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# Archive for March, 2009

## Gas prices don’t hurt me today:)

Posted by peeterjoot on March 31, 2009

Filled up the tank on my little cbr125 for the first time this season. It cost me a grand total of \$4.51 … I’ll probably have to write my self a check from the line of credit account to cover the bill!

I wanted to let as much of the gas with the fuel stabilizer from this last winterization drain out, so I I’ve been holding off the fill up to let as much as possible of this tank drain out. Now that I finally filled up on a close to empty tank, my little bike is running so much better. It really hums along so much better on a full tank.

I’ve noticed that I tense up a lot while I’m riding, and wonder if this explains the knots in my back that I had last summer. As much as I like my commute on the bike, traffic plus manual gear shifting still appears to stress me out somewhat. Tommorrow I think I’ll have to take the scenic route to work. Taking that extra 3 minutes to go North is so worth it, but I haven’t done it yet this year, perhaps because it’s still rather cold. Honestly it is still colder than it is comfortable to be out in, especially with my crappy but warm GM cavalier sitting at home in the driveway, when I’m out on the bike.

Last week riding home from a downtown visit of my Dad and grandmother I swore I was going to freeze. Each time I came to a stop, my teeth were chattering, and my hands were numb feeling by the time I got home … and that’s with a sweater, my heavy (lined) leather coat, a dicky, my warm gloves, jeans and pull over motorcycle pants. Once that Sun went down it wasn’t good weather to be going more than 40kmh.

## (non)typos in byron and fuller’s mathematics of classical and quantum physics?

Posted by peeterjoot on March 26, 2009

Dover doesn’t have an errata page for this (excellent) book. I’ve noticed a lot of i’s that look like l’s in the earlier chapters, but am not sure those are typos or just printing quirks. One typo that I was initially pretty sure of was:

page 129 (QM one dimensional harmonic oscillator solved with eigenvalue methods) the Gaussian integral evaluation isn’t right (factor of 2 error).

Suppose one wants

$I = \int_{-\infty}^\infty e^{-\alpha x^2/2} dx$

then the square, after the usual polar coordinates change of variables trick, is

$I^2 = 2 \pi \int_0^\infty e^{-\alpha r^2/2} r dr = \frac{2\pi}{\alpha}$

Doesn’t this then give you a factor of two in along with the $\hbar$ in the expression for $N_0$?

Am I mistaken? Because the book is so old I’d have thought errors like this would be fixed, which makes me doubt myself somewhat.

Ah… I’m dumb! The integral is $\int {\lvert u_0 \rvert}^2$ and that square gets rid of the factor of two in the exponential. Everything is cool in the text, and writing out my bogus argument on why it was wrong resolves the internal dispute without casualties!

## interesting lecture on classical and quantum relations for electrons.

Posted by peeterjoot on March 18, 2009

I listened to a recorded lecture today by Baylis on electron spin related via active lorentz transformations to spinors, and found his paper on arxiv. It’s timely since I’ve been working on understanding both QM and E&M, and I’d like to see some of the big picture of how these fit together.

Baylis uses a complex (quaternion like) representation that I’ll probably have to translate for myself to STA or tensor form to get a feel for things. Will see how the reading of his paper goes if this is actually neccessary.

Although he didn’t mention it, I believe that he’s also covered the acceleration bivector of GAFP in his lecture where he wrote the Lorentz force equation as $\dot{p} = scalar(\Omega u)$. Something clicked for me when I saw that … it looks just like the $\Omega \cdot \mathbf{v}$ of rotational dynamics (as in Tong’s notes, or in Goldstein). Following Goldstein or Tong, this could all probably be formulated in matrix form in index notation utilizing an antisymmetric tensor. Is there a clean representation of rotors in tensor form (ie: the exponentials)?

## damn, the universe is big!

Posted by peeterjoot on March 17, 2009

Popular science had an article on Microsoft’s world wide telescope, and I thought I’d check it out. This is like google sky, but is more feature rich. I spent a while playing with the solar system views which were pretty neat, then tried out a few of the guided tours. It didn’t take long before an hour had passed, and I think that I only just glimpsed a tiny fragment of what’s out there.

