germer experiment. The electrons are scattered preferentially toward those directions that satisfy the condition of constructive interference .
Figure 1.12 The “collapse” of the wavefunction in a position measurement: When a measurement detects a particle at position , its wavefunction collapses immediately to a highly localized form around and instantaneously vanishes elsewhere. It is as if the measurement “sucks” the wavefunction, only to concentrate it suddenly at the point where the particle was located. Clearly, such an instantaneous collapse has a nonlocal character; that is, it seems to imply some sort of
action at a distance
. But since the wavefunction is a mathematical entity—not a physical wave with energy and momentum distributed in space—this instantaneous collapse does not imply a corresponding instantaneous transfer of energy or momentum and therefore it does not violate the theory of relativity. A measurement simply “removes” all possibilities to locate the particle anywhere else than the position it was found to be.
Figure 1.13 Two alternative pictures for the double-slit experiment: (a) The
wave picture
. (b) The
particle picture
. Both pictures are legitimate. But only the wave picture—with the understanding of the wave as a probability wave—provides a qualitatively and quantitatively correct understanding of this experiment. The particle picture is just to remind us that there are only particles “behind” the wave that describes how they move in space.
Figure 1.14 Quantum mechanical explanation of the disappearance of interference fringes. Because of the measurement, the wavefunction collapses into the wave that passes through the slit where detection occurs. Simultaneous “emission” of probability waves from both slits is no longer possible, and interference fringes disappear.
Figure 1.15 The spectral distribution curve of blackbody radiation. spectral intensity radiated energy, per unit time , per unit frequency , and per unit surface of the radiating body . total intensity .
Chapter 2: The Schrödinger Equation and Its Statistical Interpretation
Figure 2.1 The Gaussian wavefunction . The mean position of the particle is zero, , and the corresponding uncertainty is . As we should expect (why?), the position uncertainty decreases as the parameter increases.
Figure 2.2 Collapse of the wavefunction upon measurement. When the particle is in the superposition state , the first measurement can yield any one of the possible outcomes with
a priori
probabilities . Once a measurement yields a particular value, however, the wavefunction “produced” by the measuring device is the eigenfunction of the eigenvalue that was just measured. And because of this fact, a second device that performs the measurement right after the first one will confirm its outcome with probability.
Figure 2.3 Dirac delta function. (a) The function . (b) The function . Both (a) and (b) describe the same function, but are centered at different points. The function is centered at , so it becomes infinite there and vanishes everywhere else, while is centered at .
Chapter 3: The Uncertainty Principle
Figure 3.1 The mathematical mechanism behind the uncertainty principle. (a) A narrow and tall wavefunction. is small, but is large because the wavefunction goes up and down very abruptly (it has high slope values). (b) A broad and short wavefunction. is clearly smaller now, since the wavefunction goes up and down more smoothly, but is large.
Conclusion
: and cannot simultaneously become small, or, even more so, vanish. In quantum mechanics, it is impossible to have a concurrent precise knowledge of the position and momentum of a particle.
Figure 3.2 Wave–particle duality and the uncertainty principle in a position measurement. Because of diffraction, any attempt to increase the accuracy of a measurement by decreasing the slit opening leads to even stronger diffraction, and, concomitantly, to a greater range of values for the particles' transverse momenta. The wave nature of the particles renders fundamentally impossible the simultaneous, exact knowledge of their position and momentum along any axis. As the accuracy in position increases , the uncertainty in momentum increases , and vice versa .
Figure 3.3 Time evolution of the mean value of a physical quantity in a superposition of energy eigenstates with energies and . The system evolves with a characteristic time on the order of a quarter or a half of the period . That is, , in agreement with the time–energy uncertainty relation .
Figure 3.4 Two typical energy levels and their energy broadenings. The excited levels have finite lifetimes, hence a finite width, while the ground state has an infinite lifetime and vanishing width.
Chapter 4: Square Potentials. I: Discrete Spectrum—Bound States
Figure 4.1 An example of a square potential.
Figure 4.2 The energy spectrum of a one-dimensional potential well. In the range of negative energies , where the classical motion is confined within the interval , the energy spectrum is discrete and the corresponding solutions represent the bound states of the particle. Conversely, in the range of positive energies , where the classical motion is unrestricted, the energy spectrum is continuous and solutions describe the scattering of the particle by the potential.
Figure 4.3 The infinite potential well. The potential function for a particle that moves freely inside a one-dimensional box (tubule) but cannot escape from it. The potential is zero inside the box and infinite outside.
Figure 4.4 General shape of a wavefunction in an infinite potential well. The wavefunction is zero everywhere outside the box and nonzero only inside.
Figure 4.5 The first three eigenvalues and eigenfunctions of an infinite potential well.
Figure 4.6 The energy of the ground state in an arbitrary potential well always lies above the bottom of the well. Falling to the bottom of a potential well is impossible in quantum mechanics. Due to the uncertainty principle, quantum particles move like “crazy” even in the state of maximal rest: the state of minimum total energy.
Figure 4.7 Quantum and classical position probability densities.
Figure 4.8 A square potential well of finite depth. The letters , , and denote the three regions of the -axis where the potential has a given constant value. Since we are interested in bound states, the energy of the particle lies below the “edge” of the well .
Figure 4.9 Graphical construction of the eigenvalues of the square well. For there are two bound states of even parity and and one of odd parity .
Figure 4.10 Eigenfunctions of a finite square well with . This particular well has only three bound states.
Chapter 5: Square Potentials. II: Continuous Spectrum—Scattering States
Figure 5.1 Square potential step and classically forbidden reflection. The particle has a finite probability to be reflected when it encounters the potential step, even though its energy is greater than the step height .
