List of Figures

1.1. Different views of the same experiment. Left is the view of observers on the planets. Right is the view of an alien space ship.
1.2. Coordinate systems for the Lorentz transformation.
1.3. Example elastic collision seen by different observers.
1.4. A completely inelastic collision.
2.1. The classical picture of a vector.
2.2. Spike diagram of a vector.
2.3. More dimensions.
2.4. Infinite dimensions.
2.5. The classical picture of a function.
2.6. Forming the dot product of two vectors.
2.7. Forming the inner product of two functions.
2.8. Illustration of the eigenfunction concept. Function $\sin(2x)$ is shown in black. Its first derivative $2\cos(2x)$ , shown in red, is not just a multiple of $\sin(2x)$ . Therefore $\sin(2x)$ is not an eigenfunction of the first derivative operator. However, the second derivative of $\sin(2x)$ is $\vphantom0\raisebox{1.5pt}{$-$}$ $4\sin(2x)$ , which is shown in green, and that is indeed a multiple of $\sin(2x)$ . So $\sin(2x)$ is an eigenfunction of the second derivative operator, and with eigenvalue $\vphantom0\raisebox{1.5pt}{$-$}$ 4.
3.1. The old incorrect Newtonian physics.
3.2. The correct quantum physics.
3.3. Illustration of the Heisenberg uncertainty principle. A combination plot of position and linear momentum components in a single direction is shown. Left: Fairly localized state with fairly low linear momentum. Right: narrowing down the position makes the linear momentum explode.
3.4. Classical picture of a particle in a closed pipe.
3.5. Quantum mechanics picture of a particle in a closed pipe.
3.6. Definitions for one-dimensional motion in a pipe.
3.7. One-dimensional energy spectrum for a particle in a pipe.
3.8. One-dimensional ground state of a particle in a pipe.
3.9. Second and third lowest one-dimensional energy states.
3.10. Definition of all variables for motion in a pipe.
3.11. True ground state of a particle in a pipe.
3.12. True second and third lowest energy states.
3.13. A combination of $\psi_{111}$ and $\psi_{211}$ seen at some typical times.
4.1. Classical picture of an harmonic oscillator.
4.2. The energy spectrum of the harmonic oscillator.
4.3. Ground state of the harmonic oscillator
4.4. Wave functions $\psi_{100}$ and $\psi_{010}$ .
4.5. Energy eigenfunction $\psi_{213}$ .
4.6. Arbitrary wave function (not an energy eigenfunction).
4.7. Spherical coordinates of an arbitrary point P.
4.8. Spectrum of the hydrogen atom.
4.9. Ground state wave function of the hydrogen atom.
4.10. Eigenfunction $\psi_{200}$ .
4.11. Eigenfunction $\psi_{210}$ , or 2p.
4.12. Eigenfunction $\psi_{211}$ (and $\psi_{21-1}$ ).
4.13. Eigenfunctions 2p, left, and 2p, right.
4.14. Hydrogen atom plus free proton far apart.
4.15. Hydrogen atom plus free proton closer together.
4.16. The electron being anti-symmetrically shared.
4.17. The electron being symmetrically shared.
5.1. State with two neutral atoms.
5.2. Symmetric sharing of the electrons.
5.3. Antisymmetric sharing of the electrons.
5.4. Approximate solutions for hydrogen (left) and helium (right) atoms.
5.5. Abbreviated periodic table of the elements. Boxes below the element names indicate the quantum states being filled with electrons in that row. Cell color indicates ionization energy. The length of a bar below an atomic number indicates electronegativity. A dot pattern indicates that the element is a gas under normal conditions and wavy lines a liquid.
5.6. Approximate solutions for lithium (left) and beryllium (right).
5.7. Example approximate solution for boron.
5.8. Periodic table of the elements. Cell color indicates ionization energy. Boxes indicate the outer electron structure. See the text for more information. [pdf]
5.9. Covalent sigma bond consisting of two 2p states.
5.10. Covalent pi bond consisting of two 2p states.
5.11. Covalent sigma bond consisting of a 2p and a 1s state.
5.12. Shape of an sp $\POW9,{3}$ hybrid state.
5.13. Shapes of the sp $\POW9,{2}$ (left) and sp (right) hybrids.
6.1. Allowed wave number vectors, left, and energy spectrum, right.
6.2. Ground state of a system of noninteracting bosons in a box.
6.3. The system of bosons at a very low temperature.
6.4. The system of bosons at a relatively low temperature.
6.5. Ground state system energy eigenfunction for a simple model system. The system has only 6 single-particle states; each of these has one of 3 energy levels. In the specific case shown here, the system contains 3 distinguishable spinless particles. All three are in the single-particle ground state. Left: mathematical form. Right: graphical representation.
