Hydrogen bonding — again, again

I’ve been thinking about hydrogen bonding ever since my senior thesis in 1959. Although its’ role in the protein alpha helix had been known since ’51 and in the DNA double helix since ’53, little did we realize at the time just how important it would be for the workings of the cell. So I was lucky Dr. Schleyer put me at an IR spectrometer and had me make a bunch of compounds, to look for hydrogen bonding of OH, NH and SH to the pi electrons of the benzene ring. I had to make a few of them, which involved getting a (CH2)n chain between the benzene ring and the hydrogen donor. Just imagine the benzene as the body of a scorpion and the (CH2) groups as the length of the tail.  The SH compounds were particularly nasty, and people would look at their shoes when I’d walk into the eating club. Naturally the college yearbook screwed things up and titled my thesis “Studies in Hydrogen Bombing”, to which my parents’ friends would say — he looks like such a nice young man, why was he doing that?

At any rate I’m going to talk about a recent paper [ Science vol. 371 pp. 160 – 164 ’21 ] on the nature of the bond in the F H F – anion.  It’s going to be pretty hard core stuff with relatively little explanatory material. You’ve either been previously exposed to this stuff or you haven’t.  So this post is for the cognoscenti.  Hold on, it’s going to be wild ride.

In conventional hydrogen bonds, the donor (D) atom is separated from the Acceptor atom (A) by 2.7 Angstroms or more, and the hydrogen nucleus is found closer to A where the potential energy minimum is found.

So it looks like this D – H . .. A

The D-H bond isn’t normal, but is stretched  and weakened.  This means that it takes less energy to stretch it meaning that it absorbs infrared radiation at a lower frequency (higher wavelength) — red shift if you will. 

Such is what we were looking for and we found it comparing 

Benzene (CH2)n OH vibrations to butanol, pentanol, hexanol, etc etc. cyclohexane (CH2)n OH.

As the D – A distance shrinks there is ultimately a flat bottomed single well potential, where H becomes a confined particle (but still delocalized) betwen D and A.

The vibrations of protons in hydrogen bonds deviate markedly from the classic quantum harmonic oscillator beloved by physicists.  Here the energy levels on solving the classic H psi = E psi equation of quantum mechanics are evenly spaced (see Lancaster & Blundell “Quantum Field Theory” p. 20.)

However in real molecules, as you ascend the vibrational ladder, conventional hydrogen bonds show a decrease in the difference between energy levels (positive anharmonicity).  By contrast, when proton confinement dictates the potential shape in short hydrogen bonds (when D and A are close together, mimicking the particle in a box model in quantum mechanics) the spacing between states increases (negative anharmonicity).

The present work shows that in FHF- the proton motion is superharmonic — https://en.wikipedia.org/wiki/Subharmonic_function — which they don’t describe very well. 

When the F F distance gets below 2.4 Angstroms, covalent bonding starts to become a notable contributor to the short hydrogen bond, and the authors actually have evidence that there is overlap in FHF- between the 3s orbital of H and the 2 Pz orbitals of the donor and the acceptor atoms, yielding a stabilization of the resulting molecular orbital. 

Is that cool or what.  The bond sits right on the borderland between a covalent bond and a hydrogen bond, taking on aspects of both. 

 

The Representation of group G on vector space V is really a left action of the group on the vector space

Say what? What does this have to do with quantum mechanics? Quite a bit. Practically everything in fact. Most chemists learn quantum mechanics because they want to see where atomic orbitals come from. So they stagger through the solution of the Schrodinger equation where the quantum numbers appear as solution of recursion equations for power series solutions of the Schrodinger equation.

Forget the Schrodinger equation (for now), quantum mechanics is really written in the language of linear algebra. Feynman warned us not to consider ‘how it can be like that’, but at least you can understand the ‘that’ — e.g. linear algebra. In fact, the instructor in a graduate course in abstract algebra I audited opened the linear algebra section with the remark that the only functions mathematicians really understand are the linear ones.

