Fundamental Quantum Mechanics for EngineersLeon van Dommele

Fundamental Quantum Mechanics for Engineers

Leon van Dommelen
12/20/07 Version 3 beta 3.4



Copyright

Copyright 2004 and on, Leon van Dommelen. You are allowed to copy or print out this work in
unmodified work for your personal use. You are allowed to attach additional notes, corrections,
and additions, as long as they are clearly identified as not being part of the original document
nor written by its author.
Distribution of this document for pay or for any other economic gain to a general audience
without permission is strictly prohibited. As an exception, its unmodified web pages may be
linked to freely, and may be displayed within your own frames, even for gain. Conversions to
html of the pdf version of this document are stupid, since there is a much better native html
version already available, so try not to do it.

iii



Dedication

To my parents

v

Preface
Why Another Book on Quantum Mechanics?

With the current emphasis on nanotechnology, quantum mechanics is becoming increasingly
essential to engineering students. Yet, the typical quantum mechanics texts for physics students are not written in a style that most engineering students would likely feel comfortable
with. Furthermore, an engineering education provides very little real exposure to modern
physics, and introductory quantum mechanics books do little to fill in the gaps. The emphasis tends to be on the computation of specific examples, rather than on discussion of the broad
picture. Undergraduate physics students may have the luxury of years of further courses to
pick up a wide physics background, engineering graduate students not really. In addition, the
coverage of typical introductory quantum mechanics books does not emphasize understanding
of the larger-scale quantum system that a density functional computation, say, would be used
for.
Hence this book, written by an engineer for engineers. As an engineering professor with an
engineering background, this is the book I wish I would have had when I started learning real
quantum mechanics a few years ago. The reason I like this book is not because I wrote it; the
reason I wrote this book is because I like it.
This book is not a popular exposition: quantum mechanics can only be described properly in
the terms of mathematics; suggesting anything else is crazy. But the assumed background in
this book is just basic undergraduate calculus and physics as taken by all engineering undergraduates. There is no intention to teach students proficiency in the clever manipulation of
the mathematical machinery of quantum mechanics. For those engineering graduate students
who may have forgotten some of their undergraduate calculus by now, there are some quick
and dirty reminders in the notations. For those students who may have forgotten some of
the details of their undergraduate physics, frankly, I am not sure whether it makes much of a
difference. The ideas of quantum mechanics are that different from conventional physics. But
the general ideas of classical physics are assumed to be known. I see no reason why a bright
undergraduate student, having finished calculus and physics, should not be able to understand
this book. A certain maturity might help, though. There are a lot of ideas to absorb.
vii

My initial goal was to write something that would “read like a mystery novel.” Something
a reader would not be able to put down until she had finished it. Obviously, this goal was
unrealistic. I am far from a professional writer, and this is quantum mechanics, after all, not
a murder mystery. But I have been told that this book is very well written, so maybe there
is something to be said for aiming high.
To prevent the reader from getting bogged down in mathematical details, I mostly avoid
nontrivial derivations in the text. Instead I have put the outlines of these derivations in notes
at the end of this document: personally, I enjoy checking the correctness of the mathematical
exposition, and I would not want to rob my students of the opportunity to do so too.
While typical physics texts jump back and forward from issue to issue, I thought that would
just be distracting for my audience. Instead, I try to follow a consistent approach, with as
central theme the method of separation-of-variables, a method that most mechanical graduate
students have seen before already. It is explained in detail anyway. To cut down on the issues
to be mentally absorbed at any given time, I purposely avoid bringing up new issues until I
really need them. Such a just-in-time learning approach also immediately answers the question
why the new issue is relevant, and how it fits into the grand scheme of things.
The desire to keep it straightforward is the main reason that topics such as Clebsch-Gordan
coefficients (except for the unavoidable introduction of singlet and triplet states) and Pauli
spin matrices have been shoved out of the way to a final chapter. My feeling is, if I can give
my students a solid understanding of the basics of quantum mechanics, they should be in a
good position to learn more about individual issues by themselves when they need them. On
the other hand, if they feel completely lost in all the different details, they are not likely to
learn the basics either.
I also try to go slow on the more abstract vector notation permeating quantum mechanics,
usually phrasing such issues in terms of a specific basis. Abstract notation may seem to be
completely general and beautiful to a mathematician, but I do not think it is going to be
intuitive to a typical engineer.
Knowledgeable readers may also note that I try to stay clear of abstract mathematical models
if I can. For example, the discussion of solids avoids the usual Kronig-Penney or Dirac combs
sort of models in favor of a physical discussion of realistic one-dimensional crystals.

I try to be as consistent as possible. Electrons are grey tones at the initial introduction of
particles, and so they stay through the rest of the book. Nuclei are red dots. Occupied
quantum states are red, empty ones grey. That of course required all figures to be custom
made. They are not intended to be fancy but consistent and clear.
When I derive the first quantum eigenfunctions, for a pipe and for the harmonic oscillator, I
make sure to emphasize that they are not supposed to look like anything that we told them
before. It is only natural for students to want to relate what we told them before about the
motion to the completely different story we are telling them now. So it should be clarified
viii


