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)

 

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.