The definitions used (vector space, inner product, matrix multiplication, Hermitian operator) are obscure and strange. You can memorize them and mumble them as incantations when needed, or you can understand why they are the way they are and where they come from. So if you are a bit rusty on your linear algebra I’ve written a series of 9 posts on the subject — here’s a link to the first https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/– just follow the links after that.

Just to whet your appetite, all of quantum mechanics consists of manipulation of a particular vector space called Hilbert space. Yes all of it.

Representations are a combination of abstract algebra and linear algebra, and are crucial in elementary particle physics. In fact elementary particles are representations of abstract symmetry groups.

So in what follows, I’ll assume you know what vector spaces, linear transformations of them, their matrix representation. I’m not going to explain what a group is, but it isn’t terribly complicated. So if you don’t know about them quit. The Wiki article is too detailed for what you need to know.

The title of the post really threw me, and understanding requires significant unpacking of the definitions, but you need to know this if you want to proceed further in physics.

So we’ll start with a Group G, its operation * and e, its identity element.

Next we have a set called X — just that a bunch of elements (called x, y, . . .), with no further structure imposed — you can’t add elements, you can’t mutiply them by real numbers. If you could with a few more details you’d have a vector space (see the survival guide)

Definition of Left Action (LA) of G on set X

LA : G x X –> X

LA : ( g, x ) |–> (g . x)

Such that the following two properties hold

l. For all x in X LA : (e, x) |–> (e.x) = x

2. For all g1 and g2 in G LA ( g1 * g2), x ) |–> ( g1 . (g2 . x )

Given vector space V define GL(V) the set of invertible linear transformations (LTs) of vector space. GL(V) becomes a group if you let composition of linear transformations become its operation (it’s all in the survival guide.

Now for the definition of representation of Group G on vector space V

It is a function

rho: G –> GL(V)

rho: g |–> LTg : V –> V linear ; LTg == Linear Transformation labeled by group element g

The representation rho defines a left group action on V

LA : (g, v) |–> LTg (V) — this satisfies the two properties above of a left action given above — think about it.

Now you’re ready for some serious study of quantum mechanics. When you read that the representation is acting on some vector space, you’ll know what they are talking about.

Math can be hard even for very smart people

50 McCosh Hall an autumn evening in 1956. The place was packed. Chen Ning Yang was speaking about parity violation. Most of the people there had little idea (including me) of what he did, but wanted to be eyewitnesses to history.. But we knew that what he did was important and likely to win him the Nobel (which happened the following year).

That’s not why Yang is remembered today (even though he’s apparently still alive at 98). Before that he and Robert Mills were trying to generalize Maxwell’s equations of electromagnetism so they would work in quantum mechanics and particle physics. Eventually this led Yang and Mills to develop the theory of nonAbelian gauge fields which pervade physics today.

Yang and James Simons (later the founder of Renaissance technologies and already a world class mathematician — Chern Simons theory) later wound up at Stony Brook. Simons, told him that gauge theory must be related to connections on fiber bundles and pointed him to Steenrod’s The Topology of Fibre Bundles. So he tried to read it and “learned nothing. The language of modern mathematics is too cold and abstract for a physicist.”

Another Yang quote “There are only two kinds of math books: Those you cannot read beyond the first sentence, and those you cannot read beyond the first page.”

So here we have a brilliant man who invented significant mathematics (gauge theory) along with Mills, unable to understand a math book written about the exact same subject (connections on fiber bundles).

247 ZeptoSeconds

247 ZeptoSeconds is not the track time of the fastest Marx brother. It is the time a wavelength of light takes to travel across a hydrogen molecule (H2) before it kicks out an electron — the photoelectric effect.