6.6. Example system energy eigenfunction with five times the single-particle ground state energy.
6.7. For distinguishable particles, there are 9 system energy eigenfunctions that have energy distribution A.
6.8. For distinguishable particles, there are 12 system energy eigenfunctions that have energy distribution B.
6.9. For identical bosons, there are only 3 system energy eigenfunctions that have energy distribution A.
6.10. For identical bosons, there are also only 3 system energy eigenfunctions that have energy distribution B.
6.11. Ground state of a system of noninteracting electrons, or other fermions, in a box.
6.12. Severe confinement in the -direction, as in a quantum well.
6.13. Severe confinement in both the and directions, as in a quantum wire.
6.14. Severe confinement in all three directions, as in a quantum dot or artificial atom.
6.15. A system of fermions at a nonzero temperature.
6.16. Particles at high-enough temperature and low-enough particle density.
6.17. Ground state of a system of noninteracting electrons, or other fermions, in a periodic box.
6.18. Conduction in the free-electron gas model.
6.19. Sketch of electron energy spectra in solids at absolute zero temperature. (No attempt has been made to picture a density of states). Far left: the free-electron gas has a continuous band of extremely densely spaced energy levels. Far right: lone atoms have only a few discrete electron energy levels. Middle: actual metals and insulators have energy levels grouped into densely spaced bands separated by gaps. Insulators completely fill up the highest occupied band.
6.20. Sketch of electron energy spectra in solids at a nonzero temperature.
6.21. Potential energy seen by an electron along a line of nuclei. The potential energy is in green, the nuclei are in red.
6.22. Potential energy seen by an electron in the one-dimensional simplified model of Kronig & Penney.
6.23. Example Kronig & Penney spectra.
6.24. Spectrum against wave number in the extended zone scheme.
6.25. Spectrum against wave number in the reduced zone scheme.
6.26. Some one-dimensional energy bands for a few basic semiconductors.
6.27. Spectrum against wave number in the periodic zone scheme.
6.28. Schematic of the zinc blende (ZnS) crystal relevant to important semiconductors including silicon.
6.29. First Brillouin zone of the FCC crystal.
6.30. Sketch of a more complete spectrum of germanium. (Based on results of the VASP 5.2 commercial computer code.)
6.31. Vicinity of the band gap in the spectra of intrinsic and doped semiconductors. The amounts of conduction band electrons and valence band holes have been vastly exaggerated to make them visible.
6.32. Relationship between conduction electron density and hole density. Intrinsic semiconductors have neither much conduction electrons nor holes.
6.33. The p-n junction in thermal equilibrium. Top: energy spectra. Quantum states with electrons in them are in red. The mean electrostatic energy of the electrons is in green. Below: Physical schematic of the junction. The dots are conduction electrons and the small circles holes. The encircled plus signs are donor atoms, and the encircled minus signs acceptor atoms. (Donors and acceptors are not as regularly distributed, nor as densely, as this greatly simplified schematic suggests.)
6.34. Schematic of the operation of an p-n junction.
6.35. Schematic of the operation of an n-p-n transistor.
6.36. Vicinity of the band gap in the electron energy spectrum of an insulator. A photon of light with an energy greater than the band gap can take an electron from the valence band to the conduction band. The photon is absorbed in the process.
6.37. Peltier cooling. Top: physical device. Bottom: Electron energy spectra of the semiconductor materials. Quantum states filled with electrons are shown in red.
6.38. An example Seebeck voltage generator.
6.39. The Galvani potential jump over the contact surface does not produce a usable voltage.
6.40. The Seebeck effect is not directly measurable.
7.1. The ground state wave function looks the same at all times.
7.2. The first excited state at all times.
7.3. Crude concept sketch of the emission of an electromagnetic photon by an atom. The initial state is left and the final state is right.