that (1) no, they are not going crazy, and (2) yes, we will eventually explain how what they
learned before fits into the grand scheme of things.
Another difference of approach in this book is the way it treats classical physics concepts
that the students are likely unaware about, such as canonical momentum, magnetic dipole
moments, Larmor precession, and Maxwell’s equations. They are largely “derived“ in quantum
terms, with no appeal to classical physics. I see no need to rub in the student’s lack of
knowledge of specialized areas of classical physics if a satisfactory quantum derivation is
readily given.
This book is not intended to be an exercise in mathematical skills. Review questions are
targeted towards understanding the ideas, with the mathematics as simple as possible. I also
try to keep the mathematics in successive questions uniform, to reduce the algebraic effort
required. I know.
Finally, this document faces the very real conceptual problems of quantum mechanics headon, including the collapse of the wave function, the indeterminacy, the nonlocality, and the
symmetrization requirements. The usual approach, and the way I was taught quantum mechanics, is to shove all these problems under the table in favor of a good sounding, but upon
examination self-contradictory and superficial story. Such superficiality put me off solidly
when they taught me quantum mechanics, culminating in the unforgettable moment when
the professor told us, seriously, that the wave function had to be symmetric with respect to
exchange of bosons because they are all truly the same, and then, when I was popping my
eyes back in, continued to tell us that the wave function is not symmetric when fermions are

exchanged, which are all truly the same. I would not do the same to my own students. And I
really do not see this professor as an exception. Other introductions to the ideas of quantum
mechanics that I have seen left me similarly unhappy on this point. One thing that really
bugs me, none had a solid discussion of the many worlds interpretation. This is obviously not
because the results would be incorrect, (they have not been contradicted for half a century,)
but simply because the teachers just do not like these results. I do not like the results myself,
but basing teaching on what the teacher would like to be true rather on what the evidence
indicates is true remains absolutely unacceptable in my book.
And I do hope this book will manage to convince you that quantum mechanics is a very fascinating subject, whatever you think of it. Our ancestors have ferreted out impressive details
about how nature works, mainly out of plain curiosity. Its benefits gave us the technological
world we live in as well as the leisure time to appreciate what they did. Going the next step
is what being an engineer is all about.

ix


Acknowledgments

This book is for a large part based on my reading of the excellent book by Griffiths, [3].
It includes a concise summary of the material of Griffiths’ chapters 1-5 (about 250 pages),
written by an engineer who was learning the material himself at the time.
Somewhat to my surprise, I find that my coverage actually tends to be closer to Yariv’s book,
[10]. I still think Griffiths is more readable for an engineer, though Yariv has some items
Griffiths does not.
The discussions on two-state systems are mainly based on Feynman’s notes, [2, chapters 811]. Since it is hard to determine the precise statements being made, much of that has been
augmented by data from web sources, mainly those referenced.
The nanomaterials lectures of colleague Anter El-Azab that I audited inspired me to add a
bit on simple quantum confinement to the first system studied, the particle in the box. That
does add a bit to a section that I wanted to keep as simple as possible, but then I figure it also
adds a sense that this is really relevant stuff for future engineers. I also added a discussion of

the effects of confinement on the density of states to the section on the free electron gas.
I thank Swapnil Jain for pointing out that the initial subsection on quantum confinement in
the pipe was definitely unclear and is hopefully better now.
I thank Johann Joss for pointing out a mistake in the formula for the averaged energy of
two-state systems.
The section on solids is mainly be based on Sproull, [8], a good source for practical knowledge
about application of the concepts. It is surprisingly up to date, considering it was written half
a century ago. Various items, however, come from Kittel [4]. The discussion of ionic solids
really comes straight from hyperphysics [3]. I prefer hyperphysics’ example of NaCl, instead
of Sproull’s equivalent discussion of KCl.
The section on the Born-Oppenheimer approximation comes from Wikipedia, [8}, with modifications including the inclusion of spin.
The section on the Hartree-Fock method is mainly based on Szabo and Ostlund [9], a wellwritten book, with some Parr and Yang [6] thrown in.
The many-worlds discussion is based on Everett’s exposition, [1]. It is brilliant but quite
impenetrable.
x


Comments and Feedback
If you find an error, please let me know. The same if you find points that are unclear to
the intended readership, ME graduate students with a typical exposure to mathematics and
physics, or equivalent. General editorial comments are also welcome. I’ll skip the philosophical
discussions. I am an engineer.
Feedback can be e-mailed to me at
This is a living document. I am still adding some things here and there, and fixing various
mistakes and doubtful phrasing. Even before every comma is perfect, I think the document
can be of value to people looking for an easy to read introduction to quantum mechanics at
a calculus level. So I am treating it as software, with version numbers indicating the level of
confidence I have in it all.

History

• The first version of this manuscript was posted Oct 24, 2004.
• A revised version was posted Nov 27, 2004, fixing a major blunder related to a nasty
problem in using classical spring potentials for more than a single particle. The fix required extensive changes. This version also added descriptions of how the wave function
of larger systems is formed.
• A revised version was posted on May 4, 2005. I finally read the paper by Everett, III on
the many worlds interpretation, and realized that I had to take the crap out of pretty
much all my discussions. I also rewrote everything to try to make it easier to follow. I
added the motion of wave packets to the discussion and expanded the one on Newtonian
motion.
• May 11 2005. I got cold feet on immediately jumping into separation of variables, so I
added a section on a particle in a pipe.
• Mid Feb, 2006. A new version was posted. Main differences are correction of a number
of errors and improved descriptions of the free electron and band spectra. There is also
a rewrite of the many worlds interpretation to be clearer and less preachy.
• Mid April, 2006. Various minor fixes. Also I changed the format from the “article” to
the “book” style.
• Mid Jan, 2007. Added sections on confinement and density of states, a commutator
reference, a section on unsteady perturbed two state systems, and an advanced chapter
on angular momentum, the Dirac equation, the electromagnetic field, and NMR. Fixed
a dubious phrasing about the Dirac equation and other minor changes.
xi