But what is a zeptosecond anyway? There are 10^21 zeptoSeconds in a second. That’s a lot. A thousand times more than the number of seconds since the big bang which is only 60 x 60 x 24 x 365 x 13.8 x 10^9 = 4. 35 x 10^17. Not that big a deal to a chemist anyway since 10^21 is 1/600th of the number of molecules in a mole.

You can read all about it in Science vol. 370 pp. 339 – 341 ’20 — https://science.sciencemag.org/content/sci/370/6514/339.full.pdf it you have a subscription.

Studying photoionization allows you to study the way light is absorbed by molecules, something important to any chemist. The 247 zeptoseconds is the birth time of the emitted electron. It depends on the travel time of the photon across the hydrogen molecule.

They don’t quite say trajectory of the photon, but it is implied even though in quantum mechanics (which we’re dealing with here), there is no such a thing as a trajectory. All we have is measurements at time t1 and time t2. We are not permitted to say what the photon is doing between these two times when we’ve done measurements. Our experience in the much larger classical physics world makes us think that there is such a thing.

It is the peculiar doublethink quantum mechanics forces on us. Chemists know this when they think about something as simple as the S2 orbital, something spherically symmetric, with electron density on either side of a node. The node is where you never find an electron. Well if you don’t, find it here, how can it have a trajectory from one side to the other.

Quantum mechanics is full of conundrums like that. Feynman warned us not to think about them, but it will take your mind off the pandemic (and if you’re good, off the election as well)..

It’s worth reading the article in Quanta which asks if wavefunctions tunnel through a barrier at speeds faster than light — here’s a link — https://www.quantamagazine.org/quantum-tunnel-shows-particles-can-break-the-speed-of-light-20201020/. It will make your head spin.

Here’s a link to an earlier post about the doublethink quantum mechanics forces on us

Doublethink and angular momentum — why chemists must be adept at it

Here’s the post itself

Doublethink and angular momentum — why chemists must be adept at it

Chemists really should know lots and lots about angular momentum which is intimately involved in 3 of the 4 quantum numbers needed to describe atomic electronic structure. Despite this, I never really understood what was going until taking the QM course, and digging into chapters 10 and 11 of Giancoli’s physics book (pp. 248 -310 4th Edition).

Quick, what is the angular momemtum of a single particle (say a planet) moving around a central object (say the sun)? Well, its magnitude is the current speed of the particle times its mass, but what is its direction? There must be a direction since angular momentum is a vector. The (unintuitive to me) answer is that the angular momemtum vector points upward (resp. downward) from the plane of motion of the planet around the center of mass of the sun planet system, if the planet is moving counterclockwise (resp. clockwise) according to the right hand rule. On the other hand, the momentum of a particle moving in a straight line is just its mass times its velocity vector (e.g. in the same direction).

Why the difference? This unintuitive answer makes sense if, instead of a single point mass, you consider the rotation of a solid (e.g. rigid) object around an axis. All the velocity vectors of the object at a given time either point in different directions, or if they point in the same direction have different magnitudes. Since the object is solid, points farther away from the axis are moving faster. The only sensible thing to do is point the angular momentum vector along the axis of rotation (it’s the only thing which has a constant direction).

Mathematically, this is fairly simple to do (but only in 3 dimensions). The vector from the axis of rotation to the planet (call it r), and the vector of instantaneous linear velocity of the planet (call it v) do not point in the same direction, so they define a plane (if they do point in the same direction the planet is either hurtling into the sun or speeding directly away, hence not rotating). In 3 dimensions, there is a unique direction at 90 degrees to the plane. The vector cross product of r and v gives a vector pointing in this direction (to get a unique vector, you must use the right or the left hand rule). Nicely, the larger r and v, the larger the angular momentum vector (which makes sense). In more than 3 dimensions there isn’t a unique direction away from a plane, which is why the cross product doesn’t work there (although there are mathematical analogies to it).