Figure 5.2 Reflection and transmission coefficients for the square potential step: quantum case (a,b) and classical case (c,d).
Figure 5.3 Scattering by a rectangular potential barrier. A quantum mechanical particle has a finite probability to be reflected even when its energy compels it classically to go through the barrier . Conversely, it has a finite probability to go through even when its energy forbids this within classical mechanics .
Figure 5.4 The transmission coefficient as a function of the energy of the particle for a square potential barrier with . The nonvanishing values of for confirm the
tunneling effect
, while the values of smaller than 1 for show that
“forbidden reflection”
also takes place. The instances of for correspond to the so-called
resonances
.
Figure 5.5 Classical “analog” of the tunneling effect. The spherical ball does not have the energy required to cross over the hill, and yet it does emerge on the other side using the “tunnel” on the hillside. The probabilistic nature of this quantum phenomenon—sometimes the particle crosses the hill, sometimes it does not—is described classically with a random opening or closing of the tunnel's entrance!
Figure 5.6 General shape of the wavefunction of a particle that crosses a classically forbidden region. Even though the amplitude of the wavefunction decays exponentially inside the barrier region, there is a finite probability for the particle to reach the other side and continue its motion as a wave with a significantly reduced amplitude.
Figure 5.7 The potential experienced by an alpha particle in a radioactive nucleus. Inside the nucleus , the potential has the form of an attractive well due to the strong nuclear forces. Outside, it has the form of a repulsive Coulomb potential due to the electrostatic repulsion between the escaping alpha particle and the rest of the nucleus. Since the alpha particle stays for a long time inside the nucleus, we must have , so the only way for the particle to escape the nucleus is via the tunneling effect.
Figure 5.8 Tunneling through a potential barrier of arbitrary shape. The classically forbidden region is the interval between points and .
Chapter 6: The Harmonic Oscillator
Figure 6.1 The potential of a harmonic oscillator.
Figure 6.2 The parabolic approximation. Any potential can be approximated by a harmonic oscillator potential in the vicinity of a local minimum ( in this case).
Figure 6.3 The first four eigenfunctions of the harmonic oscillator. The first eigenfunction is even with zero nodes, the second eigenfunction odd with one node, the third again even with two nodes, and so on.
Figure 6.4 The harmonic oscillator eigenfunction with a quantum number = 20. As we move away from the origin, the local wavelength of successive half-waves increases, while their corresponding “height” also increases. Thus, in the limit of large , classical behavior is restored: The probability of finding the particle increases in regions where the local speed is small, that is, near the boundaries of the classical oscillation.
Figure 6.5 Comparison between the quantum and classical probability densities for the harmonic oscillator state with . The quantum distribution oscillates symmetrically about the corresponding classical curve. In the limit of large , where quantum oscillations cease to be observable, the classical curve can be regarded as a kind of average of the quantum distribution.
Figure 6.6 Comparison between the quantum and classical probability densities for the ground state of the harmonic oscillator. In contrast to the classical case, the quantum particle is much more likely to be found near the origin than near the endpoints of the classical oscillation. Thus, when the particle is in the ground state, it behaves in the most extreme anticlassical manner.
Figure 6.7 How the state of minimum energy is reached for the harmonic oscillator.
Scenario
(a): In order for the particle to get closer to the bottom of the potential and thus minimize its potential energy, it “forms” a strongly localized wavefunction. But in doing so, its kinetic energy increases so much that it cancels out the benefit of the lowering of the potential energy.
Scenario
(b): In order for the particle to avoid excessive localization—and the associated increase of its kinetic energy—it “forms” a very broad wavefunction. Such a wavefunction, however, raises the potential energy excessively, since it enhances the probability of finding the particle away from the bottom of the potential. The total energy is not minimized now either.
Scenario
(c): Finally, the minimization of the total energy is achieved by balancing the competing requirements of the potential and kinetic energy terms. The actual wavefunction of the particle is neither too narrow nor too broad. It has the optimal extent.
Figure 6.8 The mean total energy of a quantum oscillator as a function of the “size” of its wavefunction. If the wavefunction is very localized (small ), the potential energy decreases, but then the kinetic energy increases excessively. Conversely, if the wavefunction is very extended (large ), the kinetic energy goes down, but the potential energy grows greatly. The minimum total energy is attained for an intermediate value of and the corresponding minimum energy is equal to .
Figure 6.9 Allowed and forbidden transitions in the harmonic oscillator. The only allowed transitions are between neighboring levels (). All other transitions () are forbidden; they are depicted here with “crossed-out” arrows and without any indication of emitted photons. Since the oscillator levels are equidistant, the allowed transitions produce radiation of a single frequency, as in the corresponding classical problem.
Chapter 8: The Hydrogen Atom. I: Spherically Symmetric Solutions
Figure 8.1 Cartesian and spherical coordinates, and the relation between them. The proton () is considered fixed at the origin, while the electron () is at some point with spherical coordinates .
Figure 8.2 The Coulomb potential. Bound states exist only when the total energy of the electron lies below the limiting value of the Coulomb potential at infinity. In other words, for bound states, the total energy must be
negative
.
Figure 8.3 Energy-level diagram for the hydrogen atom. The diagram shows the energy levels of the hydrogen atom and the two basic quantities derived from it: the minimum excitation energy and the ionization energy . The atom in its ground state is stable against thermal collisions, since : It can move neither down in energy (there is no available state) nor up (due to the large energy gap separating the ground state from the first excited state).
Figure 8.4 How the electron chooses its ground state in the hydrogen atom.