I have to say, it’s amazing that astronomers can make sense of everything out there. Looking at some of the zoom-ins in an xray image tour, what looked like layer after layer of possibly interesting stuff was passed by and somehow the supernova was found hiding in there among all the other stuff!

## chapter 1 of ‘Mathematics of Classical and Quantum Physics’

Posted by peeterjoot on March 15, 2009

This is probably the largest Dover book on my shelf, weighing in at about 650 pages. I bought it for the Green’s function content. I’d gotten through bits of it, but reading it out of sequence turns out to be hard. Some of what I’m seeing now in QM books I know is covered nicely here, and on whim I grabbed it off my bookshelf in the morning this weekend, and started giving it a read from the beginning.

Now, I’d intended to skip the first chapter on Vectors in classical physics, but I am glad that I didn’t. This chapter is actually packed with a lot of good stuff.

– Rotations. Calculation of coordinates in original and rotated coordinate systems are covered. The difference between the rotation matrix for active alteration of a vector and the change of basis (expressing a vector in a rotated coordinate system) is nicely explained. I’d become confused about this a few times. Each time usually resulted in me re-deriving the rotation identities, to make sure that I got them right for the use I intended.
– A nice example of a Newton’s law (projectile trajectories) done in two coordinate systems. Working through that in detail was an interesting exercise.
– I even found the section on dot product interesting. As an exercise I did the proof that the dot product in cosine form is linear, which was fun, and illuminating.
– Levi-Civita tensor use in some cross product identities. A summation identity that looked familiar was covered, but I hadn’t ever used in a cross product context. I did the proof initially with a perl script, then realized that it was really just the dot product of bivectors expressed in coordinates.
– grad, div, curl, and Laplacian in curvalinear coordinates. This is something that I’d tried to blunder through myself, and is also covered in Geometric Algebra for Physicists, but like so much in that book, not covered in a form that one could learn from if it wasn’t already known. This is an excellent treatment of both the general curvilinear case as well as the case of an orthonormal basis. One aspect of this treatment that I found interesting was that both divergence and curl are defined in terms of integrals. Hestenes did something like this for the gradient, in one of the papers on his website, and I’d like to dig that up to compare.
– The Laplacian as a measure of the difference of the average value of the field in a neighborhood and at a point is interesting. One of the problems I noticed is to give a physical interpretation of the Helmholtz equation, and I’ll try to apply that idea to the problem. I also didn’t fully understand the description of the wave equation in terms of this average and will have to think that through a bit better.
– There’s some nice examples of tensors here that would have been good to see before trying to struggle through GAFP, and Hestenes’s NFCM. Many of the examples are the same as those covered in a GA context in those books, but learning of them without having to simultaneously learn the GA stuff would have been easier.

## some random thoughts and notes on Liboff’s and Pauli’s QM books (so far).

Posted by peeterjoot on March 12, 2009

I borrowed Liboff’s book from the public library again, figuring I’d take a second look. My initial impression of this book was that it was garbage, at least for a new learner.

This time around I know a bit more. I’ve now listened to Susskind’s QM lectures, which give a top down approach that turns out to be similar to Liboff’s (both pulling the approach out of thin air and magic hats instead of trying to show a logical progression).

The real key to understanding things was a good read of Bohm’s Quantum theory book, in particular chapters 3 and 9, and I’ve blogged about those separately.

Complementing Bohm’s book I’ve read the first two chapters of Pauli’s wave mechanics book.

Pauli’s book isn’t particularily easy to follow, but there are a number of aspects of it that I do like. One is the mini relativity primer in one page. He really knows that subject. This book is similar to Bohm’s in that all the integrals are written out in full. It has an old fashioned-ness that gets the core ideas across in a straightforward way.

