List of Figures

1.1. The classical picture of a vector.
1.2. Spike diagram of a vector.
1.3. More dimensions.
1.4. Infinite dimensions.
1.5. The classical picture of a function.
1.6. Forming the dot product of two vectors.
1.7. Forming the inner product of two functions.
2.1. A visualization of an arbitrary wave function.
2.2. Combined plot of position and momentum components.
2.3. The uncertainty principle illustrated.
2.4. Classical picture of a particle in a closed pipe.
2.5. Quantum mechanics picture of a particle in a closed pipe.
2.6. Definitions.
2.7. One-dimensional energy spectrum for a particle in a pipe.
2.8. One-dimensional ground state of a particle in a pipe.
2.9. Second and third lowest one-dimensional energy states.
2.10. Definition of all variables.
2.11. True ground state of a particle in a pipe.
2.12. True second and third lowest energy states.
2.13. A combination of $\psi_{111}$ and $\psi_{211}$ seen at some typical times.
2.14. The harmonic oscillator.
2.15. The energy spectrum of the harmonic oscillator.
2.16. Ground state $\psi_{000}$ of the harmonic oscillator
2.17. Wave functions $\psi_{100}$ and $\psi_{010}$ .
2.18. Energy eigenfunction $\psi_{213}$ .
2.19. Arbitrary wave function (not an energy eigenfunction).
3.1. Spherical coordinates of an arbitrary point P.
3.2. Spectrum of the hydrogen atom.
3.3. Ground state wave function $\psi_{100}$ of the hydrogen atom.
3.4. Eigenfunction $\psi_{200}$ .
3.5. Eigenfunction $\psi_{210}$ , or 2p.
3.6. Eigenfunction $\psi_{211}$ (and $\psi_{21-1}$ ).
3.7. Eigenfunctions 2p, left, and 2p, right.
3.8. Hydrogen atom plus free proton far apart.
3.9. Hydrogen atom plus free proton closer together.
3.10. The electron being anti-symmetrically shared.
3.11. The electron being symmetrically shared.
4.1. State with two neutral atoms.
4.2. Symmetric sharing of the electrons.
4.3. Antisymmetric sharing of the electrons.
4.4. Approximate solutions for hydrogen (left) and helium (right).
4.5. Approximate solutions for lithium (left) and beryllium (right).
4.6. Example approximate solution for boron.
4.7. Periodic table, from NIST.
4.8. Covalent sigma bond consisting of two 2p states.
4.9. Covalent pi bond consisting of two 2p states.
4.10. Covalent sigma bond consisting of a 2p and a 1s state.
4.11. Shape of an sp hybrid state.
4.12. Shapes of the sp (left) and sp (right) hybrids.
5.1. The ground state wave function looks the same at all times.
5.2. The first excited state at all times.
5.3. A combination of $\psi_{111}$ and $\psi_{211}$ seen at some typical times.
5.4. Emission and absorption of radiation by an atom.
5.5. Triangle inequality.
5.6. 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 $\protect{\underline x}$ .
5.7. The real part (red) and envelope (black) of an example wave.
5.8. The wave moves with the phase speed.
5.9. The real part (red) and magnitude or envelope (black) of a wave packet. (Schematic).
5.10. The velocities of wave and envelope are not equal.
5.11. A particle in free space.
5.12. An accelerating particle.
5.13. An decelerating particle.
5.14. Unsteady solution for the harmonic oscillator. The third picture shows the maximum distance from the nominal position that the wave packet reaches.
5.15. Harmonic oscillator potential energy , eigenfunction $h_{50}$ , and its energy $E_{50}$ .
5.16. A partial reflection.
5.17. An tunneling particle.
5.18. Penetration of an infinitely high potential energy barrier.
5.19. Schematic of a scattering potential and the asymptotic behavior of an example energy eigenfunction for a wave packet coming in from the far left.