This also explains why I never understood momentum (angular or otherwise) till now. It’s very easy to conflate linear momentum with force and I did. Get hit by a speeding bullet and you feel a force in the same direction as the bullet — actually the force you feel is what you’ve done to the bullet to change its momentum (force is basically defined as anything that changes momentum).

So the angular momentum of an object is never in the direction of its instantaneous linear velocity. But why should chemists care about angular momentum? Solid state physicists, particle physicists etc. etc. get along just fine without it pretty much, although quantum mechanics is just as crucial for them. The answer is simply because the electrons in a stable atom hang around the nucleus and do not wander off to infinity. This means that their trajectories must continually bend around the nucleus, giving each trajectory an angular momentum.

Did I say trajectory? This is where the doublethink comes in. Trajectory is a notion of the classical world we experience. Consider any atomic orbital containing a node (e.g. everything but a 1 s orbital). Zeno would have had a field day with them. Nodes are surfaces in space where the electron is never to be found. They separate the various lobes of the orbital from each other. How does the electron get from one lobe to the other by a trajectory? We do know that the electron is in all the lobes because a series of measurements will find the electron in each lobe of the orbital (but only in one lobe per measurement). The electron can’t make the trip, because there is no trip possible. Goodbye to the classical notion of trajectory, and with it the classical notion of angular momentum.

But the classical notions of trajectory and angular momentum still help you think about what’s going on (assuming anything IS in fact going on down there between measurements). We know quite a lot about angular momentum in atoms. Why? Because the angular momentum operators of QM commute with the Hamiltonian operator of QM, meaning that they have a common set of eigenfunctions, hence a common set of eigenvalues (e.g. energies). We can measure these energies (really the differences between them — that’s what a spectrum really is) and quantum mechanics predicts this better than anything else.

Further doublethink — a moving charge creates a magnetic field, and a magnetic field affects a moving charge, so placing a moving charge in a magnetic field should alter its energy. This accounts for the Zeeman effect (the splitting of spectral lines in a magnetic field). Trajectories help you understand this (even if they can’t really exist in the confines of the atom).

The pleasures of reading Feynman on Physics — III

The more I read volume III of the Feynman Lectures on Physics about Quantum Mechanics the better I like it.  Even having taken two courses in it 60 and 10 years ago, Feynman takes a completely different tack, plunging directly into what makes quantum mechanics different than anything else.

He starts by saying “Traditionally, all courses in quantum mechanics have begun in the same way, retracing the path followed in the historical development of the subject.  One first learns a great deal about classical mechanics so that he will be able to understand how to solve the Schrodinger equation.  Then he spends a long time working out various solutions.  Only after a detailed study of this equation does he get to the advanced subject of the electron’s spin.”

Not to worry, he gets to the Hamiltonian on p. 85 and  the Schrodinger equation p. 224.   But he is blunt about it “We do not intend to have you think we have derived the Schrodinger equation but only wish to show you one way of thinking abut it.  When Schrodinger first wrote it down, he gave a kind of derivation based on some heuristic arguments and some brilliant intuitive guesses.  Some of the arguments he used were even false, but that does not matter. “

When he gives the law correct of physics for a particle moving freely in space with no forces, no disturbances (basically the Hamiltonian), he says “Where did we get that from”  Nowhere. It’s not possible to derive it from anything you know.  It came out of the mind of Schrodinger, invented in his struggle to find an understanding of the experimental observations of the real world.”  How can you not love a book written like this?

Among the gems are the way the conservation laws of physics arise in a very deep sense from symmetry (although he doesn’t mention Noether’s name).   He shows that atoms radiate photons because of entropy (p. 69).

Then there is his blazing honesty “when philosophical ideas associated with science are dragged into another field, they are usually completely distorted.”  

He spends a lot of time on the Stern Gerlach experiment and its various modifications and how they put you face to face with the bizarrities of quantum mechanics.