7.4. Addition of angular momenta in classical physics.
7.5. Longest and shortest possible magnitudes of the final atomic angular momentum in classical physics.
7.6. A combination of two energy eigenfunctions seen at some typical times.
7.7. Energy slop diagram.
7.8. Schematized energy slop diagram.
7.9. Emission and absorption of radiation by an atom.
7.10. Approximate Dirac delta function $\delta_\varepsilon(x-\protect{\underline x})$ is shown left. The true delta function $\delta(x-\protect{\underline x})$ is the limit when $\varepsilon$ becomes zero, and is an infinitely high, infinitely thin spike, shown right. It is the eigenfunction corresponding to a position $\protect{\underline x}$ .
7.11. The real part (red) and envelope (black) of an example wave.
7.12. The wave moves with the phase speed.
7.13. The real part (red) and magnitude or envelope (black) of a wave packet. (Schematic).
7.14. The velocities of wave and envelope are not equal.
7.15. A particle in free space.
7.16. An accelerating particle.
7.17. A decelerating particle.
7.18. Unsteady solution for the harmonic oscillator. The third picture shows the maximum distance from the nominal position that the wave packet reaches.
7.19. A partial reflection.
7.20. An tunneling particle.
7.21. Penetration of an infinitely high potential energy barrier.
7.22. Schematic of a scattering potential and the asymptotic behavior of an example energy eigenfunction for a wave packet coming in from the far left.
8.1. Separating the hydrogen ion.
8.2. The Bohm experiment before the Venus measurement (left), and immediately after it (right).
8.3. Spin measurement directions.
8.4. Earth’s view of events (left), and that of a moving observer (right).
8.5. The space-time diagram of Wheeler’s single electron.
8.6. Bohm’s version of the Einstein, Podolski, Rosen Paradox.
8.7. Nonentangled positron and electron spins; up and down.
8.8. Nonentangled positron and electron spins; down and up.
8.9. The wave functions of two universes combined
8.10. The Bohm experiment repeated.
8.11. Repeated experiments on the same electron.
10.1. Billiard-ball model of the salt molecule.
10.2. Billiard-ball model of a salt crystal.
10.3. The salt crystal disassembled to show its structure.
10.4. The lithium atom, scaled more correctly than before.
10.5. Body-centered-cubic (BCC) structure of lithium.
10.6. Fully periodic wave function of a two-atom lithium “crystal.”
10.7. Flip-flop wave function of a two-atom lithium “crystal.”
10.8. Wave functions of a four-atom lithium “crystal.” The actual picture is that of the fully periodic mode.
10.9. Reciprocal lattice of a one-dimensional crystal.
10.10. Schematic of energy bands.
10.11. Schematic of merging bands.
10.12. A primitive cell and primitive translation vectors of lithium.
10.13. Wigner-Seitz cell of the BCC lattice.
10.14. Schematic of crossing bands.
10.15. Ball and stick schematic of the diamond crystal.
10.16. Assumed simple cubic reciprocal lattice, shown as black dots, in cross-section. The boundaries of the surrounding primitive cells are shown as thin red lines.
10.17. Occupied states for one, two, and three free electrons per physical lattice cell.
10.18. Redefinition of the occupied wave number vectors into Brillouin zones.
10.19. Second, third, and fourth Brillouin zones seen in the periodic zone scheme.
10.20. The red dot shows the wavenumber vector of a sample free electron wave function. It is to be corrected for the lattice potential.
10.21. The grid of nonzero Hamiltonian perturbation coefficients and the problem sphere in wave number space.
10.22. Tearing apart of the wave number space energies.
10.23. Effect of a lattice potential on the energy. The energy is represented by the square distance from the origin, and is relative to the energy at the origin.
10.24. Bragg planes seen in wave number space cross section.
10.25. Occupied states for the energies of figure 10.23 if there are two valence electrons per lattice cell. Left: energy. Right: wave numbers.
10.26. Smaller lattice potential. From top to bottom shows one, two and three valence electrons per lattice cell. Left: energy. Right: wave numbers.