• Mid Feb 2007. There are now lists of key points and review questions for chapter 1.
Answers are in the new solution manual.
• 4/2 2007. There are now lists of key points and review questions for chapter 2. That
makes it the 3 beta 2 version. So I guess the final beta version will be 3 beta 6. Various
other fixes. I also added, probably unwisely, a note about zero point energy.
• 5/5 2007. There are now lists of key points and review questions for chapter 3. That
makes it the 3 beta 3 version. Various other fixes, like spectral line broadening, Helium’s

refusal to take on electrons, and countless other less than ideal phrasings. And full
solutions of the harmonic oscillator, spherical harmonics, and hydrogen wave function
ODEs, Mandelshtam-Tamm energy-time uncertainty, (all in the notes.) A dice is now a
die, though it sounds horrible to me. Zero point energy went out again as too speculative.
• 5/21 2007. An updated version 3 beta 3.1 to correct a poorly written subsection on
quantum confinement for the particle in a pipe. Thanks to Swapnil Jain for pointing
out the problem. I do not want people to get lost so early in the game, so I made it a
priority correction. In general, I do think that the sections added later to the document
are not of the same quality as the original with regard to writing style. The reason is
simple. When I wrote the original, I was on a sabbatical and had plenty of time to
think and rethink how it would be clearest. The later sections are written during the
few spare hours I can dig up. I write them and put them in. I would need a year off to
do this as it really should be done.
• 7/19 2007. Version 3 beta 3.2 adds a section on Hartree-Fock. It took forever. My
main regret is that most of them who wasted my time in this major way are probably
no longer around to be properly blasted. Writing a book on quantum mechanics by an
engineer for engineers is a minefield of having to see through countless poor definitions
and dubious explanations. It takes forever. In view of the fact that many of those
physicist were probably supported by tax payers much of the time, it should not be such
an absolute mess!
There are some additions on Born-Oppenheimer and the variational formulation that
were in the Hartree-Fock section, but that I took out, since they seemed to be too general
to be shoved away inside an application. Also rewrote section 4.9 and subsection 4.11.2
to be consistent, and in particular in order to have a single consistent notation. Zero
point energy (the vacuum kind) is back. What the heck.
• 9/9 2007. Version 3 beta 3.3 mainly adds sections on solids, that have been combined
with rewritten free and nearly free electron gas sections into a full chapter on solids. The
rest of the old chapter on examples of multiple particle systems has been pushed back
into the basic multiple particle systems chapter. A completely nonsensical discussion in
a paragraph of the free electron gas section was corrected; I cannot believe I have read

over that several times. I probably was reading what I wanted to say instead of what I
said. The alternative name “twilight terms” has been substituted for “exchange terms.”
Many minor changes.
xii


• 12/20 2007. Version 3 beta 3.4 cleans up the format of the “notes.” No more need for
loading an interminable web of 64 notes all at the same time over your phone line to read
20 words. It also corrects a few errors, one pointed out by Johann Joss, thanks. (It is
good to know some people are actually looking at any of this.) It also also extends some
further griping about correlation energy to all three web locations. You may surmise
from the lack of progress that I have been installing linux on my home PC. You are
right.

Wish List
Donald Knuth, in his versions of TeX, was approaching π. I seem to be approaching π + ε
with ε < 0.26. Geez, I hope not!
I would like to add key points and review questions to all basic sections. I am inching up to
it. Very slowly. If I ever get chapter 4 done, it will be version 3.4, no kidding. Up to version
3.6. (3.7 will be left without for now, as being all advanced, elective material.)
After that, the idea is to run all this text through a style checker to eliminate the dead wood.
Also, ispell seems to be missing misspelled words. Probably thinks they are TeX.
It would be nice to put frames around all key formulae. Many are already there.
There is supposed to be a second volume or additional chapter on computational methods,
in particular density-functional theory. Actually, at the time of this writing, I would already
be in heaven if I managed to get around to writing a small section on DFT. In short, don’t
hold your breath. But something may be there eventually. How old are you, and how is your
health?

xiii

Contents
Dedication

v

Preface
Why another book on quantum mechanics?
Acknowledgments . . . . . . . . . . . . . . .
Comments and Feedback . . . . . . . . . . .
History . . . . . . . . . . . . . . . . . . . . .
Wish list . . . . . . . . . . . . . . . . . . . .

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vii
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xi
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List of Figures

xxiii

List of Tables

xxvii

1 Mathematical Prerequisites
1.1 Complex Numbers . . . . . . . . . . . .
1.2 Functions as Vectors . . . . . . . . . . .
1.3 The Dot, oops, INNER Product . . . . .
1.4 Operators . . . . . . . . . . . . . . . . .
1.5 Eigenvalue Problems . . . . . . . . . . .
1.6 Hermitian Operators . . . . . . . . . . .
1.7 Additional Points . . . . . . . . . . . . .
1.7.1 Dirac notation . . . . . . . . . . .
1.7.2 Additional independent variables
2 Basic Ideas of Quantum Mechanics
2.1 The Revised Picture of Nature . . . . .
2.2 The Heisenberg Uncertainty Principle .
2.3 The Operators of Quantum Mechanics
2.4 The Orthodox Statistical Interpretation
2.4.1 Only eigenvalues . . . . . . . .
2.4.2 Statistical selection . . . . . . .
2.5 Schr¨odinger’s Cat [Background] . . . .