Scenario
(a): To “capitalize” on the attraction from the nucleus—that is, to minimize its potential energy—the electron “forms” a very localized wavefunction, which, however, results in a large increase of its kinetic energy due to the uncertainty principle.
Scenario
(b): To avoid being highly localized, the electron “blows up” its wavefunction and thus lowers its kinetic energy. But it then increases its potential energy by being far away from the nucleus.
Scenario
(c): The atom finally achieves the state of minimum total energy by balancing the competing demands of the kinetic and potential energy terms. In this balanced state, its wavefunction is neither too “tight” nor too “loose”: it has the optimum size. As we see, in choosing its ground state, the electron obeys a kind of “Goldilocks principle”: Its wavefunction is at first too localized, then too extended, and finally, about right.
Figure 8.5 Total energy of the hydrogen atom as a function of its possible size. Becoming either too small or too large is energetically costly for the atom. When it gets too small, it lowers its potential energy, but the kinetic energy increases dramatically; when it gets too large, it lowers its kinetic energy, but its potential energy increases rapidly (it becomes less negative). The total energy is minimized when the atomic radius becomes equal to the Bohr radius.
Figure 8.6 Radial probability density in the ground state of the hydrogen atom. The most probable distance of the electron from the nucleus is , which is equal to the radius of the first Bohr orbit in the old quantum theory.
Figure 8.7 Resolution of a paradox. Even though the probability per unit volume to find the electron is greater near the nucleus (the “dots” get much denser there), the most probable distance is actually away from the origin (at ) because the radial probability is being maximized there. All the dots that lie inside the volume of a spherical shell extending from to contribute to the value of the radial probability. Since the volume of this shell increases with , it overcompensates, up to some value , the loss of probability per unit volume.
Chapter 9: The Hydrogen Atom. II: Solutions with Angular Dependence
Figure 9.1 Equipotential surfaces and forces in a central potential. The equipotential surfaces ( constant) are spheres centered at the origin . Since the forces are normal to the equipotential surfaces, the force lines always pass through the center. In other words, these are central forces.
Figure 9.2 Why wavefunctions should be periodic in . When changes by we are back to the starting point. Therefore, for the wavefunction to be a single-valued function of space, it has to remain the same under the change .
Figure 9.3 Effective potential for the hydrogen atom. The region of bound states lies below the value of the potential at infinity, that is, at negative energies.
Figure 9.4 The first few radial functions with .
Figure 9.5 Energy-level diagram for the hydrogen atom. Each column in the diagram represents the energy levels for a given and can be regarded as the separate energy diagram for the corresponding effective potential .
Figure 9.6 1
s
and 2
s
orbitals. Both orbitals are spherically symmetric, but the 2
s
orbital has a “rarefied probability” in the vicinity of the node, , of the corresponding wavefunction, and it changes sign across this node. But such differences among s orbitals in how rarefied they locally are, are secondary compared to their common feature of
spherical symmetry
. It is thus customary to graphically depict all s orbitals using only the sketch of their first member: the 1
s
orbital.
Figure 9.7 Construction of the 2
p
orbital from its wavefunction. Due to the factor, only two lobes (of opposite sign) survive around the -axis. Thus, the orbital has strong directionality along the -axis, a fact denoted by the index in 2
p
. All other states with and (but with greater ) have the same general form, but with more lobes on each side of the -axis. Again, since the dominant feature is the strong directionality around the -axis (the factor is the same in all cases), it is customary to graphically represent all such orbitals with the 2
p
orbital, and use the notation without reference to the quantum number .
Figure 9.8 The triplet of , , orbitals.
Figure 9.9 The quantized vector of angular momentum for and . Due to the quantization of its projection onto a particular axis, the angular momentum “vector” (to the extent that depicting it as a vector has a meaning) can only take discrete orientations in space. We call this effect “quantization of orientation” or “space quantization.” Note that the vector is never aligned with the -axis, which is a rather thought-provoking feature. Even when the projection of the vector onto the -axis is maximized, it forms a nonzero angle with it, since we always have .
Figure 9.10 Allowed and forbidden transitions in the hydrogen atom. Allowed transitions satisfy the selection rule and are shown with solid lines, while forbidden transitions do not obey the above rule and are shown with dashed lines. The Figure shows only transitions originating from the 2
s
, 2
p
, 3
p
, and 3
d
levels.
Figure 9.11 The “forbidden” transition 2
s
1
s
. The transition is actually possible by emission of two photons of opposite spin , so as not to violate angular momentum conservation. But since the emission of two photons is a “difficult” process, the lifetime of the electron in the 2
s
state is huge by atomic measures: approximately !
Chapter 10: Atoms in a Magnetic Field and the Emergence of Spin
Figure 10.1 Lifting of the rotational degeneracy when an atom is placed in a magnetic field. A single level gives rise to a triplet of levels that are symmetrically arranged with respect to the initial level (Zeeman splitting).
Figure 10.2 The magnetic dipole in its traditional and modern depiction (bar magnet and current-carrying loop, respectively). (a) If magnetic charges existed, we would define the magnetic moment as a vector that points from the south to the north pole (inside the magnet) and has a magnitude , where are the magnetic charges at the edges of the dipole and is the dipole length. (b) The magnetic moment is perpendicular to the plane of the current-carrying loop and has a magnitude in units— in SI—where is the current and is the surface area of the loop.
Figure 10.3 Relation between magnetic moment and angular momentum for a charged particle. The vectors and are parallel and .
Figure 10.4 Zeeman splitting for a level with . When we place the atom in a magnetic field, the rotational degeneracy of a level with quantum number is lifted and new levels appear, symmetrically arranged with respect to the original level. The distance between adjacent levels is .
