Tag Archives: Schrodinger equation

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

“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.

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.