He doesn’t shy away from dealing with ‘spooky action at a distance’ although he calls it the Einstein Podolsky Rosen paradox.  He shows why if you accept the way quantum mechanics works, it isn’t a paradox at all (this takes a lot of convincing).

He ends up with “Do you think that it is not a paradox, but that it is still very peculiar?  On that we can all agree. It is what makes physics fascinating”

There are tons more but I hope this whets your appetite

The pleasures of reading Feynman on Physics – II

If you’re tired of hearing and thinking about COVID-19 24/7 even when you don’t want to, do what I did when I was a neurology resident 50+ years ago making clever diagnoses and then standing helplessly by while patients died.  Back then I read topology and the intense concentration required to absorb and digest the terms and relationships, took me miles and miles away.  The husband of one of my interns was a mathematician, and she said he would dream about mathematics.

Presumably some of the readership are chemists with graduate degrees, meaning that part of their acculturation as such was a course in quantum mechanics.  Back in the day it was pretty much required of chemistry grad students — homage a Prof. Marty Gouterman who taught the course to us 3 years out from his PhD in 1961.  Definitely a great teacher.  Here he is now, a continent away — http://faculty.washington.edu/goutermn/.

So for those happy souls I strongly recommend volume III of The Feynman Lectures on Physics.  Equally strongly do I recommend getting the Millennium Edition which has been purged of the 1,100 or so errors found in the 3 volumes over the years.

“Traditionally, all courses in quantum mechanics have begun in the same way, retracing the path followed in the historical development of the subject.  One first learns a great deal about classical mechanics so that he will be able to understand how to solve the Schrodinger equation.  Then he spends a long time working out various solutions.  Only after a detailed study of this equation does he get to the advanced subject of the electron’s spin.”

The first half of volume III is about spin

Feynman doesn’t even get to the Hamiltonian until p. 88.  I’m almost half through volume III and there has been no sighting of the Schrodinger equation so far.  But what you will find are clear explanations of Bosons and Fermions and why they are different, how masers and lasers operate (they are two state spin systems), how one electron holds two protons together, and a great explanation of covalent bonding.  Then there is great stuff beyond the ken of most chemists (at least this one) such as the Yukawa explanation of the strong nuclear force, and why neutrons and protons are really the same.  If you’ve read about Bell’s theorem proving that ‘spooky action at a distance must exist’, you’ll see where the numbers come from quantum mechanically that are simply impossible on a classical basis.  Zeilinger’s book “The Dance of the Photons” goes into this using .75 (which Feynman shows is just cos(30)^2.

Although Feynman doesn’t make much of a point about it, the essentiality of ‘imaginary’ numbers (complex numbers) to the entire project of quantum mechanics impressed me.  Without them,  wave interference is impossible.

I’m far from sure a neophyte could actually learn QM from Feynman, but having mucked about using and being exposed to QM and its extensions for 60 years, Feynman’s development of the subject is simply a joy to read.

So get the 3 volumes and plunge in.  You’ll forget all about the pandemic (for a while anyway)

 

The pleasures of reading Feynman on Physics

“Traditionally, all courses in quantum mechanics have begun in the same way, retracing the path followed in the historical development of the subject.  One first learns a great deal about classical mechanics so that he will be able to understand how to solve the Schrodinger equation.  Then he spends a long time working out various solutions.  Only after a detailed study of this equation does he get to the advanced subject of the electron’s spin.”

From vol. III of the Feynman lectures on physics  p. 3 – 1.

Certainly that’s the way I was taught QM as a budding chemist in 1961. Nothing wrong with that.  For a chemist it is very useful to see how all those orbitals pop out of series solutions to the Schrodinger equation for the hydrogen atom.

“We have come to the conclusion that what are usually called the advanced parts of quantum mechanics are in fact, quite simple. The mathematics that is involved is particularly simple, involving simple algebraic operations and no differential equations or at most only very simple ones.”

