Category Archives: Quantum Mechanics

Tensors — again, again, again

“A tensor is something that transforms like a tensor” — and a duck is something that quacks like a duck. If you find this sort of thing less than illuminating, I’ve got the book for you — “An Introduction to Tensors and Group Theory for Physicists” by Nadir Jeevanjee.

He notes that many physics books trying to teach tensors start this way, without telling you what a tensor actually is.  

Not so Jeevanjee — right on the first page of text (p. 3) he says “a tensor is a function which eats a certain number of vectors (known as the rank r of the tensor) and produces a number.  He doesn’t say what that number is, but later we are told that it is either C or R.

Then comes  the crucial fact that tensors are multilinear functions. From that all else flows (and quickly).

This means that you know everything you need to know about a tensor if you know what it does to its basis vectors.  

He could be a little faster about what these basis vectors actually are, but on p. 7 you are given an example explicitly showing them.

To keep things (relatively) simple the vector space is good old 3 dimensional space with basis vectors x, y and z.

His rank 2 tensor takes two vectors from this space (u and v) and produces a number.  There are 9 basis vectors not 6 as you might think — x®x, x®y, x®z, y®x, y®y, y®z, z®x, z®y, and z®z.    ® should be read as x inside a circle

Tensor components are the (real) numbers the tensor assigns to the 9 — these are written T(x®x) , T(x®y) T( x®z), T(y®x), T(y®y), T(y®z), T(z®x), T(z®y), and T(z®z)– note that there is no reason that T(x®y) should equal T(y®x) any more than a function R^2 –> R should give the same values for (1, 2) and (2, 1).

One more complication — where do the components of u and v fit in?  u is really (u^1, u^2, u^3) and v is really (v^1, v^2, v^3)

They multiply each other and the T’s  — so the first term of the tensor (sometimes confusingly called a tensor component)

is u^1 * v^1 * T(x®x)  and the last is u^3 * v^3 T(z®z).  Then the 9 tensor terms/components are summed giving a number. 

Then on pp. 7 and 8 he shows how a change of basis matrix (a 3 x 3 matrix written A^rs where rs, is one of 1, 2, 3) with nonZero determinant) gives the (usually incomprehensible) formula 
 
T^i’j’ = A^ik * A^jl T * (k, l)  where i, j, k, l are one of x, y, and z (or 1, 2, 3 as usually written)
 
So now you have a handle on the cryptic algebraic expression for tensors and what happens to them on a change of basis (e.g. how they transform).  Not bad for 5 pages of work — certainly not everything, but enough to make you comfortable with what follows — dual vectors, invariance, symmetric etc. etc.
 
Just knowing the multilinearity of tensors and just 2 postulates of quantum mechanics is all you need to understand entanglement — yes truly.  Yes, and you don’t need the Schrodinger equation, or differential equations at all, just linear algebra. 
 
Here is an old post to show you exactly how this works
 

How formal tensor mathematics and the postulates of quantum mechanics give rise to entanglement

Tensors continue to amaze. I never thought I’d get a simple mathematical explanation of entanglement, but here it is. Explanation is probably too strong a word, because it relies on the postulates of quantum mechanics, which are extremely simple but which lead to extremely bizarre consequences (such as entanglement). As Feynman famously said ‘no one understands quantum mechanics’. Despite that it’s never made a prediction not confirmed by experiments, so the theory is correct even if we don’t understand ‘how it can be like that’. 100 years of correct prediction of experimentation are not to be sneezed at.

If you’re a bit foggy on just what entanglement is — have a look at https://luysii.wordpress.com/2010/12/13/bells-inequality-entanglement-and-the-demise-of-local-reality-i/. Even better; read the book by Zeilinger referred to in the link (if you have the time).

Actually you don’t even need all the postulates for quantum mechanics (as given in the book “Quantum Computation and Quantum Information by Nielsen and Chuang). No differential equations. No Schrodinger equation. No operators. No eigenvalues. What could be nicer for those thirsting for knowledge? Such a deal ! ! ! Just 2 postulates and a little formal mathematics.

Postulate #1 “Associated to any isolated physical system, is a complex vector space with inner product (that is a Hilbert space) known as the state space of the system. The system is completely described by its state vector which is a unit vector in the system’s state space”. If this is unsatisfying, see an explication of this on p. 80 of Nielson and Chuang (where the postulate appears)

Because the linear algebra underlying quantum mechanics seemed to be largely ignored in the course I audited, I wrote a series of posts called Linear Algebra Survival Guide for Quantum Mechanics. The first should be all you need. https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/ but there are several more.

Even though I wrote a post on tensors, showing how they were a way of describing an object independently of the coordinates used to describe it, I did’t even discuss another aspect of tensors — multi linearity — which is crucial here. The post itself can be viewed at https://luysii.wordpress.com/2014/12/08/tensors/

Start by thinking of a simple tensor as a vector in a vector space. The tensor product is just a way of combining vectors in vector spaces to get another (and larger) vector space. So the tensor product isn’t a product in the sense that multiplication of two objects (real numbers, complex numbers, square matrices) produces another object of the exactly same kind.

So mathematicians use a special symbol for the tensor product — a circle with an x inside. I’m going to use something similar ‘®’ because I can’t figure out how to produce the actual symbol. So let V and W be the quantum mechanical state spaces of two systems.

Their tensor product is just V ® W. Mathematicians can define things any way they want. A crucial aspect of the tensor product is that is multilinear. So if v and v’ are elements of V, then v + v’ is also an element of V (because two vectors in a given vector space can always be added). Similarly w + w’ is an element of W if w an w’ are. Adding to the confusion trying to learn this stuff is the fact that all vectors are themselves tensors.

Multilinearity of the tensor product is what you’d think

(v + v’) ® (w + w’) = v ® (w + w’ ) + v’ ® (w + w’)

= v ® w + v ® w’ + v’ ® w + v’ ® w’

You get all 4 tensor products in this case.

This brings us to Postulate #2 (actually #4 on the book on p. 94 — we don’t need the other two — I told you this was fairly simple)

Postulate #2 “The state space of a composite physical system is the tensor product of the state spaces of the component physical systems.”

http://planetmath.org/simpletensor

Where does entanglement come in? Patience, we’re nearly done. One now must distinguish simple and non-simple tensors. Each of the 4 tensors products in the sum on the last line is simple being the tensor product of two vectors.

What about v ® w’ + v’ ® w ?? It isn’t simple because there is no way to get this by itself as simple_tensor1 ® simple_tensor2 So it’s called a compound tensor. (v + v’) ® (w + w’) is a simple tensor because v + v’ is just another single element of V (call it v”) and w + w’ is just another single element of W (call it w”).

So the tensor product of (v + v’) ® (w + w’) — the elements of the two state spaces can be understood as though V has state v” and W has state w”.

v ® w’ + v’ ® w can’t be understood this way. The full system can’t be understood by considering V and W in isolation, e.g. the two subsystems V and W are ENTANGLED.

Yup, that’s all there is to entanglement (mathematically at least). The paradoxes entanglement including Einstein’s ‘creepy action at a distance’ are left for you to explore — again Zeilinger’s book is a great source.

But how can it be like that you ask? Feynman said not to start thinking these thoughts, and if he didn’t know you expect a retired neurologist to tell you? Please.

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 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 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

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

https://luysii.wordpress.com/2009/12/10/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.