7.1. Billiard-ball model of the salt molecule.
7.2. Billiard-ball model of a salt crystal.
7.3. The salt crystal disassembled to show its structure.
7.4. Sketch of electron energy spectra in solids.
7.5. The lithium atom, scaled more correctly than in chapter 4.9
7.6. Body-centered-cubic (bcc) structure of lithium.
7.7. Fully periodic wave function of a two-atom lithium “crystal.”
7.8. Flip-flop wave function of a two-atom lithium “crystal.”
7.9. Wave functions of a four-atom lithium “crystal.” The actual picture is that of the fully periodic mode.
7.10. Reciprocal lattice of a one-dimensional crystal.
7.11. Schematic of energy bands.
7.12. Energy versus linear momentum.
7.13. Schematic of merging bands.
7.14. A primitive cell and primitive translation vectors of lithium.
7.15. Wigner-Seitz cell of the bcc lattice.
7.16. Schematic of crossing bands.
7.17. Ball and stick schematic of the diamond crystal.
7.18. Allowed wave number vectors.
7.19. Schematic energy spectrum of the free electron gas.
7.20. Occupied wave number states and Fermi surface in the ground state
7.21. Density of states for the free electron gas.
7.22. Energy states, top, and density of states, bottom, when there is severe confinement in the -direction, as in a quantum well.
7.23. Energy states, top, and density of states, bottom, when there is severe confinement in both the - and -directions, as in a quantum wire.
7.24. Energy states, top, and density of states, bottom, when there is severe confinement in all three directions, as in a quantum dot or artificial atom.
7.25. Wave number vectors seen in a cross section of constant . Top: sinusoidal solutions. Bottom: exponential solutions.
7.26. 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.
7.27. Occupied states for one, two, and three free electrons per physical lattice cell.
7.28. Redefinition of the occupied wave number vectors into Brillouin zones.
7.29. Second, third, and fourth Brillouin zones seen in the periodic zone scheme.
7.30. The red dot shows the wavenumber vector of a sample free electron wave function. It is to be corrected for the lattice potential.
7.31. The grid of nonzero Hamiltonian perturbation coefficients and the problem sphere in wave number space.
7.32. Tearing apart of the wave number space energies.
7.33. 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.
7.34. Bragg planes seen in wave number space cross section.
7.35. Occupied states for the energies of figure 7.33 if there are two valence electrons per lattice cell. Left: energy. Right: wave numbers.
7.36. Smaller lattice potential. From top to bottom shows one, two and three valence electrons per lattice cell. Left: energy. Right: wave numbers.
7.37. Sketch of electron energy spectra in solids at absolute zero temperature.
7.38. Sketch of electron energy spectra in solids at a nonzero temperature.
7.39. Specific heat at constant volume of gases. Temperatures from absolute zero to 1200 K. Data from NIST-JANAF and AIP.
7.40. Specific heat at constant pressure of solids. Temperatures from absolute zero to 1200 K. Carbon is diamond; graphite is similar. Water is ice and liquid. Data from NIST-JANAF, CRC, AIP, Rohsenow et al.
7.41. Depiction of an electromagnetic ray.
7.42. Law of reflection in elastic scattering from a plane.
7.43. Scattering from multiple “planes of atoms”.
7.44. Difference in travel distance when scattered from P rather than O.
8.1. Graphical depiction of an arbitrary system energy eigenfunction for 95 distinguishable particles.
8.2. Graphical depiction of an arbitrary system energy eigenfunction for 95 identical bosons.
8.3. Graphical depiction of an arbitrary system energy eigenfunction for 31 identical fermions.
8.4. Illustrative small model system having 4 distinguishable particles. The particular eigenfunction shown is arbitrary.
8.5. The number of system energy eigenfunctions for a simple model system with only three energy buckets. Positions of the squares indicate the numbers of particles in buckets 2 and 3; darkness of the squares indicates the relative number of eigenfunctions with those bucket numbers. Left: system with 4 distinguishable particles, middle: 16, right: 64.
8.6. Number of energy eigenfunctions on the oblique energy line in 8.5. (The curves are mathematically interpolated to allow a continuously varying fraction of particles in bucket 2.) Left: 4 particles, middle: 64, right: 1024.
8.7. Probabilities for bucket-number sets for the simple 64 particle model system if there is uncertainty in energy. More probable bucket-number distributions are shown darker. Left: identical bosons, middle: distinguishable particles, right: identical fermions. The temperature is the same as in figure 8.5.
8.8. Probabilities of bucket-number sets for the simple 64 particle model system if bucket 1 is a non-degenerate ground state. Left: identical bosons, middle: distinguishable particles, right: identical fermions. The temperature is the same as in figure 8.7.