2.6 A Particle Confined Inside a Pipe . . .
2.6.1 The physical system . . . . . .
2.6.2 Mathematical notations . . . .
xv

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4
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15
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2.7

2.6.3 The Hamiltonian . . . . . . . . . . . .
2.6.4 The Hamiltonian eigenvalue problem .
2.6.5 All solutions of the eigenvalue problem
2.6.6 Discussion of the energy values . . . .
2.6.7 Discussion of the eigenfunctions . . . .
2.6.8 Three-dimensional solution . . . . . . .
2.6.9 Quantum confinement . . . . . . . . .
The Harmonic Oscillator . . . . . . . . . . . .
2.7.1 The Hamiltonian . . . . . . . . . . . .
2.7.2 Solution using separation of variables .
2.7.3 Discussion of the eigenvalues . . . . . .
2.7.4 Discussion of the eigenfunctions . . . .
2.7.5 Degeneracy . . . . . . . . . . . . . . .
2.7.6 Non-eigenstates . . . . . . . . . . . . .

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3 Single-Particle Systems
3.1 Angular Momentum . . . . . . . . . . . . . . . . . . .
3.1.1 Definition of angular momentum . . . . . . . .
3.1.2 Angular momentum in an arbitrary direction . .
3.1.3 Square angular momentum . . . . . . . . . . . .
3.1.4 Angular momentum uncertainty . . . . . . . . .
3.2 The Hydrogen Atom . . . . . . . . . . . . . . . . . . .
3.2.1 The Hamiltonian . . . . . . . . . . . . . . . . .
3.2.2 Solution using separation of variables . . . . . .
3.2.3 Discussion of the eigenvalues . . . . . . . . . . .
3.2.4 Discussion of the eigenfunctions . . . . . . . . .
3.3 Expectation Value and Standard Deviation . . . . . . .
3.3.1 Statistics of a die . . . . . . . . . . . . . . . . .
3.3.2 Statistics of quantum operators . . . . . . . . .
3.3.3 Simplified expressions . . . . . . . . . . . . . . .
3.3.4 Some examples . . . . . . . . . . . . . . . . . .
3.4 The Commutator . . . . . . . . . . . . . . . . . . . . .
3.4.1 Commuting operators . . . . . . . . . . . . . .
3.4.2 Noncommuting operators and their commutator
3.4.3 The Heisenberg uncertainty relationship . . . .
3.4.4 Commutator reference [Reference] . . . . . . . .
3.5 The Hydrogen Molecular Ion . . . . . . . . . . . . . . .
3.5.1 The Hamiltonian . . . . . . . . . . . . . . . . .
3.5.2 Energy when fully dissociated . . . . . . . . . .
3.5.3 Energy when closer together . . . . . . . . . . .
3.5.4 States that share the electron . . . . . . . . . .
3.5.5 Comparative energies of the states . . . . . . .

3.5.6 Variational approximation of the ground state .
3.5.7 Comparison with the exact ground state . . . .
xvi

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4 Multiple-Particle Systems
4.1 Wave Function for Multiple Particles . . . . . . . . . . . .
4.2 The Hydrogen Molecule . . . . . . . . . . . . . . . . . . .
4.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . .
4.2.2 Initial approximation to the lowest energy state . .
4.2.3 The probability density . . . . . . . . . . . . . . . .
4.2.4 States that share the electrons . . . . . . . . . . . .
4.2.5 Variational approximation of the ground state . . .

4.2.6 Comparison with the exact ground state . . . . . .
4.3 Two-State Systems . . . . . . . . . . . . . . . . . . . . . .
4.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Instantaneous Interactions [Background] . . . . . . . . . .
4.6 Multiple-Particle Systems Including Spin . . . . . . . . . .
4.6.1 Wave function for a single particle with spin . . . .
4.6.2 Inner products including spin . . . . . . . . . . . .
4.6.3 Wave function for multiple particles with spin . . .
4.6.4 Example: the hydrogen molecule . . . . . . . . . .
4.6.5 Triplet and singlet states . . . . . . . . . . . . . . .
4.7 Identical Particles . . . . . . . . . . . . . . . . . . . . . . .
4.8 Global Symmetrization [Background] . . . . . . . . . . . .
4.9 Ways to Symmetrize the Wave Function . . . . . . . . . .
4.10 Matrix Formulation [Advanced] . . . . . . . . . . . . . . .
4.11 Atoms Heavier Than Hydrogen . . . . . . . . . . . . . . .
4.11.1 The Hamiltonian eigenvalue problem . . . . . . . .
4.11.2 Approximate solution using separation of variables
4.11.3 Hydrogen and helium . . . . . . . . . . . . . . . . .
4.11.4 Lithium to neon . . . . . . . . . . . . . . . . . . . .
4.11.5 Sodium to argon . . . . . . . . . . . . . . . . . . .
4.11.6 Kalium to krypton . . . . . . . . . . . . . . . . . .
4.12 Exclusion-Principle Repulsion . . . . . . . . . . . . . . . .
4.13 Chemical Bonds . . . . . . . . . . . . . . . . . . . . . . . .
4.13.1 Covalent sigma bonds . . . . . . . . . . . . . . . .
4.13.2 Covalent pi bonds . . . . . . . . . . . . . . . . . . .
4.13.3 Polar covalent bonds and hydrogen bonds . . . . .
4.13.4 Promotion and hybridization . . . . . . . . . . . . .
4.13.5 Ionic bonds . . . . . . . . . . . . . . . . . . . . . .
4.13.6 Limitations of valence bond theory . . . . . . . . .
5 Solids