I liked the way Pauli motivates the non-relativistic wave equation by making a $\sqrt{p^2 + m^2 c^2}$ approximation starting from the relativistic scalar wave equation. That approximation ends up with an extra $\partial^2 \psi/\partial t^2$ term that the normal Schrodinger equation doesn’t have, and later when he switches to the regular non-relativistic Schrodinger equation he just drops this term. It will be interesting once I learn about the Dirac equation to see how the transition between the Klien Gordon and that equation works.

Pauli’s treatment of the uncertaintly relation is direct and clever and a classic example of working backwards from an answer to get to the solution in the fastest possible way. The neat thing about his way of doing it is how easily it shows how equality follows only for the Gaussian distribution.

I have to admit that I didn’t fully understand his measuring arrangement examples (the microscope and coherency parts in particular). Perhaps I got hung up on not understanding where the Abbe sine condition $\Delta x \approx \lambda/\sin\epsilon$ came from. The relativistic particle and light wave collision example is quite neat, but also highlights the fact that I really have to go and give that Hamilton’s equation treatment in Goldstein a good read. He used those ideas in his relativity review as well.

In Chapter II of Pauli’s book he has got a pretty slick treatment of 3D eigenfunction orthogonality for a boundary where the probability current is zero.

His completeness relation coverage was not particularily easy to follow. I wrote up my own notes on this separately, and managed to make sense of it for myself.

The final bits in chapter II, the “initial value problem and fundamental solution” I’m going to have to come back to. I can follow it step by step, but it’s not particularily easy. The end result makes sense. He comes up with a Gaussian form of the delta function by considering an impulse response. I’m not sure exactly how to relate all that to things that I currently know, and will probably have to try to work this problem myself at my own pace before I’ll really get it. Perhaps to make it different and avoid just spitting out exactly what he did I should try it myself in 3D.

I had intended to drop a few thoughts about Liboff’s book and instead dug out all my thoughts on Pauli’s chapter 1 and 2. Ah well. So, back to Liboff…

In this second look, after picking up some motivation and details in other sources, I rather like what I’ve read so far. It’s true that he has absolutely no attempt to motivate the ideas, but if you get that elsewhere what he does cover is done well. What did I like about his book?

The one dimensional momentum operator, so well motivated by Bohm, and so randomly stated by Liboff is actually a nice example of a one dimensional wave equation (ie: when stated in eigenvalue equation form). How he uses this as a simple example of the Helmholtz equation is also nice. That you can get the Helmholtz equation from separation of variables of the regular mechanical (or E&M) wave equation as well as from the QM wave equation makes some sense, so it’s nice to see this pointed out to see commonalities between the different parts physics.

Liboff’s treatment of the initial value problem is very nice. Much easier to understand than Pauli’s delta function variant and it shows a concrete example of a unitary operator $\hat{U} = \exp(-i t \hat{H}/\hbar$ that clarifies what Susskind was getting at in his lectures. Here the unitary operator (although it wasn’t named as such) is expressed as an integrating factor, and encodes all the time evolution of the initial state. Quite slick, and his example where he considers the eigenstates of the Hamiltonian really firms it up and takes the abstraction out of things.

Some notes on problems in Liboff:

Problem 3.5 is kind of nice, put a number to the energy. I calculated a value of about 150 eV for the 10^-8 cm wavelength electron. Wonder if I got that right? I got lazy with units and did some of the calculation with google calculator.

Problem 3.6 on the delta function properties. I’m unsure about how to show (b). $\delta'(x) = -\delta'(-y)$. The others I did, but didn’t see how to do (e) until tacking the more general and abstract varient of it (i), which is amusing.

Problems 3.16-3.23 (all the end of chapter problems). Have only done the first so far (that one I could do in my head without paper while I was in the hot tub;) The rest will have to wait for tommorrow.

## Some comments on Bohm’s Quantum Theory text.

Posted by peeterjoot on March 2, 2009

Did some more reading of Bohm’s book Bohm’s book this weekend.