Figure 10.5 The Zeeman effect. Owing to the selection rules , , each spectral line becomes a triplet, with the middle line positioned at the original one and the other two lines placed symmetrically on either side. The distance between the “Zeeman components” of each spectral line is proportional to the magnetic field.
Figure 10.6 Three groups of spectral lines of a many-electron atom in the presence of a magnetic field. The left “line” is related to the
normal Zeeman spectrum
, while the other two pertain to the
“anomalous” Zeeman spectrum
. The distances between the original lines, as well as the distances between the Zeeman components of each line, are shown out of scale.
Figure 10.7 The Stern–Gerlach experiment for the hydrogen atom. Because the field is inhomogeneous, atoms in the beam experience a force in the vertical (
z
) direction that is proportional to the quantum number . Consequently, the beam splits into components, where is the quantum number of the electronic spin. In the actual experiment, the beam splits in
two
components. Therefore, the electron carries an intrinsic angular momentum (spin) with quantum number .
Figure 10.8 The actual trace of the beam in the Stern–Gerlach experiment: (a) in quantum physics, (b) in classical physics.
Figure 10.9 Time evolution of spin in a magnetic field. : The probability to find an electron having again spin up along the -axis as it evolves in a magnetic field that points along the -axis. : The probability of a “spin flip” after the lapse of time . Both probabilities oscillate periodically with frequency and period .
Chapter 11: Identical Particles and the Pauli Principle
Figure 11.1 The fundamental difference between classical and quantum mechanics with respect to the distinguishability of identical particles. The electrons of a classical atom in (a) can always be distinguished owing to the uniqueness of their trajectory. In contrast, it is impossible to distinguish them in the corresponding quantum atom (b), since the electrons are now described by overlapping wavefunctions, and can thus be found at the same point in space.
Figure 11.2 Density plots for the probability distributions and corresponding to parallel and antiparallel spin arrangement for two particles that populate the first two levels of an infinite potential well. In the parallel arrangement, “hills” are found along the
second diagonal
(where the distance is large), while in the antiparallel arrangement, the probability density is maximized when the interparticle distance is small, that is, along the
first diagonal
. Therefore, parallel spins (a) like to stay apart, while antiparallel spins (b) prefer to get together. Note that the horizontal and vertical axes represent and , respectively.
Chapter 12: Atoms: The Periodic Table of the Elements
Figure 12.1 Screened and bare Coulomb potentials. The two potentials coincide near the origin, while at greater distances, the screened potential is always
higher
than the bare potential, since it corresponds to a decreased electrostatic attraction from the nucleus.
Figure 12.2 Hydrogen-like energy levels (bare Coulomb potential). The main feature of this diagram is the hydrogen-like degeneracy, that is, the coincidence of energy levels with ranging from zero to for a particular .
Figure 12.3 Energy-level diagram for many-electron atoms. Because of screening, the hydrogen-like degeneracy is lifted and the levels for a particular are ordered in increasing values of the quantum number .
Figure 12.4 The modern quantum mechanical picture for the first two atoms of the periodic table. In hydrogen, the 1
s
orbital is occupied by one electron. In helium, the 1
s
orbital is occupied by two electrons of opposite spins, which thus form a
closed shell
. The two orbitals have the same shape but differ in size, since the corresponding Bohr radius—assuming we can ignore electron–electron repulsions—depends on through the formula . Therefore, the 1
s
orbital of helium is about two times smaller than the corresponding orbital of hydrogen.
Figure 12.5 Energy-level diagram for the ground state of the elements with atomic numbers from to . Because of the Pauli principle, the placement of electrons in the degenerate 2
p
, 2
p
, 2
p
states has to be done as shown in the Figure We thus successively place one electron on each level with spins in parallel, and when we are done with parallel arrangement we continue with antiparallel spins. This way of filling the levels—known empirically as
“Hund's first rule”
—is imposed by the minimization of the atomic energy, since parallel spins keep electrons further apart and thus decrease their electrostatic repulsions.
Figure 12.6 Quantum mechanical explanation of the geometric shape of the water molecule. Given that the valence orbitals of oxygen are 2
p
and 2
p
, oxygen is chemically reactive in two orthogonal directions, along which it can form bonds with the 1
s
valence orbitals of the two hydrogen atoms. According to this scenario, the (i.e., water) molecule has the shape of an isosceles right triangle with an oxygen atom on its apex and two hydrogen atoms at the ends of its base. Actually—for reasons we will explain later—the apex angle is not but approximately .
Figure 12.7 Ionization energy as a function of atomic number for the elements of the small periodic table.
Figure 12.8 Occupied energy-level diagram for the element with (nitrogen). The Pauli principle imposes parallel arrangement of electronic spins. As a result, the total spin of the atom is and its valence is equal to 3, since there are three empty states in the 2
p
valence shell.
Figure 12.9 The NH molecule according to quantum mechanics. Since the nitrogen valence orbitals are 2
p
, 2
p
, 2
p
, the bonding of nitrogen with the three hydrogen atoms takes place along three orthogonal directions, as shown in (a). As a result, the molecule has the pyramid-like shape of (b).
Chapter 13: Molecules. I: Elementary Theory of the Chemical Bond
Figure 13.1 The double-well problem as a simple model for the quantum mechanical description of chemical bonds: (a) The symmetric double well. (b) The asymmetric double well. In both cases, energy minimization forces the particle's wavefunction to have “hills” (i.e., higher values) on the wells—to take advantage of low potential energy regions—and “valleys” (i.e., lower values) between the wells, where the potential energy is high. Owing to symmetry, the molecular wavefunction in the first case is an equal-weight superposition of the wavefunctions for single wells. In the asymmetric double well, the wavefunction of the deeper well dominates (i.e., ) because it is energetically favorable for the particle to spend more “time” in this region.