5.1 Molecular solids . . . . . . . . .
5.2 Ionic solids . . . . . . . . . . . .
5.3 Introduction to band theory . .
5.4 Metals . . . . . . . . . . . . . .
5.4.1 Lithium . . . . . . . . .
5.4.2 One-dimensional crystals

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5.5
5.6

5.7

5.8
5.9

5.4.3 Wave functions of one-dimensional crystals . . . .
5.4.4 Analysis of the wave functions . . . . . . . . . . .
5.4.5 Floquet (Bloch) theory . . . . . . . . . . . . . . .
5.4.6 Fourier analysis . . . . . . . . . . . . . . . . . . .
5.4.7 The reciprocal lattice . . . . . . . . . . . . . . . .
5.4.8 The energy levels . . . . . . . . . . . . . . . . . .
5.4.9 Electrical conduction . . . . . . . . . . . . . . . .
5.4.10 Merging and splitting bands . . . . . . . . . . . .
5.4.11 Three-dimensional metals . . . . . . . . . . . . .
Covalent Materials . . . . . . . . . . . . . . . . . . . . .
Confined Free Electrons . . . . . . . . . . . . . . . . . .
5.6.1 The Hamiltonian eigenvalue problem . . . . . . .
5.6.2 Solution by separation of variables . . . . . . . .
5.6.3 Discussion of the solution . . . . . . . . . . . . .
5.6.4 A numerical example . . . . . . . . . . . . . . . .
5.6.5 The density of states and confinement [Advanced]
5.6.6 Relation to Bloch functions . . . . . . . . . . . .

Nearly-Free Electrons [Advanced] . . . . . . . . . . . . .
5.7.1 The lattice structure . . . . . . . . . . . . . . . .
5.7.2 The small perturbation approach . . . . . . . . .
5.7.3 Zeroth order solution . . . . . . . . . . . . . . . .
5.7.4 First order solution . . . . . . . . . . . . . . . . .
5.7.5 Second order solution . . . . . . . . . . . . . . . .
5.7.6 Discussion of the energy changes . . . . . . . . .
Quantum Statistical Mechanics . . . . . . . . . . . . . .
Additional Points [Advanced] . . . . . . . . . . . . . . .
5.9.1 Thermal properties . . . . . . . . . . . . . . . . .
5.9.2 Ferromagnetism . . . . . . . . . . . . . . . . . . .
5.9.3 X-ray diffraction . . . . . . . . . . . . . . . . . .

6 Time Evolution
6.1 The Schr¨odinger Equation . . . . . . . . . . . . . . . .
6.1.1 Energy conservation . . . . . . . . . . . . . . .
6.1.2 Stationary states . . . . . . . . . . . . . . . . .
6.1.3 Time variations of symmetric two-state systems
6.1.4 Time variation of expectation values . . . . . .
6.1.5 Newtonian motion . . . . . . . . . . . . . . . .
6.2 Unsteady perturbations of two-state systems . . . . . .
6.2.1 Schr¨odinger equation for a two-state system . .
6.2.2 Stimulated and spontaneous emission . . . . . .
6.2.3 Absorption of radiation . . . . . . . . . . . . . .
6.3 Conservation Laws and Symmetries [Background] . . .
6.4 The Position and Linear Momentum Eigenfunctions . .
6.4.1 The position eigenfunction . . . . . . . . . . . .
6.4.2 The linear momentum eigenfunction . . . . . .
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6.5

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241
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7 Some Additional Topics
7.1 All About Angular Momentum [Advanced] . . . . . . . .
7.1.1 The fundamental commutation relations . . . . .
7.1.2 Ladders . . . . . . . . . . . . . . . . . . . . . . .
7.1.3 Possible values of angular momentum . . . . . . .
7.1.4 A warning about angular momentum . . . . . . .
7.1.5 Triplet and singlet states . . . . . . . . . . . . . .

7.1.6 Clebsch-Gordan coefficients . . . . . . . . . . . .
7.1.7 Pauli spin matrices . . . . . . . . . . . . . . . . .
7.2 The Relativistic Dirac Equation [Advanced] . . . . . . .
7.2.1 The Dirac idea . . . . . . . . . . . . . . . . . . .
7.2.2 Emergence of spin from relativity . . . . . . . . .
7.3 The Electromagnetic Field [Advanced] . . . . . . . . . .
7.3.1 The Hamiltonian . . . . . . . . . . . . . . . . . .
7.3.2 Maxwell’s equations . . . . . . . . . . . . . . . .
7.3.3 Electrons in magnetic fields . . . . . . . . . . . .
7.4 Nuclear Magnetic Resonance [Advanced] . . . . . . . . .
7.4.1 Description of the method . . . . . . . . . . . . .
7.4.2 The Hamiltonian . . . . . . . . . . . . . . . . . .
7.4.3 The unperturbed system . . . . . . . . . . . . . .
7.4.4 Effect of the perturbation . . . . . . . . . . . . .
7.5 The Variational Method [Advanced] . . . . . . . . . . . .
7.5.1 Basic variational statement . . . . . . . . . . . .
7.5.2 Differential form of the statement . . . . . . . . .
7.5.3 Example application using Lagrangian multipliers
7.6 The Born-Oppenheimer Approximation [Advanced] . . .
7.6.1 The Hamiltonian . . . . . . . . . . . . . . . . . .
7.6.2 The basic Born-Oppenheimer approximation . . .
7.6.3 Going one better . . . . . . . . . . . . . . . . . .
7.7 The Hartree-Fock Approximation [Advanced] . . . . . . .