10.27. Depiction of an electromagnetic ray.
10.28. Law of reflection in elastic scattering from a plane.
10.29. Scattering from multiple “planes of atoms”.
10.30. Difference in travel distance when scattered from P rather than O.
11.1. Graphical depiction of an arbitrary system energy eigenfunction for 36 distinguishable particles.
11.2. Graphical depiction of an arbitrary system energy eigenfunction for 36 identical bosons.
11.3. Graphical depiction of an arbitrary system energy eigenfunction for 33 identical fermions.
11.4. Illustrative small model system having 4 distinguishable particles. The particular eigenfunction shown is arbitrary.
11.5. The number of system energy eigenfunctions for a simple model system with only three energy shelves. Positions of the squares indicate the numbers of particles on shelves 2 and 3; darkness of the squares indicates the relative number of eigenfunctions with those shelf numbers. Left: system with 4 distinguishable particles, middle: 16, right: 64.
11.6. Number of energy eigenfunctions on the oblique energy line in the previous figure. (The curves are mathematically interpolated to allow a continuously varying fraction of particles on shelf 2.) Left: 4 particles, middle: 64, right: 1,024.
11.7. Probabilities of shelf-number sets for the simple 64 particle model system if there is uncertainty in energy. More probable shelf-number distributions are shown darker. Left: identical bosons, middle: distinguishable particles, right: identical fermions. The temperature is the same as in the previous two figures.
11.8. Probabilities of shelf-number sets for the simple 64 particle model system if shelf 1 is a nondegenerate ground state. Left: identical bosons, middle: distinguishable particles, right: identical fermions. The temperature is the same as in the previous figures.
11.9. Like the previous figure, but at a lower temperature.
11.10. Like the previous figures, but at a still lower temperature.
11.11. Schematic of the Carnot refrigeration cycle.
11.12. Schematic of the Carnot heat engine.
11.13. A generic heat pump next to a reversed Carnot one with the same heat delivery.
11.14. Comparison of two different integration paths for finding the entropy of a desired state. The two different integration paths are in black and the yellow lines are reversible adiabatic process lines.
11.15. Specific heat at constant volume of gases. Temperatures from absolute zero to 1,200 K. Data from NIST-JANAF and AIP.
11.16. Specific heat at constant pressure of solids. Temperatures from absolute zero to 1,200 K. Carbon is diamond; graphite is similar. Water is ice and liquid. Data from NIST-JANAF, CRC, AIP, Rohsenow et al.
12.1. Example bosonic ladders.
12.2. Example fermionic ladders.
12.3. Triplet and singlet states in terms of ladders
12.4. Clebsch-Gordan coefficients of two spin one half particles.
12.5. Clebsch-Gordan coefficients when the second angular momentum contribution has azimuthal quantum number 1/2.
12.6. Clebsch-Gordan coefficients when the second angular momentum contribution has azimuthal quantum number 1.
13.1. Relationship of Maxwell’s first equation to Coulomb’s law.
13.2. Maxwell’s first equation for a more arbitrary region. The figure to the right includes the field lines through the selected points.
13.3. The net number of field lines leaving a region is a measure for the net charge inside that region.
13.4. Since magnetic monopoles do not exist, the net number of magnetic field lines leaving a region is always zero.
13.5. Electric power generation.
13.6. Two ways to generate a magnetic field: using a current (left) or using a varying electric field (right).
13.7. Electric field and potential of a charge that is distributed uniformly within a small sphere. The dotted lines indicate the values for a point charge.
13.8. Electric field of a two-dimensional line charge.
13.9. Field lines of a vertical electric dipole.
13.10. Electric field of a two-dimensional dipole.
13.11. Field of an ideal magnetic dipole.
13.12. Electric field of an almost ideal two-dimensional dipole.
13.13. Magnetic field lines around an infinite straight electric wire.
13.14. An electromagnet consisting of a single wire loop. The generated magnetic field lines are in blue.
13.15. A current dipole.
13.16. Electric motor using a single wire loop. The Lorentz forces (black vectors) exerted by the external magnetic field on the electric current carriers in the wire produce a net moment on the loop. The self-induced magnetic field of the wire and the corresponding radial forces are not shown.