Figure 13.2 A plausible approximate form for the wavefunction of the first excited state for a symmetric double well. Since the wavefunction must resemble the local solution or in the vicinity of the corresponding well, the first excited state ought to be the
odd superposition
of and , in order to have the expected symmetry and the required one node.
Figure 13.3 Approximate solution (eigenvalues and eigenfunctions) for the symmetric double well, with the LCAO method. The main conclusion applies also to higher states and can be stated as follows. When we bring close enough two identical wells (i.e., two identical attractive centers) then each state of the single wells gives rise to two states for the double well, symmetrically positioned with respect to the original state. The corresponding eigenfunctions are given by the even and odd superpositions of the single-well eigenfunctions, respectively.
Figure 13.4 Energy levels of an asymmetric double well in the LCAO approximation. By moving in both wells at the same time, the particle lowers its energy even more than if it remained localized in the deeper well. We thus have .
Figure 13.5 Quantum mechanical picture of the molecule. (a) The overlapping atomic orbitals participating in the bond. (b) A simplified sketch of the molecular orbital produced by combining (“conjoining”) the atomic orbitals, where the two valence electrons occupy the molecular orbital with opposite spins. The molecule is a classic example of
covalent bonding
.
Figure 13.6 Energy-level diagram of the molecule. On either side we show the (ground state) levels of the free atoms, while the first two molecular states are shown in the middle. The energy gain due to molecule formation is equal to .
Figure 13.7 The total energy of the hydrogen molecule. is a sum of the term for repulsion between nuclei, and the term for attraction of the valence electrons by the protons. The minimum of the curve determines the length of the molecule in its equilibrium configuration.
Figure 13.8 Energy-level diagram of a hypothetical He molecule. The molecule cannot exist because the energy gained by placing two electrons in the
bonding state
is cancelled by the necessary placement of the other two electrons in the
antibonding state
.
Figure 13.9 Energy-level diagram for the lithium molecule. Because inner atomic levels are fully occupied, any energy gain can only result from the outer level 2
s
that is half-filled. Therefore the inner, filled levels do not affect the chemical behavior of the atom—since they do not contribute energywise. Only the outer, half-filled 2
s
level (the so-called valence orbital) is important in this respect.
Figure 13.10 Two possible ways to combine atomic orbitals and produce a double bond in an oxygen molecule. (a) The orbitals of one pair bind
head-on
(i.e., axially) with a
strong overlap
, thus forming a very
strong bond
, while the orbitals of the other pair bind sideways (i.e., laterally) with very weak overlap and form a second,
much weaker bond
. Note that we have used a series of vertical lines to denote the weak lateral overlap of orbitals in this case. (b) Another way to combine orbitals, where both bonds have intermediate strength. General energy considerations show that the bonding configuration (a) is energetically favorable, since in configuration (b), the produced molecular orbitals form an angle—that is, the corresponding molecular wavefunction has abrupt spatial variations—and cause an excessive increase of the kinetic energy of valence electrons. Simple symmetry arguments lead to the same conclusion. For instance, that the ground state always has the full symmetry of the problem (in our case, rotational symmetry about the molecular axis) and no nodes, while the first excited state has one node—or one nodal surface in a three–dimensional problem. The bonding configuration (a) satisfies these general requirements, while (b) does not.)
Figure 13.11 The formation of molecular orbitals in the molecule.
Figure 13.12 The quantum mechanical explanation for the shape of the water molecule. Owing to the directionality of the valence orbitals in the O atom, its binding with two H atoms can take place along two perpendicular directions, as in (a). As a result, the water molecule looks (to first approximation) like an isosceles orthogonal triangle, with the O atom at its apex and the two H species at the vertices of its base, as in (b).
Figure 13.13 Actual shape of the water molecule. Because oxygen is more electronegative than hydrogen, valence electrons move toward the O atom and produce an excess of negative charge there, and a deficit of negative charge—that is, positive charge—around the H atom. The repulsion between the positively charged H atoms widens the angle of the molecule to its final value of . For even greater angles, the energy gained by further distancing the H atoms from each other is overcompensated by the energy penalty associated with the deformation of the and valence orbitals of the O atom.
Figure 13.14 Chemical notation for the dipole moments of bonds and the total dipole moment of molecules (here, for O). The dipole moments point in the direction of motion of the valence electrons along the bond, which is opposite to the direction used by physicists.
Figure 13.15 A spatial ordering of molecules that brings their oppositely charged regions to proximity. Solid lines denote the standard polar covalent O−H bonds, while dashed lines show the
weaker coupling
between an oxygen atom from one molecule and a hydrogen atom from another, caused by the electrostatic attraction of their opposite charges. Through this dual action, each hydrogen atom acts as a kind of
bridge
between two oxygen atoms, hence the term
hydrogen bond
.
Figure 13.16 The quantum mechanical mechanism of hydrogen bonding. (a) Under the simultaneous attraction of two centers (the two oxygen atoms), the hydrogen atom performs a delocalized motion between the corresponding wells, interchanging continuously the type of its “binding” to each oxygen ion (continuous line). Thus a hydrogen bond is actually a quantum superposition of forms I and II, as shown in (b). A more realistic description should also take into account the lack of symmetry between the positions of the two oxygen atoms. The simplified picture above is thus only meant to explain the basic idea of hydrogen bonding.
Figure 13.17 Hydrogen bonds between bases in the double helix of DNA. Adenine binds to thymine (A–T) and guanine binds to cytosine (G–C). Dashed lines denote hydrogen bonds.
Figure 13.18 Conventions for the structural formulas of large organic molecules. The complete structural formula of adenine (to be compared with Figure 13.17).