Quite true, but when, 50 years or so later,  I audited a QM course at an elite woman’s college, the underlying linear algebra wasn’t taught — so I wrote a series of posts giving the basics of the linear algebra used in QM — start at https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/ and follow the links (there are 8 more posts).

Even more interesting was the way Mathematica had changed the way quantum mechanics was taught — see https://luysii.wordpress.com/2009/09/22/what-hath-mathematica-wrought/

But back to Feynman:  I’m far from sure a neophyte could actually learn QM this way, but having mucked about using and being exposed to QM and its extensions for 60 years, Feynman’s development of the subject is simply a joy to read. Feynman starts out as a good physicist should with the experiments.  Nothing fancy, bullets are shot at a screen through a slit, then electrons then two slits, and the various conundrums arising when one slit is closed.

Onward and upward through the Stern Gerlach experiments and how matrices are involved (although Feynman doesn’t call them that).  The only flaw in what I’ve found so far is his treatment of phase factors (p. 4 -1 ).  They aren’t really defined, but they are crucial as phase factors are what breaks the objects of physics into fermions and bosons.

If you’ve taken any course in QM and have some time (who doesn’t now that we’re all essentially inmates in our own homes/apartments) than have a look.   You’ll love it.  As Bill Gates said about the books “It is good to sit at the feet of the master”.

One piece of advice — get the new Millennium edition — it has removed some 1,100 errors and misprints found over the decades, so if you’re studying it by yourself, you won’t be tripped up by a misprint in the text when you don’t understand something.

Want to understand Quantum Computing — buy this book

As quantum mechanics enters its second century, quantum computing has been hot stuff for the last third of it, beginning with Feynman’s lectures on computation in 84 – 86.  Articles on quantum computing  appear all the time in Nature, Science and even the mainstream press.

Perhaps you tried to understand it 20 years ago by reading Nielsen and Chuang’s massive tome Quantum Computation and Quantum information.  I did, and gave up.  At 648 pages and nearly half a million words, it’s something only for people entering the field.  Yet quantum computers are impossible to ignore.

That’s where a new book “Quantum Computing for Everyone” by Chris Bernhardt comes in.  You need little more than high school trigonometry and determination to get through it.  It is blazingly clear.  No term is used before it is defined and there are plenty of diagrams.   Of course Bernhardt simplifies things a bit.  Amazingly, he’s able to avoid the complex number system. At 189 pages and under 100,000 words it is not impossible to get through.

Not being an expert, I can’t speak for its completeness, but all the stuff I’ve read about in Nature, Science is there — no cloning, entanglement, Ed Frenkin (and his gate), Grover’s algorithm,  Shor’s algorithm, the RSA algorithm.  As a bonus there is a clear explanation of Bell’s theorem.

You don’t need a course in quantum mechanics to get through it, but it would make things easier.  Most chemists (for whom this blog is basically written) have had one.  This plus a background in linear algebra would certainly make the first 70 or so pages a breeze.

Just as a book on language doesn’t get into the fonts it can be written in, the book doesn’t get into how such a computer can be physically instantiated.  What it does do is tell you how the basic guts of the quantum computer work. Amazingly, they are just matrices (explained in the book) which change one basis for representing qubits (explained) into another.  These are the quantum gates —  ‘just operations that can be described by orthogonal matrices” p. 117.  The computation comes in by sending qubits through the gates (operating on vectors by matrices).

Be prepared to work.  The concepts (although clearly explained) come thick and fast.

