Quantum Mechanics for Engineers
© Leon van Dommelen
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List of Figures
1.1.
Different views of the same experiment.
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.
3.1.
The old incorrect Newtonian physics.
3.2.
The correct quantum physics.
3.3.
Illustration of the Heisenberg uncertainty principle.
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
and
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
and
.
4.5.
Energy eigenfunction
.
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
.
4.11.
Eigenfunction
,
or 2
p
.
4.12.
Eigenfunction
(and
)
.
4.13.
Eigenfunctions 2
p
,
left, and 2
p
,
right.
4.14.
Hydrogen atom plus free proton far apart.
4.15.
Hydrogen atom plus free proton closer together.
4.16.
The electron being antisymmetrically 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 and helium.
5.5.
Abbreviated periodic table of the elements.
5.6.
Approximate solutions for lithium (left) and beryllium (right).
5.7.
Example approximate solution for boron.
5.8.
Periodic table of the elements.
5.9.
Covalent sigma bond consisting of two 2
p
states.
5.10.
Covalent pi bond consisting of two 2
p
states.
5.11.
Covalent sigma bond consisting of a 2
p
and a 1s state.
5.12.
Shape of an s
p
hybrid state.
5.13.
Shapes of the s
p
and sp 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 energy eigenfunction for a simple system.
6.6.
State with 5 times the single-particle ground state energy.
6.7.
Distinguishable particles: eigenfunctions for distribution A.
6.8.
Distinguishable particles: eigenfunctions for distribution B.
6.9.
Bosons: only 3 energy eigenfunctions for distribution A.
6.10.
Bosons: also only 3 energy eigenfunctions for distribution B.
6.11.
Ground state of noninteracting electrons (fermions) in a box.
6.12.
Severe confinement in the
-
direction.
6.13.
Severe confinement in both the
and
directions.
6.14.
Severe confinement in all three directions.
6.15.
A system of fermions at a nonzero temperature.
6.16.
Particles at high-enough temperature and volume.
6.17.
Ground state of noninteracting electrons in a periodic box.
6.18.
Conduction in the free-electron gas model.
6.19.
Sketch of electron spectra in solids at zero temperature.
6.20.
Sketch of electron spectra in solids at nonzero temperature.
6.21.
Potential energy seen by an electron along a line of nuclei.
6.22.
Potential energy in the one-dimensional Kronig & Penney model.
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.
One-dimensional energy bands for basic semiconductors.
6.27.
Spectrum against wave number in the periodic zone scheme.
6.28.
Schematic of the zinc blende (ZnS) crystal.
6.29.
First Brillouin zone of the FCC crystal.
6.30.
Sketch of a more complete spectrum of germanium.
6.31.
Vicinity of the band gap of intrinsic and doped semiconductors.
6.32.
Relationship between conduction electron density and hole density.
6.33.
The
p-n
junction in thermal equilibrium.
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 of an insulator.
6.37.
Peltier cooling.
6.38.
An example Seebeck voltage generator.
6.39.
The Galvani potential 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.
Concept sketch of the emission of a photon by an atom.
7.4.
Addition of angular momenta in classical physics.
7.5.
Longest and shortest possible final momenta in classical physics.
7.6.
A combination of two eigenfunctions 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.
Dirac delta function.
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 and magnitude or envelope of a wave packet.
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.
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 wave packet.
8.1.
Separating the hydrogen ion.
8.2.
Before the Venus measurement and immediately after it.
8.3.
Spin measurement directions.
8.4.
Earth's view of events and that of a moving observer.
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.''
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 in cross-section.
10.17.
Occupied states for one, two, and three electrons per lattice cell.
10.18.
Redefining the occupied wave numbers into Brillouin zones.
10.19.
Second, third, and fourth zones in the periodic zone scheme.
10.20.
The wavenumber vector of a sample free electron wave function.
10.21.
The grid of nonzero Hamiltonian perturbation coefficients.
10.22.
Tearing apart of the wave number space energies.
10.23.
Effect of a lattice potential on the energy.
10.24.
Bragg planes seen in wave number space cross section.
10.25.
Occupied states if there are two valence electrons per lattice cell.
10.26.
Smaller lattice potential.
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 distance when scattered from P rather than O.
11.1.
An arbitrary system eigenfunction for 36 distinguishable particles.
11.2.
An arbitrary system eigenfunction for 36 identical bosons.
11.3.
An arbitrary system eigenfunction for 33 identical fermions.
11.4.
Illustrative small model system having 4 distinguishable particles.
11.5.
The number of system eigenfunctions for a model system.
11.6.
Number of energy eigenfunctions on the oblique energy line.
11.7.
Probabilities if there is uncertainty in energy.
11.8.
Probabilities if shelf 1 is a nondegenerate ground state.
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.
11.14.
Comparison of integration paths for finding the entropy.
11.15.
Specific heat at constant volume of gases.
11.16.
Specific heat at constant pressure of solids.
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 for second momentum one-half.
12.6.
Clebsch-Gordan coefficients for second angular momentum one.
13.1.
Relationship of Maxwell's first equation to Coulomb's law.
13.2.
Maxwell's first equation for a more arbitrary region.
13.3.
The net number of outgoing field lines indicates net charge.
13.4.
The net number of outgoing magnetic field lines is zero.
13.5.
Electric power generation.
13.6.
Two ways to generate a magnetic field.