13.17. Variables for the computation of the moment on a wire loop in a magnetic field.
13.18. Larmor precession of the expectation spin (or magnetic moment) vector around the magnetic field.
13.19. Probability of being able to find the nuclei at elevated energy versus time for a given perturbation frequency $\omega$ .
13.20. Maximum probability of finding the nuclei at elevated energy.
13.21. A perturbing magnetic field, rotating at precisely the Larmor frequency, causes the expectation spin vector to come cascading down out of the ground state.
14.1. Nuclear decay modes. [pdf]
14.2. Binding energy per nucleon. [pdf]
14.3. Proton separation energy. [pdf]
14.4. Neutron separation energy. [pdf]
14.5. Proton pair separation energy. [pdf]
14.6. Neutron pair separation energy. [pdf]
14.7. Error in the von Weizsäcker formula. [pdf]
14.8. Half-life versus energy release for the atomic nuclei marked in NUBASE 2003 as showing pure alpha decay with unqualified energies. Top: only the even values of the mass and atomic numbers cherry-picked. Inset: really cherry-picking, only a few even mass numbers for thorium and uranium! Bottom: all the nuclei except one. [pdf]
14.9. Schematic potential for an alpha particle that tunnels out of a nucleus.
14.10. Half-life predicted by the Gamow / Gurney & Condon theory versus the true value. Top: even-even nuclei only. Bottom: all the nuclei except one. [pdf]
14.11. Example average nuclear potentials: (a) harmonic oscillator, (b) impenetrable surface, (c) Woods-Saxon, (d) Woods-Saxon for protons.
14.12. Nuclear energy levels for various assumptions about the average nuclear potential. The signs indicate the parity of the states.
14.13. Schematic effect of spin-orbit interaction on the energy levels. The ordering within bands is realistic for neutrons. The designation behind the equals sign is the “official” one. (Assuming counting starts at 1).
14.14. Energy levels for doubly-magic oxygen-16 and neighbors. [pdf]
14.15. Nucleon pairing effect. [pdf]
14.16. Energy levels for neighbors of doubly-magic calcium-40. [pdf]
14.17. 2 $\POW9,{+}$ excitation energy of even-even nuclei. [pdf]
14.18. Collective motion effects. [pdf]
14.19. Failures of the shell model. [pdf]
14.20. An excitation energy ratio for even-even nuclei. [pdf]
14.21. Textbook vibrating nucleus tellurium-120. [pdf]
14.22. Rotational bands of hafnium-177. [pdf]
14.23. Ground state rotational band of tungsten-183. [pdf]
14.24. Rotational bands of aluminum-25. [pdf]
14.25. Rotational bands of erbium-164. [pdf]
14.26. Ground state rotational band of magnesium-24. [pdf]
14.27. Rotational bands of osmium-190. [pdf]
14.28. Simplified energetics for fission of fermium-256. [pdf]
14.29. Spin of even-even nuclei. [pdf]
14.30. Spin of even-odd nuclei. [pdf]
14.31. Spin of odd-even nuclei. [pdf]
14.32. Spin of odd-odd nuclei. [pdf]
14.33. Selected odd-odd spins predicted using the neighbors. [pdf]
14.34. Selected odd-odd spins predicted from theory. [pdf]
14.35. Parity of even-even nuclei. [pdf]
14.36. Parity of even-odd nuclei. [pdf]
14.37. Parity of odd-even nuclei. [pdf]
14.38. Parity of odd-odd nuclei. [pdf]
14.39. Parity versus the shell model. [pdf]
14.40. Magnetic dipole moments of the ground-state nuclei. [pdf]
14.41. 2 $\POW9,{+}$ magnetic moment of even-even nuclei. [pdf]
14.42. Electric quadrupole moment. [pdf]
14.43. Electric quadrupole moment corrected for spin. [pdf]
14.44. Isobaric analog states. [pdf]
14.45. Energy release in beta decay of even-odd nuclei. [pdf]
14.46. Energy release in beta decay of odd-even nuclei. [pdf]
14.47. Energy release in beta decay of odd-odd nuclei. [pdf]
14.48. Energy release in beta decay of even-even nuclei. [pdf]
14.49. Examples of beta decay. [pdf]
14.50. The Fermi integral. It shows the effects of energy release and nuclear charge on the beta decay rate of allowed transitions. Other effects exists. [pdf]
14.51. Beta decay rates. [pdf]
14.52. Beta decay rates as fraction of a ballparked value. [pdf]
14.53. Parity violation. In the beta decay of cobalt-60, left, the electron preferentially comes out in the direction that a left-handed screw rotating with the nuclear spin would move. Seen in the mirror, right, that becomes the direction of a right-handed screw.