Linear algebra is basic to quantum mechanics.  Superposition of quantum states is nothing more than a linear combination of vectors.  When I audited a course on QM 10 years ago to see what had changed in 50 years, I was amazed at how little linear algebra was emphasized.  You could do worse that read a series of posts on my blog titled “Linear Algebra Survival Guide for Quantum Mechanics” — There are 9 — start here and follow the links — you may find it helpful — https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/

From a mathematical point of view entanglement (discussed extensively in the book) is fairly simple -philosophically it’s anything but – and the following was described by a math prof as concise and clear– https://luysii.wordpress.com/2014/12/28/how-formal-tensor-mathematics-and-the-postulates-of-quantum-mechanics-give-rise-to-entanglement/

The book is a masterpiece — kudos to Bernhardt

Feynman and Darwin

What do Richard Feynman and Charles Darwin have in common?  Both have written books which show a brilliant mind at work.  I’ve started reading the New Millennium Edition of Feynman’s Lectures on Physics (which is the edition you should get as all 1165 errata found over the years have been corrected), and like Darwin his thought processes and their power are laid out for all to see.  Feynman’s books are far from F = ma.  They are basically polished versions of lectures, so it reads as if Feynman is directly talking to you.  Example: “We have already discussed the difference between knowing the rules of the game of chess and being able to play.”  Another: talking about Zeno  “The Greeks were somewhat confused by such problems, being helped, of course, by some very confusing Greeks.”

He’s always thinking about the larger implications of what we know.  Example: “Newton’s law has the peculiar property that if it is right on a certain small scale, then it will be right on a larger scale”

He then takes this idea and runs with it.  “Newton’s laws are the ‘tail end’ of the atomic laws extrapolated to a very large size”  The fact that they are extrapolatable and the fact that way down below are the atoms producing them means, that extrapolatable laws are the only type of physical law which could be discovered by us (until we could get down to the atomic level).  Marvelous.  Then he notes that the fundamental atomic laws (e.g. quantum mechanics) are NOTHING like what we see in the large scale environment in which we live.

If you like this sort of thing, you’ll love the books.  I don’t think they would be a good way to learn physics for the first time however.  No problems, etc. etc.  But once you’ve had exposure to some physics “it is good to sit at the feet of the master” — Bill Gates.

Most of the readership is probably fully engaged with work, family career and doesn’t have time to actually read “The Origin of Species”. In retirement, I did,and the power of Darwin’s mind is simply staggering. He did so much with what little information he had. There was no clear idea of how heredity worked and at several points he’s a Lamarckian — inheritance of acquired characteristics. If you do have the time I suggest that you read the 1859 book chapter by chapter along with a very interesting book — Darwin’s Ghost by Steve Jones (published in 1999) which update’s Darwin’s book to contemporary thinking chapter by chapter.  Despite the advances in knowledge in 140 years, Darwin’s thinking beats Jones hands down chapter by chapter.

Book recommendation

Tired of reading books about physics?  Want the real McCoy”?  Well written and informal?  Contains stuff whose names you know but don’t understand — Jones Polynomial, Loop Quantum Gravity, Quantum field theory, Gauge groups and transformations —  etc. etc.

Up to date?  Well no, it’s 25 years old but still very much worth a read, so very unlike molecular biology, chemistry, computer science etc. etc.

Probably you should know as much physics and math as a beginning chemistry grad student. If you studied electromagnetism through Maxwell’s equations it would be a plus.  I stopped at Coulomb’s Law, and picked up enough to understand NMR.

This will give you a sample of the way it is written

“Much odder is that we are saying the vector field v is the linear combination of . .  partial derivatives.  What we are doing might be regarded as rather sloppy, since we are identifying two different although related things: the vector  field and the operator v^i * d-/dx^i which takes a directional derivative in the direction of v.”

“Now let us define vector fields on a manifold M. .. . these will be entities whose sole ambition in life is to differentiate functions”

The book is “Gauge FIelds, Knots and Gravity” by John Baez and Javier P. Muniain.

The writing, although clear has a certain humility.  “Unfortunately understanding these new ideas depends on a through mastery of quantum field theory, general relativity, geometry, topology and algebra.  Indeed, it is almost certain that nobody is sufficiently prepared to understand these ideas fully.”

I’m going to take it with me to the amateur chamber music festival.  As usual, at least 2 math full professors will be there to help me out.  Buy it and enjoy