13.7.
Electric field and potential of a uniform spherical 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.
13.15.
A current dipole.
13.16.
Electric motor using a single wire loop.
13.17.
Computation of the moment on a wire loop in a magnetic field.
13.18.
Larmor precession of the expectation spin.
13.19.
Probability of being able to find the nuclei at elevated energy.
13.20.
Maximum probability of finding the nuclei at elevated energy.
13.21.
Effect of a magnetic field rotating at the Larmor frequency.
14.1.
Chart of the nuclides.
14.2.
Nuclear decay modes.
14.3.
Nuclear half-lifes.
14.4.
Binding energy per nucleon.
14.5.
Proton separation energy.
14.6.
Neutron separation energy.
14.7.
Proton pair separation energy.
14.8.
Neutron pair separation energy.
14.9.
Error in the von Weizsäcker formula.
14.10.
Half-life versus energy release in alpha decay.
14.11.
Schematic potential for a tunneling alpha particle.
14.12.
Half-life predicted by the Gamow / Gurney & Condon theory.
14.13.
Example average nuclear potentials.
14.14.
Nuclear energy levels for various average nuclear potentials.
14.15.
Schematic effect of spin-orbit interaction on the energy levels.
14.16.
Energy levels for doubly-magic oxygen-16 and neighbors.
14.17.
Nucleon pairing effect.
14.18.
Energy levels for neighbors of doubly-magic calcium-40.
14.19.
2
excitation energy of even-even nuclei.
14.20.
Collective motion effects.
14.21.
Failures of the shell model.
14.22.
An excitation energy ratio for even-even nuclei.
14.23.
Textbook vibrating nucleus tellurium-120.
14.24.
Rotational bands of hafnium-177.
14.25.
Ground state rotational band of tungsten-183.
14.26.
Rotational bands of aluminum-25.
14.27.
Rotational bands of erbium-164.
14.28.
Ground state rotational band of magnesium-24.
14.29.
Rotational bands of osmium-190.
14.30.
Simplified energetics for fission of fermium-256.
14.31.
Spin of even-even nuclei.
14.32.
Spin of even-odd nuclei.
14.33.
Spin of odd-even nuclei.
14.34.
Spin of odd-odd nuclei.
14.35.
Odd-odd spins predicted using the neighbors.
14.36.
Odd-odd spins predicted from theory.
14.37.
Parity of even-even nuclei.
14.38.
Parity of even-odd nuclei.
14.39.
Parity of odd-even nuclei.
14.40.
Parity of odd-odd nuclei.
14.41.
Parity versus the shell model.
14.42.
Magnetic dipole moments of the ground-state nuclei.
14.43.
2
magnetic moment of even-even nuclei.
14.44.
Electric quadrupole moment.
14.45.
Electric quadrupole moment corrected for spin.
14.46.
Isobaric analog states.
14.47.
Energy release in beta decay of even-odd nuclei.
14.48.
Energy release in beta decay of odd-even nuclei.
14.49.
Energy release in beta decay of odd-odd nuclei.
14.50.
Energy release in beta decay of even-even nuclei.
14.51.
Examples of beta decay.
14.52.
The Fermi integral.
14.53.
Beta decay rates.
14.54.
Beta decay rates as fraction of a ballparked value.
14.55.
Parity violation.
14.56.
Energy levels of tantalum-180.
14.57.
Half-life of the longest-lived even-odd isomers.
14.58.
Half-life of the longest-lived odd-even isomers.
14.59.
Half-life of the longest-lived odd-odd isomers.
14.60.
Half-life of the longest-lived even-even isomers.
14.61.
Weisskopf ballpark half-lifes for electromagnetic transitions.
14.62.
Moszkowski ballpark half-lifes for magnetic transitions.
14.63.
Comparison of electric gamma decay rates with theory.
14.64.
Comparison of magnetic gamma decay rates with theory.
14.65.
Decay rates between the same initial and final states.
A.1.
Analysis of conduction.
A.2.
A system eigenfunction for 36 distinguishable particles.
A.3.
A system energy eigenfunction for 36 identical bosons.
A.4.
A system energy eigenfunction for 33 identical fermions.
A.5.
Wave functions with just one type of single particle state.
A.6.
Creation and annihilation operators for one single particle state.
A.7.
Effect of coordinate system rotation on spherical coordinates
A.8.
Effect of coordinate system rotation on a vector.
A.9.
Example energy eigenfunction for the particle in free space.
A.10.
Example energy eigenfunction for an accelerating force.
A.11.
Example energy eigenfunction for a decelerating force.
A.12.
Example energy eigenfunction for the harmonic oscillator.
A.13.
Example energy eigenfunction for a brief accelerating force.
A.14.
Example energy eigenfunction for tunneling through a barrier.
A.15.
Tunneling through a delta function barrier.
A.16.
Harmonic oscillator potential energy and example eigenfunction.
A.17.
The Airy Ai and Bi functions.
A.18.
Connection formulae for going from normal to tunneling.
A.19.
Connection formulae for going from tunneling to normal.
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.
A.25.
Crude deuteron model with a 0.5 fm repulsive core.
A.26.
Effects of uncertainty in orbital angular momentum.
A.27.
Possible momentum states for a particle in a periodic box.
D.1.
Blunting of the absolute potential.
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.
N.3.
Spherical coordinates of an arbitrary point P.
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