Figure 13.19 Quantum mechanical analysis of the molecule. (a) Pairing between the , and valence orbitals of N and the 1
s
valence orbitals of the three H atoms. (b) The final geometry of the molecule. The molecule has the shape of a pyramid with the N atom at the apex and the three H atoms at the vertices of the base. Since nitrogen is clearly more electronegative than hydrogen, N–H bonds are polar and is a
polar molecule
. Owing to repulsions between hydrogen atoms, the HNH angles exceed (the experimental value is ).
Figure 13.20 Nitrogen inversion in the ammonia molecule. Owing to the finite energy barrier that separates its two mirror configurations, the ammonia molecule's true ground state is the even superposition of these configurations, while the odd superposition represents an excited state of the molecule. As a result, the NH vibrational levels split in
two
, a typical feature for particle motion in a double well. (The matrix element of the Hamiltonian between the two localized vibrational states is .)
Figure 13.21 Time evolution of the state of an ammonia molecule that was originally in its left polar form. is the probability of finding the molecule in the same state L after time , while is the probability for the molecule to undergo
inversion
and end up in its right polar state. (See also Figure 13.20.)
Figure 13.22 Mechanical model for a diatomic molecule. A weightless bar of length (equal to the bond length) with atomic masses and is attached to its ends. The system is known as a rigid rotor. The center of mass (CM) divides the line segment joining the atoms into intervals that are inversely proportional to the atomic masses at its respective ends.
Figure 13.23 The quantized rotational levels of a diatomic molecule and the allowed transitions between them, according to the selection rule . The energies of the emitted photons are integer multiples of , and thus the rotational spectrum consists of a fundamental frequency and its integer multiples.
Figure 13.24 The molecular potential and the corresponding vibrational spectrum for a diatomic molecule (in this case, HCl). Since the molecular potential coincides with that of a harmonic oscillator (dashed line) in the vicinity of its minimum, the first few vibrational energy levels are almost equidistant. Only as we go to higher states do the energy levels become gradually denser.
Figure 13.25 The three types of molecular spectra and their position in the electromagnetic spectrum measured in eV. (The spectra are shown out of scale.)
Figure 13.26 Absorption lines of the vibrational–rotational spectrum for a typical diatomic molecule. Transitions with and produce two groups of spectral lines, positioned symmetrically on either side of a hypothetical spectral line , which corresponds to a
forbidden transition
between the purely vibrational levels and . This transition is not observed in the spectrum because it violates the selection rule , which has to be satisfied simultaneously with the rule , for reasons we explained before.
Figure 13.27 The vibrational–rotational absorption spectrum of the molecule. The appearance of a second peak for each line is due to the Cl isotope that constitutes of the Cl in the HCl sample. We can clearly see two groups of lines (R-branch and L-branch) corresponding, respectively, to transitions and . In the middle of the curve we show the “central frequency” that corresponds to the forbidden purely vibrational transition , , .
Chapter 14: Molecules. II: The Chemistry of Carbon
Figure 14.1 The molecule according to the elementary theory of the chemical bond. (a): Coupling between atomic orbitals of C and H. Since the orbital of C contains no electrons, it couples to the orbitals of two H atoms, which provide the two electrons that occupy the molecular orbital that is formed. (b): The geometric shape of CH. Because the dipole moments of the two bonds on the – plane do not cancel each other out, the CH molecule ought to be
polar
, with all the associated physical properties. And yet, CH is
nonpolar
. The elementary theory of the chemical bond fails spectacularly here.
Figure 14.2 Actual structure of the methane molecule. The carbon atom sits at the center of a tetrahedron whose four vertices are occupied by the H atoms. Owing to the symmetric arrangement of the H atoms, the dipole moments of the four C–H bonds cancel each other out, and the CH molecule is
nonpolar
.
Figure 14.3 The concept of hybridization. The superposition (i.e., hybridization) of the
s
and
p
orbitals produces a strongly unidirectional orbital that is much more suitable for chemical interactions with other atoms. Of course, this sketch reflects a convention used in the literature rather than an accurate depiction, which would require two equal-sized lobes but with much more intense shading on the upper lobe than the lower one.
Figure 14.4 A “pure”
p
orbital (a) and a hybridized orbital (b) couple with the 1
s
orbital of another atom. The coupling is much stronger in case (b) because of the much higher intensity of the probability cloud in the upper lobe of the hybridized orbital. Even though this is not very clearly shown in the figure, we could say that in case (b) we have
enhanced overlap
between the participating orbitals, since in the overlap integral the intensity of the probability clouds also plays a role, not only the spatial extent of their overlap. The chemical bond formed from hybridized orbitals as depicted in (b) is thus much stronger.
Figure 14.5 Occupied energy-level diagram in a hybridized state. A carbon atom can achieve hybridization between its
s
and
p
orbitals by “lifting” one of the two 2
s
electrons up to the empty level. Thus—assuming we ignore the energy difference between the 2
s
and 2
p
states—all states , and , and all their linear combinations, are equally available for chemical bonding.
Figure 14.6 The methane molecule according to hybridization theory. The four hybridized orbitals of C point from the center of a tetrahedron—shown in dashed lines—toward its four vertices, where they meet with the orbitals of the four hydrogen atoms.
Figure 14.7 Tetrahedral geometry. Four unit vectors form a regular tetrahedron when they have a common origin and their mutual angles are all equal.
Figure 14.8 The
sp
and
sp
hybridized orbitals.
Figure 14.9 The need for partial hybridization in multiple bonds. The
sp
hybridization does not result in “good quality” bonds for a molecule such as ethene (CH) owing to the double bond between the carbon atoms. Partial hybridization is necessary in this case.