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255
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256
257
260
262

263
265
269
271
271
274
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285
287
287
288
290
292
294
294
295
296
298
298
300
302
305

6.6

6.7

Wave Packets in Free Space . . . . . . . . .

6.5.1 Solution of the Schr¨odinger equation.
6.5.2 Component wave solutions . . . . . .
6.5.3 Wave packets . . . . . . . . . . . . .
6.5.4 The group velocity . . . . . . . . . .
Motion near the Classical Limit . . . . . . .
6.6.1 General procedures . . . . . . . . . .
6.6.2 Motion through free space . . . . . .
6.6.3 Accelerated motion . . . . . . . . . .
6.6.4 Decelerated motion . . . . . . . . . .
6.6.5 The harmonic oscillator . . . . . . .
Scattering . . . . . . . . . . . . . . . . . . .
6.7.1 Partial reflection . . . . . . . . . . .
6.7.2 Tunneling . . . . . . . . . . . . . . .

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305
310
311
314
316
322
325
325
327

A Notes
A.1 Notes on the Mathematical Prerequisites . . . . . . . . . . . . . . . .
A.1.1 Derivation of the Euler identity . . . . . . . . . . . . . . . . .

A.1.2 Nature and real eigenvalues . . . . . . . . . . . . . . . . . . .
A.1.3 Are Hermitian operators really like that? . . . . . . . . . . . .
A.2 Notes on the Basic Ideas of Quantum Mechanics . . . . . . . . . . . .
A.2.1 Why the linear momentum operators are Hermitian . . . . . .
A.2.2 Why boundary conditions are tricky . . . . . . . . . . . . . .
A.2.3 Three-dimensional solutions from one-dimensional ones . . . .
A.2.4 Derivation of the harmonic oscillator solution . . . . . . . . .
A.2.5 More on the harmonic oscillator and uncertainty . . . . . . . .
A.3 Notes on the Single-Particle Systems . . . . . . . . . . . . . . . . . .
A.3.1 Derivation of a vector identity . . . . . . . . . . . . . . . . . .
A.3.2 Derivation of the spherical harmonics . . . . . . . . . . . . . .
A.3.3 Derivation of the hydrogen radial wave functions . . . . . . . .
A.3.4 Definitions of the Laguerre polynomials . . . . . . . . . . . . .
A.3.5 Justification of the expression for the expectation value . . . .
A.3.6 Why commuting operators have a common set of eigenvectors
A.3.7 Derivation of the generalized uncertainty relationship . . . . .
A.3.8 Derivation of the commutator rules . . . . . . . . . . . . . . .
A.3.9 How the hydrogen molecular ion integrals are done . . . . . .
A.3.10 In what sense the variational approximation is best . . . . . .
A.3.11 More on the accuracy of variational approximation . . . . . .
A.3.12 Why the hydrogen molecular ion wave function is positive . .
A.3.13 Symmetries of the hydrogen molecular ion wave function . . .
A.4 Notes on the Multiple-Particle Systems . . . . . . . . . . . . . . . . .
A.4.1 Bell’s actual analysis . . . . . . . . . . . . . . . . . . . . . . .
A.4.2 Why spin does not change the hydrogen molecule ground state
A.4.3 Limitations of the shielding approximation . . . . . . . . . . .
A.4.4 Why the s states get the closest to the origin . . . . . . . . . .
A.5 Notes on Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.5.1 Ambiguities in the definition of electron affinity . . . . . . . .
A.5.2 Why Floquet theory should be called so . . . . . . . . . . . .

A.6 Notes on Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . .
A.6.1 Why energy eigenstates are physically stationary . . . . . . .

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335
335

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335
336
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339
342
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347
347
348
349
349
350
350
351
352
352
352
353
353
354
354
354

356
357
357

7.8
7.9

7.7.1 Wave function approximation . . . . . . . .
7.7.2 The Hamiltonian . . . . . . . . . . . . . . .
7.7.3 The expectation value of energy . . . . . . .
7.7.4 The canonical Hartree-Fock equations . . . .
7.7.5 Additional points . . . . . . . . . . . . . . .
Some Topics Not Covered [Advanced] . . . . . . . .
The Meaning of Quantum Mechanics [Background]
7.9.1 Failure of the Schr¨odinger Equation? . . . .
7.9.2 The Many-Worlds Interpretation . . . . . .