8.9. Like figure 8.8, but at a lower temperature.
8.10. Like figure 8.8, but at a still lower temperature.
8.11. Schematic of the Carnot refrigeration cycle.
8.12. Schematic of the Carnot heat engine.
8.13. A generic heat pump next to a reversed Carnot one with the same heat delivery.
8.14. 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.
9.1. Example bosonic ladders.
9.2. Example fermionic ladders.
9.3. Triplet and singlet states in terms of ladders
9.4. Clebsch-Gordan coefficients of two spin one half particles.
9.5. Clebsch-Gordan coefficients for equal to one half.
9.6. Clebsch-Gordan coefficients for equal to one.
9.3. Relationship of Maxwell’s first equation to Coulomb’s law.
9.4. Maxwell’s first equation for a more arbitrary region. The figure to the right includes the field lines through the selected points.
9.9. The net number of field lines leaving a region is a measure for the net charge inside that region.
9.10. Since magnetic monopoles do not exist, the net number of magnetic field lines leaving a region is always zero.
9.11. Electric power generation.
9.12. Two ways to generate a magnetic field: using a current (left) or using a varying electric field (right).
9.13. 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.
9.14. Electric field of a two-dimensional line charge.
9.15. Field lines of a vertical electric dipole.
9.16. Electric field of a two-dimensional dipole.
9.17. Field of an ideal magnetic dipole.
9.18. Electric field of an almost ideal two-dimensional dipole.
9.19. Magnetic field lines around an infinite straight electric wire.
9.20. An electromagnet consisting of a single wire loop. The generated magnetic field lines are in blue.
9.21. A current dipole.
9.22. 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.
9.23. Variables for the computation of the moment on a wire loop in a magnetic field.
9.24. Larmor precession of the expectation spin (or magnetic moment) vector around the magnetic field.
9.25. Probability of being able to find the nuclei at elevated energy versus time for a given perturbation frequency $\omega$ .
9.26. Maximum probability of finding the nuclei at elevated energy.
9.27. A perturbing magnetic field, rotating at precisely the Larmor frequency, causes the expectation spin vector to come cascading down out of the ground state.
10.1. Graphical depiction of an arbitrary system energy eigenfunction for 95 distinguishable particles.
10.2. Graphical depiction of an arbitrary system energy eigenfunction for 95 identical bosons.
10.3. Graphical depiction of an arbitrary system energy eigenfunction for 31 identical fermions.
10.4. Example wave functions for a system with just one type of single particle state. Left: identical bosons; right: identical fermions.
10.5. Annihilation and creation operators for a system with just one type of single particle state. Left: identical bosons; right: identical fermions.
11.1. Separating the hydrogen ion.
11.2. The Bohm experiment before the Venus measurement (left), and immediately after it (right).
11.3. Spin measurement directions.
11.4. Earth's view of events (left), and that of a moving observer (right).
11.5. Bohm's version of the Einstein, Podolski, Rosen Paradox
11.6. Non entangled positron and electron spins; up and down.
11.7. Non entangled positron and electron spins; down and up.
11.8. The wave functions of two universes combined
11.9. The Bohm experiment repeated.
11.10. Repeated experiments on the same electron.
A.1. Coordinate systems for the Lorentz transformation.
A.2. Example elastic collision seen by different observers.
A.3. A completely inelastic collision.
A.4. Bosons in single-particle-state boxes.
A.5. Example energy eigenfunction for the particle in free space.
A.6. Example energy eigenfunction for a particle entering a constant accelerating force field.
A.7. Example energy eigenfunction for a particle entering a constant decelerating force field.
A.8. Example energy eigenfunction for the harmonic oscillator.
A.9. Example energy eigenfunction for a particle encountering a brief accelerating force.
A.10. Example energy eigenfunction for a particle tunneling through a barrier.
A.11. Example energy eigenfunction for tunneling through a delta function barrier.
A.12. 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.13. Connection formulae for a turning point from classical to tunneling.
A.14. Connection formulae for a turning point from tunneling to classical.
A.15. WKB approximation of tunneling.
A.16. Scattering of a beam off a target.
A.17. Graphical interpretation of the Born series.
A.18. Possible polarizations of a pair of hydrogen atoms.
A.19. Schematic of an example boson distribution in a bucket.
A.20. Schematic of the Carnot refrigeration cycle.