Figure 14.10 Hybridization in the molecule. The two C atoms hybridize in an
sp
state, which results in a
planar
molecule.
Figure 14.11 Hybridization in the acetylene molecule. The hybridization state of C atoms is
sp
, since a triple bond can contain only one bond, with the other two bonds being necessarily -type. As a result of this hybridization, the CH molecule is
linear
.
Figure 14.12 In the molecule, the two hydrogen “tripods” are in a crosswise (staggered) arrangement in order to minimize their electrostatic repulsions. Note, by the way, that the small lobes of the hybrid orbital are not shown anymore, for simplicity.
Figure 14.13 The triple potential well experienced by each hydrogen “tripod” of the ethane molecule, as it rotates with respect to the other tripod. Because of this potential, the molecular spectrum for torsional oscillations around the molecule's axis consists of triplets of adjacent levels.
Figure 14.14 Structural formula (a) and configuration of valence orbitals (b) in the benzene molecule. Owing to the double bonds, carbon hybridizes into an
sp
state, while the need for sideways overlap of the unhybridized orbitals makes the molecule
planar
.
Figure 14.15 The structural formula (a) and the configuration of valence orbitals (b) in the hexatriene molecule. The hybridization state of carbon is
sp
, for the same reason as in the benzene molecule. While this does not guarantee the planarity of the molecule—since it is still possible for parts of the molecule to rotate around the single bonds—the molecule is actually planar, for reasons we will explain later. Moreover, owing to the angle between the strong bonds, the actual spatial arrangement of the carbon atoms—the “spine” of the molecule, so to speak—does not follow a straight line (as implied by the simplified sketch of the structural formula) but “zigzags” at angles.
Figure 14.16 Application of hybridization theory to the molecules and . (a) Using
sp
hybridization for the HO molecule. Four of the six electrons in the shell of O ([O] = [He]) fill two of the four
sp
orbitals. The other two electrons half-fill the remaining two orbitals, which therefore become the
valence orbitals
of O and bond with the 1
s
orbitals of H to form the HO molecule. (b) An analogous picture for the NH molecule. Since there are five available electrons here—remember that [N]=[He] 2
s
2
p
—only one
sp
orbital is filled; the other three hybrid orbitals remain half-filled, and bind with the 1
s
orbitals of H to form the NH molecule.
Figure 14.17 The four possible isomers of dibromobenzene. Since we can detect only one dibromobenzene of the 1,2- type in the laboratory, we are led to conclude that edges with single and double bonds are equivalent, as are therefore all the hexagon edges. The benzene molecule is thus a
regular hexagon
.
Figure 14.18 The delocalization mechanism in the benzene molecule. (a) Imagine stripping the molecule momentarily from its six electrons, leaving behind an equal number of carbon ions; this is how the molecule would look then. Subsequently, the six ions act as attractive centers—i.e., potential wells—that draw the electrons to perform delocalized motion along the entire hexagonal chain, to achieve the lowest possible energy. (b) The six orbitals of the molecule form a hexagonal array, along which delocalized motion is bound to occur, as the electrons can “hop” from one orbital to another with the same probability.
Figure 14.19 Why delocalization is impossible for electrons in the benzene molecule. There is considerable overlap between atomic orbitals forming a bond, but negligible overlap between atomic orbitals of different bonds. Consequently, the probability for electrons to hop onto a neighboring bond (dotted arrows) is far smaller than the probability of them staying on the same bond and hopping from one of its orbitals onto the other (solid arrows). So the bonds are localized and thus independent from one another, precisely as predicted by traditional chemistry.
Figure 14.20 The occupied energy-level diagram of electrons in the benzene molecule according to the free-electron model. The first excitation energy is equal to , where is the electron mass.
Figure 14.21 (a) Graphical construction of the energy eigenvalues for delocalized motion in the benzene molecule. (b) The occupied energy-level diagram of the molecule.
Figure 14.22 The molecular orbital for the ground state of the delocalized motion of electrons in a benzene molecule. (a) The probability amplitude of the pair of electrons occupying this orbital resembles two tori that lie above and below the molecule's hexagon. The torus above (below) the hexagon has positive (negative) sign. (b) The modern chemical symbol for delocalized chemical bonds in the benzene molecule.
Figure 14.23 The concept of resonance. When a conjugated molecule has more than one possible locations for its bonds—that is, more than one possible structures, or forms—then its actual state is a suitable quantum superposition of these forms. In the case of benzene, where there are two, entirely symmetrical, possible forms—also known as
Kekulé structures
—the actual state of the molecule is described by their symmetric superposition. Moreover, because none of these forms is an eigenstate of the molecular Hamiltonian, if the molecule is found momentarily in either of these, it will start to
oscillate periodically
between the two forms—hence the term
resonance
to describe this motion. After spending some time in this oscillatory “motion,” the molecule will emit the extra energy of the initial form—via a photon—and fall to its ground state whereby no periodic oscillation takes place. While in the ground state, the molecule has an equal probability to be in form I or form II at any given moment.
Figure 14.24
p
orbitals of the butadiene molecule in its planar form. In this arrangement, all
p
orbitals are parallel to each other and thus have the same sideways overlap. This means that delocalized motion throughout the chain is not only possible but also inevitable. As for the zigzag shape of the chain, this clearly does not affect the probability of an electron hopping from one
p
orbital to another.
Figure 14.25 The occupied energy-level diagram for the delocalized motion of electrons in butadiene.