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A.6.2 More precise description of two-state systems . . . . . . . . . .
A.6.3 Derivation of the evolution of expectation values . . . . . . . . .
A.6.4 The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . .
A.6.5 The energy-time uncertainty relationship . . . . . . . . . . . . .
A.6.6 Justification of the two-state approximation in atom radiation .
A.6.7 About spectral broadening . . . . . . . . . . . . . . . . . . . . .
A.6.8 Why symmetry eigenvalues are preserved . . . . . . . . . . . . .
A.6.9 Remark on the edges of the wave packet . . . . . . . . . . . . .
A.7 Notes on the Additional Topics . . . . . . . . . . . . . . . . . . . . . .
A.7.1 Physical justification of the fundamental commutation relations

A.7.2 Angular momentum components have only zero in common . . .
A.7.3 Components of vectors are less than the total vector . . . . . . .
A.7.4 Finding the spherical harmonics using ladder operators . . . . .
A.7.5 Why angular momenta components can be added . . . . . . . .
A.7.6 Why the Clebsch-Gordan tables can be read either way . . . . .
A.7.7 How to make your very own Clebsch-Gordan tables . . . . . . .
A.7.8 Machine language version of the Clebsch-Gordan tables . . . . .
A.7.9 The triangle inequality in quantum mechanics . . . . . . . . . .
A.7.10 Awkward questions about spin . . . . . . . . . . . . . . . . . . .
A.7.11 More awkwardness about spin . . . . . . . . . . . . . . . . . . .
A.7.12 Derivation of a vectorial triple product property . . . . . . . . .
A.7.13 More on Maxwell’s third law . . . . . . . . . . . . . . . . . . . .
A.7.14 Setting the record straight on alignment . . . . . . . . . . . . .
A.7.15 Solving the NMR equations . . . . . . . . . . . . . . . . . . . .
A.7.16 A basic description of Lagrangian multipliers . . . . . . . . . . .
A.7.17 The generalized variational principle . . . . . . . . . . . . . . .
A.7.18 Born-Oppenheimer approximation and spin-degeneracy . . . .
A.7.19 Derivation of the Born-Oppenheimer approximation . . . . . . .
A.7.20 Why a single Slater determinant does not work . . . . . . . . .
A.7.21 Simplification of the Hartree-Fock energy . . . . . . . . . . . . .
A.7.22 Constraints on the Coulomb and exchange integrals . . . . . . .
A.7.23 Generalized orbitals . . . . . . . . . . . . . . . . . . . . . . . . .
A.7.24 Derivation of the Hartree-Fock equations . . . . . . . . . . . . .
A.7.25 Why the Fock operator is Hermitian . . . . . . . . . . . . . . .
A.7.26 Basic science (BS) behind “correlation energy” . . . . . . . . . .
A.7.27 Everett’s theory and vacuum energy . . . . . . . . . . . . . . .
A.7.28 A tenth of a googol in universes . . . . . . . . . . . . . . . . . .

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357
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367
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377

380
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393
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Bibliography

395

Web Pages

397

Notations

399

Index

421
xxi



List of Figures
1.1
1.2

1.3
1.4
1.5
1.6
1.7

The classical picture of a vector. . . . . . . .
Spike diagram of a vector. . . . . . . . . . .
More dimensions. . . . . . . . . . . . . . . .
Infinite dimensions. . . . . . . . . . . . . . .
The classical picture of a function. . . . . .
Forming the dot product of two vectors. . .
Forming the inner product of two functions.

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4
4
4
5
5
6
7

2.1
2.2

2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19

A visualization of an arbitrary wave function. . . . . . . .
Combined plot of position and momentum components. . .
The uncertainty principle illustrated. . . . . . . . . . . . .
Classical picture of a particle in a closed pipe. . . . . . . .
Quantum mechanics picture of a particle in a closed pipe. .
Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . .
One-dimensional energy spectrum for a particle in a pipe. .
One-dimensional ground state of a particle in a pipe. . . .
Second and third lowest one-dimensional energy states. . .
Definition of all variables. . . . . . . . . . . . . . . . . . .
True ground state of a particle in a pipe. . . . . . . . . . .
True second and third lowest energy states. . . . . . . . . .

A combination of ψ111 and ψ211 seen at some typical times.
The harmonic oscillator. . . . . . . . . . . . . . . . . . . .
The energy spectrum of the harmonic oscillator. . . . . . .
Ground state ψ000 of the harmonic oscillator . . . . . . . .
Wave functions ψ100 and ψ010 . . . . . . . . . . . . . . . . .
Energy eigenfunction ψ213 . . . . . . . . . . . . . . . . . . .
Arbitrary wave function (not an energy eigenfunction). . .

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16
18
18
25
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32
34
34
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38

40
41
47
49
50
50
54

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

Spherical coordinates of an arbitrary point P. . . . . .
Spectrum of the hydrogen atom. . . . . . . . . . . . . .
Ground state wave function ψ100 of the hydrogen atom.
Eigenfunction ψ200 . . . . . . . . . . . . . . . . . . . . .
Eigenfunction ψ210 , or 2pz . . . . . . . . . . . . . . . . .
Eigenfunction ψ211 (and ψ21−1 ). . . . . . . . . . . . . .
Eigenfunctions 2px , left, and 2py , right. . . . . . . . . .
Hydrogen atom plus free proton far apart. . . . . . . .

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72
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75
91

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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
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17

State with two neutral atoms. . . . . . . . . . . . . . . . . . .
Symmetric state . . . . . . . . . . . . . . . . . . . . . . . . . .
Antisymmetric state . . . . . . . . . . . . . . . . . . . . . . .
Separating the hydrogen ion. . . . . . . . . . . . . . . . . . . .
The Bohm experiment . . . . . . . . . . . . . . . . . . . . . .
The Bohm experiment, after the Venus measurement. . . . . .
Spin measurement directions. . . . . . . . . . . . . . . . . . .
Earth’s view of events. . . . . . . . . . . . . . . . . . . . . . .