14.54. Energy levels of tantalum-180. [pdf]
14.55. Half-life of the longest-lived even-odd isomers. [pdf]
14.56. Half-life of the longest-lived odd-even isomers. [pdf]
14.57. Half-life of the longest-lived odd-odd isomers. [pdf]
14.58. Half-life of the longest-lived even-even isomers. [pdf]
14.59. Weisskopf ballpark half-lifes for electromagnetic transitions versus energy release. Broken lines include ballparked internal conversion. [pdf]
14.60. Moszkowski ballpark half-lifes for magnetic transitions versus energy release. Broken lines include ballparked internal conversion. [pdf]
14.61. Comparison of electric gamma decay rates with theory. [pdf]
14.62. Comparison of magnetic gamma decay rates with theory. [pdf]
14.63. Comparisons of decay rates between the same initial and final states.
A.1. Analysis of conduction.
A.2. Graphical depiction of an arbitrary system energy eigenfunction for 36 distinguishable particles.
A.3. Graphical depiction of an arbitrary system energy eigenfunction for 36 identical bosons.
A.4. Graphical depiction of an arbitrary system energy eigenfunction for 33 identical fermions.
A.5. Example wave functions for a system with just one type of single particle state. Left: identical bosons; right: identical fermions.
A.6. Creation and annihilation operators for a system with just one type of single particle state. Left: identical bosons; right: identical fermions.
A.7. Effect of a rotation of the coordinate system on the spherical coordinates of a particle at an arbitrary location P.
A.8. Effect of rotation of the coordinate system on a vector. The vector is physically the same, but it has a different mathematical representation, different components, in the two coordinate systems.
A.9. Example energy eigenfunction for the particle in free space.
A.10. Example energy eigenfunction for a particle entering a constant accelerating force field.
A.11. Example energy eigenfunction for a particle entering a constant decelerating force field.
A.12. Example energy eigenfunction for the harmonic oscillator.
A.13. Example energy eigenfunction for a particle encountering a brief accelerating force.
A.14. Example energy eigenfunction for a particle tunneling through a barrier.
A.15. Example energy eigenfunction for tunneling through a delta function barrier.
A.16. Harmonic oscillator potential energy , eigenfunction $h_{50}$ , and its energy $E_{50}$ .
A.17. The Airy Ai and Bi functions that solve the Hamiltonian eigenvalue problem for a linearly varying potential energy. Bi very quickly becomes too large to plot for positive values of its argument.
A.18. Connection formulae for a turning point from classical to tunneling.
A.19. Connection formulae for a turning point from tunneling to classical.
A.20. WKB approximation of tunneling.
A.21. Scattering of a beam off a target.
A.22. Graphical interpretation of the Born series.
A.23. Possible polarizations of a pair of hydrogen atoms.
A.24. Crude deuteron model. The potential is in green. The relative probability of finding the nucleons at a given spacing is in black.
A.25. Crude deuteron model with a 0.5 fm repulsive core. Thin grey lines are the model without the repulsive core. Thin red lines are more or less comparable results from the Argonne $v_{18}$ potential.
A.26. Effects of uncertainty in orbital angular momentum.
A.27. Possible momentum states for a particle confined to a periodic box. The states are shown as points in momentum space. States that have momentum less than some example maximum value are in red.
D.1. Right: the absolute value of the wave function has a kink at a zero crossing. Middle: the kink has been slightly blunted. Right: an alternate way of blunting.
D.2. Bosons in single-particle-state boxes.
D.3. Schematic of an example boson distribution on a shelf.
D.4. Schematic of the Carnot refrigeration cycle.
N.1. Spectrum for a weak potential.
N.2. The 17 real wave functions of lowest energy for a small one-dimensional periodic box with only 12 atomic cells. Black curves show the square wave function, which gives the relative probability of finding the electron at that location.
N.3. Spherical coordinates of an arbitrary point P.