A couple weeks ago I had initially been super impressed with his Wave Packets and De Boglie Waves chapter (ch. 3), where he motivates the free particle wave equation. Unlike so many other Schrodinger equation motivations I’d actually call this one a derivation. It is such a logical sequence of arguments showing how to build on ideas already familiar and produce the QM equation. All the prerequisite ideas are covered, and each one so well. A small example is his excellent coverage of what group velocity is mathematically (first order term in a Taylor expansion of $\omega(k)$) … reading that and his subsequent gaussian pulse deformation example made things so much clearer than the fuzzy ideas I can remember from my study of French’s vibrations and waves in school.

Based on $E = h nu$ for the energy of a wave packet, and the requirement that the group velocity of the wave packet matches the classical particle velocity, he shows how the angular frequency of the wave packet must be $\omega = \hbar k^2/2m$. This defines the time dependent phase change $exp(i \omega t)$ of the wave packet. Based on a Fourier propagator expression of the wave packet based on the wave packet at an initial time, and the angular frequency just determined he shows how the potential free wave equation describes this wave packet. Looking back on this treatment, it is perhaps not the most natural thing to do if one hadn’t seen the wave equation before … in a sense he is working backwards from the answer. Everybody else does this too in their “derivations” of the wave equation, but this one is, by far, the best approach.

Now, would I have liked this as much if I wasn’t comfortable with Fourier transforms (I’ve been playing with them recently in 1, 3 and 4 dimensions in the context of calculating propagator solutions to Maxwell’s equations). No. The whole book justifiably requires Fourier theory as a prerequisite, so this would have not likely been satisfying back in school in our dinky engineering intro QM course since I took that well before my systems and signals course where we covered that in detail (or as much detail as you can do without covering distributions).

Now, after deriving the free particle wave equation, Bohm takes FIVE chapters to discuss this and some of the related ideas, mostly in words and minimal equations before moving on to the more general wave equation in chapter 9. I’ve never seen any physics text do anything like that (with the exception of Feynman’s Lectures, which aren’t a traditional text).

After these five chapters comes the masterpiece, his Wave functions and operators chapter (ch 9). Here we have an extension of the free particle treatment, and the construction of the mathematical toolbox required to get there.

There’s a fair amount in the toolbox that’s required, but Bohm’s treatment cuts to the heart of it. Rather than focus on the cute notational gimmicks that you’ll find treatments by those already comfortable by all the ideas (eg: Susskind’s QM lectures, or a text like Liboff‘s), this treatment is minimal. Average values are all written out in full using integrals, just probability weighted sums.

The operator form of momentum, $p \sim -i \hbar \partial/{\partial x}$, isn’t pulled out of a magic hat, but just shown to be a consequence of an inverse Fourier transform representation given the momentum space wave function. Interesting, he shows the same thing for the position operator when expressed in the momentum space ($x \sim i \hbar \partial/\partial k$). One immediately sees the hints of the Hamiltonian/Poisson-bracket structure that Susskind talks about in his lectures. Bohm introduces the commutator, Hermitian operators, and Hermitian conjugate nicely. Like the averages, these are all expressed in full integral form, which is very clear … no notational gunk to get in the way. Having seen this the long way the value of the fancier notation becomes clear, but for a first time through the content (or a time through the content after struggling to make sense of things in the bra/ket formalism) this just makes so much sense.

Seeing the Hermitian conjuate expressed this way, I can see why there’s both a conjugate and Hermitian conjugate in QM (with distinct symbols). Soon I may be equipt to go back to Doran/Lasenby and make some sense of it.

I enjoyed seeing where the requirement for $H$ to be Hermitian came from (in order to conserve probability), and also seeing how naturally the Hamiltonian/Operator commutator arrives from looking at the time rate of change of an operator’s average value. This is the Poisson structure mentioned in Susskind’s classical lectures, and it’s not actually all that complex to get to it.

The finale of the chapter was using the idea of Ehrenfest’s theorem (average wave packet behaviour in classical correspondence with Newton’s laws) to complete the QM wave equation derivation is just masterful. Seeing it done this way without the obstruction of the bra/ket notation is really nice, and to simultaneously get the Schrodinger equation for free tops the cake!

All in all this book is a supreme pleasure to read! Other than the Student’s guide to Maxwell’s equations, I’m not sure I’ve seen a better crafted physics text.