Figure 14.26
The light-collecting molecule for vision (structural formula)
. It is a typical conjugated system with eleven carbon atoms on the main chain. The molecule is originally attached on an enzymatic catalyst from which it detaches upon absorption of a photon. The detachment activates the enzyme and triggers a series of chemical reactions, whose final product is an electrical signal that is transmitted to the brain via the optical nerve.
Figure 14.27 The two possible conformations of the CH molecule (structural formula (14.35)). Conformation (a): – delocalization system. The molecule is planar. Conformation (b): – delocalization system. The end C–H bonds form planes that are perpendicular to each other.
Figure 14.28 Graphene (a single sheet of graphite). Carbon atoms are
sp
hybridized, and, accordingly, form a two-dimensional hexagonal “honeycomb” lattice. Perpendicular to the plane of the honeycomb lattice lie the unhybridized orbitals, forming also a two-dimensional
network
on which the electrons can become completely delocalized. Graphene was isolated and characterized for the first time in 2004 by Andre Geim and Konstantin Novoselov (Nobel Prize, 2010), and its spectacular properties are currently one of the most active fields of research.
Figure 14.29 The C fullerene. The carbon atoms lie on the vertices of a polyhedron that consists of 12 regular pentagons and 20 regular hexagons. Carbon is in an (almost) hybridization state and forms three strong bonds with neighboring C atoms. The (almost) unhybridized
p
orbitals are perpendicular to the molecule's circumscribing sphere. Thus a “forest” of
p
orbitals is formed, on which complete delocalization of electrons takes place.
Figure 14.30 Why pentagons are needed. Five hexagons around a
pentagon
leave gaps among them. When this planar structure is folded to become part of a convex polyhedron, the hexagons can touch each other without being distorted. In contrast, structural distortions will arise if we try the same folding with a lattice of regular hexagons that fill the plane.
Chapter 15: Solids: Conductors, Semiconductors, Insulators
Figure 15.1 Band structure of the energy spectrum in a periodic potential. Bands form around the discrete energy levels of the single well. But they also form in the range of the (single well's) continuous spectrum.
Figure 15.2 Energy bands in an insulator (a) and a conductor (b). The highest occupied band in an insulator is completely filled with electrons, but in a conductor, it is only partially occupied, thus enabling the movement of electrons, that is, current flow.
Figure 15.3 Energy bands in a semiconductor for and . When the energy gap between the valence and conduction bands is sufficiently small , a non-negligible fraction of valence electrons is thermally excited to the conduction band, where the electrons can easily move, as can the
holes
they leave behind. The material behaves then as a
semiconductor
.
Figure 15.4 Dependence of the electronic energy on the continuous parameter that characterizes energy states in a one-dimensional crystal. Since is linked to the crystal momentum through the relation , the Figure also shows the relation between energy and momentum in the crystal.
Figure 15.5 How oblique scattering is avoided during wave propagation in a periodic medium. The secondary quantum waves emitted from the atoms reconstruct the original plane wave and the electrons keep moving forward without oblique scattering.
Figure 15.6 Fermi energy of a crystalline solid at absolute zero temperature.
Figure 15.7 A cubic-shaped solid for the free-electron model. Electrons are assumed to move as free particles inside a cubic box that represents the bulk of the solid body.
Figure 15.8 “Allowed points” for and in space, for a free particle in a two-dimensional box. The “volume” (actually, here an “area”) that corresponds to each point is equal to .
Figure 15.9 The density of states as a function of energy in the free-electron model.
Figure 15.10 Typical experimental curve for the density of states in the conduction band of a three-dimensional conductor.
Figure 15.11 Experimental determination of the density of states. Data from x-ray spectra of atoms in a crystal allow us to determine the electronic density of states of the band where the transitions originated from.
Figure 15.12 Electronic specific heat of metals. Comparison of the classical prediction and a typical experimental curve for the electronic specific heat of metals as a function of temperature.
Figure 15.13 The occupancy of the states at a temperature above absolute zero.
Chapter 16: Matter and Light: The Interaction of Atoms with Electromagnetic Radiation
Figure 16.1 The three fundamental processes that take place when a photon hits an atom that is initially in its ground state.
Figure 16.2 The two fundamental processes that take place when the atom is already in an excited state.
Figure 16.3 Time evolution of the ground-state population for a two-level system, which undergoes resonant absorption and stimulated emission at rate and spontaneous emission at rate .
Figure 16.4 The cross section of a process. The effective area through which the atom collects the necessary “reaction” energy of one photon in the characteristic time of the particular process. This implies that
Figure 16.5 Proof of the exponential decay law . By the time the light beam has traversed a slice of thickness and area of the material, the fractional decrease of its intensity must be equal to the fractional area “blocked” by the cross sections of atoms contained in this slice.
Figure 16.6 Effective cross section of the atom–photon interaction as a function of incident photon energy. The effective cross section skyrockets to high values in the immediate vicinity of atomic bound states (resonances), while it almost vanishes away from them, in which case the (very weak) process of scattering occurs (hence the index “sc”). Only near the ionization threshold the effective cross section coincides, roughly, with the geometric cross section of the atom. Beyond this threshold, there is the continuum of unbound states and the process that takes place is the ionization or photoelectric effect, hence the index ‘p’ (photoelectric) in the corresponding cross section. Note also that resonances become gradually weaker and broader. Can you explain why?
Figure 16.7 Qualitative estimate of the matrix element . The superposition of the two states involved in the transition produces an asymmetric “hybrid” for which the mean position of the electron is no longer at the origin, as was the case for each of the two states of the transition that had mirror symmetry. The mean displacement of the electronic cloud due to superposition is of the same order as the spatial extent of the participating orbitals, and hence comparable to a Bohr radius .
Figure 16.8 The operation principle of a laser cavity. If there is
population inversion