A moving observer’s view of events. . . . . . . . . . . . . . . .
Approximate solutions for hydrogen (left) and helium (right).
Approximate solutions for lithium (left) and beryllium (right).
Example approximate solution for boron. . . . . . . . . . . . .
Covalent sigma bond consisting of two 2pz states. . . . . . . .
Covalent pi bond consisting of two 2px states. . . . . . . . . .
Covalent sigma bond consisting of a 2pz and a 1s state. . . . .
Shape of an sp3 hybrid state. . . . . . . . . . . . . . . . . . . .
Shapes of the sp2 (left) and sp (right) hybrids. . . . . . . . . .

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10

Possible polarizations of a pair of hydrogen atoms. . . . . . . . . . .
Billiard-ball model of the salt molecule. . . . . . . . . . . . . . . . .
Billiard-ball model of a salt crystal. . . . . . . . . . . . . . . . . . .
The salt crystal disassembled to show its structure. . . . . . . . . .
Sketch of electron energy spectra in solids. . . . . . . . . . . . . . .
The lithium atom, scaled more correctly than in chapter 4.11 . . . .
Body-centered-cubic (bcc) structure of lithium. . . . . . . . . . . .
Fully periodic wave function of a two-atom lithium “crystal.” . . . .
Flip-flop wave function of a two-atom lithium “crystal.” . . . . . . .

Wave functions of a four-atom lithium “crystal.” The actual picture
the fully periodic mode. . . . . . . . . . . . . . . . . . . . . . . . .
Reciprocal lattice of a one-dimensional crystal. . . . . . . . . . . . .
Schematic of energy bands. . . . . . . . . . . . . . . . . . . . . . . .
Energy versus linear momentum. . . . . . . . . . . . . . . . . . . .
Schematic of merging bands. . . . . . . . . . . . . . . . . . . . . . .
A primitive cell and primitive translation vectors of lithium. . . . .
Wigner-Seitz cell of the bcc lattice. . . . . . . . . . . . . . . . . . .
Schematic of crossing bands. . . . . . . . . . . . . . . . . . . . . . .
Ball and stick schematic of the diamond crystal. . . . . . . . . . . .
Allowed wave number vectors. . . . . . . . . . . . . . . . . . . . . .
Schematic energy spectrum of the free electron gas. . . . . . . . . .
Occupied wave number states and Fermi surface in the ground state
Density of states for the free electron gas. . . . . . . . . . . . . . . .

5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22

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5.23 Energy states, top, and density of states, bottom, when there is severe confinement in the y-direction, as in a quantum well. . . . . . . . . . . . . . . . . . .
5.24 Energy states, top, and density of states, bottom, when there is severe confinement in both the y- and z-directions, as in a quantum wire. . . . . . . . . . . .
5.25 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. . . . . . .
5.26 Wave number vectors seen in a cross section of constant kz . Top: sinusoidal
solutions. Bottom: exponential solutions. . . . . . . . . . . . . . . . . . . . . .
5.27 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. .
5.28 Occupied states for one, two, and three free electrons per physical lattice cell. .
5.29 Redefinition of the occupied wave number vectors into Brillouin zones. . . . .
5.30 Second, third, and fourth Brillouin zones seen in the periodic zone scheme. . .
5.31 The typical exponential free electron wave function to be corrected for the
lattice potential is shown as a red dot. . . . . . . . . . . . . . . . . . . . . . .
5.32 The k-lattice and k-sphere in wave number space. . . . . . . . . . . . . . . . .
5.33 Tearing apart of the wave number space energies. . . . . . . . . . . . . . . . .
5.34 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. . .
5.35 Bragg planes seen in wave number space cross section. . . . . . . . . . . . . .
5.36 Occupied states for the energies of figure 5.34 if there are two valence electrons
per lattice cell. Left: energy. Right: wave numbers. . . . . . . . . . . . . . . .
5.37 Smaller lattice potential. From top to bottom shows one, two and three valence
electrons per lattice cell. Left: energy. Right: wave numbers. . . . . . . . . . .
5.38 Sketch of electron energy spectra in solids. . . . . . . . . . . . . . . . . . . . .
5.39 Specific heat at constant volume of gases. Temperatures from absolute zero to
1200 K. Data from NIST-JANAF and AIP. . . . . . . . . . . . . . . . . . . . .
5.40 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. . . . . . . . . . . . . . .
5.41 Depiction of an electromagnetic ray. . . . . . . . . . . . . . . . . . . . . . . . .
5.42 Law of reflection in elastic scattering from a plane. . . . . . . . . . . . . . . .
5.43 Scattering from multiple “planes of atoms”. . . . . . . . . . . . . . . . . . . .
5.44 Difference in travel distance when scattered from P rather than O. . . . . . . .
6.1
6.2

6.3
6.4
6.5
6.6
6.7
6.8

Emission and absorption of radiation by an atom. . . . . . . . . . . . . . . . .
Approximate Dirac delta function δε (x−ξ) is shown left. The true delta function
δ(x − ξ) is the limit when ε becomes zero, and is an infinitely high, infinitely
thin spike, shown right. It is the eigenfunction corresponding to a position ξ. .

The real part (red) and envelope (black) of an example wave. . . . . . . . . . .
The wave moves with the phase speed. . . . . . . . . . . . . . . . . . . . . . .
The real part (red) and magnitude or envelope (black) of a typical wave packet
The velocities of wave and envelope are not equal. . . . . . . . . . . . . . . . .
A particle in free space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An accelerating particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxv

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