Archive for the ‘Quantum Mechanics’ Category

Lectures on Classical and Quantum Physics

December 13, 2009

The Indian Institute of Technology Madras has some online collections of lectures. Among them, a Quantum Physics course imparted by Prof. V. Balakrishnan. You can see the first lecture below, devoted to an introduction to the conceptual underpinnings of quantum mechanics and, more specifically, Heisenberg’s principle. Prof. Balakrishnan does a remarkable job at explaining these horny issues, avoiding common pitfalls; and, as the students’ questions make him wax philosophical, he shows a very refreshing honesty in not trying to sell the theory as the end of the path, avoiding the “shut up and calculate” stance that was so common during the second half of the past century (and which i still suffered during my undergraduate and graduate years). The exposition doesn’t use any advanced mathematics and keeps at a conceptual level, and i think it should be understandable by anyone with a very modest background (perhaps some acquaintance with classical mechanics will help at some point, but it’s not a requirement to enjoy the lecture).

There’re are thirty more lectures in the course, all of them available for your learning pleasure. Although i haven’t had the time to watch them all, if Prof. Balakrishnan keeps their quality as high as in the first installment, i’m pretty sure they’ll make up for many hours of fun.

And, if you feel like learning classical physics, i’ve got good news for you too: there’s also a series by Balakrishnan on Classical Physics! Here’s the first lecture:

and here you can find 37 more!

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Quantum Theory at the Crossroads  Conference

September 29, 2006

Guido Bacciagaluppi (from Berkeley’s Philosophy Department) and Antony Valentini (from the Imperial College of London) are about to publish a 500 pages long book entitled Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, and they’ve been kind enough to make a draft copy publicly available: just follow the link. Not that i’ve had time to read it yet, but its abstract looks all but promising:

We reconsider the crucial 1927 Solvay conference in the context of current research in the foundations of quantum theory. Contrary to folklore, the interpretation question was not settled at this conference and no consensus was reached […] [W]e provide a complete English translation of the original proceedings (lectures and discussions), and give background essays on the three main interpretations presented: de Broglie’s pilot-wave theory, Born and Heisenberg’s quantum mechanics, and Schroedinger’s wave mechanics. We provide an extensive analysis of the lectures and discussions that took place, in the light of current debates about the meaning of quantum theory. The proceedings contain much unexpected material, including extensive discussions of de Broglie’s pilot-wave theory (which de Broglie presented for a many-body system), and a “quantum mechanics” apparently lacking in wave function collapse or fundamental time evolution.

Chances are this book will make its way into any recommendation on required QM readings in no time!

Students and quantum mechanics

July 28, 2006

The latest issue of Physics Today has a freely available article that, under the title Improving students’ understanding of quantum mechanics, gives a very interesting analysis of student’s difficulties when faced with university courses in Quantum Mechanics:

Extensive testing and interviews demonstrate that a significant fraction of advanced undergraduate and beginning graduate students, even after one or two full years of instruction in quantum mechanics, still are not proficient at those functional skills. They often possess deep-rooted misconceptions about such features as the meaning and significance of stationary states, the meaning of an expectation value, properties of wavefunctions, and quantum dynamics. Even students who excel at solving technically difficult questions are often unable to answer qualitative versions of the same questions.

The article goes on describing little problems posed to students and how they revealed fundamental misconceptions (if you know a bit about QM you may find interesting to try your hand at them too), and proposing ways of improving their understanding by means of interactive software and tutorials (more fun ahead). Recommended.

Litmus test for QM interpretations

June 29, 2006

Chances are that, if you are interested in the foundations of Quantum Mechanics, you already now about Matt Leifer and his excellent blog, Quantum Quandaries. But just in case, i wanted to point out (and strongly recommend) his latest post about the criteria that any interpretation of QM should meet to be considered as a worthy contender in this crowded arena.

Matt makes his points so well that i won’t belabour further on what he says, but just comment his mentioning one of the issues that have been worrying me as of late, namely, the correct interpretation of probability in QM (or what he calls the great probabilty debate). The issue is also discussed in other recent post by Matt himself, which mentions a recent and interesting paper (also discussed over at Guide to Reality) sort of making the case for Bayesian approaches and debunking frequentist ones. The post goes on to tackle a (to me) intriguing issue: the connection between many-worlds interpretations and probability. For many years i had been reading popular accounts about Everett’s interpretation without realizing that nobody was explaining me a fundamental issue: how does the constant unfolding of new worlds account for Born’s rule of probability distribution? What makes some words more probable than others if all of them exist? When later on i read more technical descriptions i found the, so to speak, interpretations of the interpretation less than compeling. But, again, i don’t need to further comment on them: Matt is already giving a pretty good account, with pointers to further reading.

Quantum mechanical quotes

June 9, 2006

From the revamped This Quantum World site, an all but sobering quotes collection:

  • Quantum mechanics is magic. Daniel Greenberger.
  • Everything we call real is made of things that cannot be regarded as real. Niels Bohr.
  • Those who are not shocked when they first come across quantum theory cannot possibly have understood it. Niels Bohr.
  • If you are not completely confused by quantum mechanics, you do not understand it. John Wheeler.
  • It is safe to say that nobody understands quantum mechanics. Richard Feynman.
  • If [quantum theory] is correct, it signifies the end of physics as a science. Albert Einstein.
  • I do not like [quantum mechanics], and I am sorry I ever had anything to do with it. Erwin Schrödinger.
  • Quantum mechanics makes absolutely no sense. Roger Penrose.

Serendipity

June 3, 2006

Reading the recent article Electrons Act Like Waves (from the Physical Review Focus series, a highly recommended, lay[wo]man-friendly feed for your newsreader), i've discovered one of those peculiar stories that make the history of physics even more enjoyable than it would be by its purely scientific side alone.

The article tells the story of Davisson and Germer's discovery of the so-called wave-like nature of electrons. As explained in every textbook, they set up an experiment consisting in scattering electrons through a (nickel) crystal, and observed the familiar fringes that one obtains when a wave crosses a gratting with alternating slots in a wall.

SlitIt was 1927, and i always pictured Davisson and Germer as intrepid experimenters boldly trying to confirm de Broglie's 1924 ideas about the wave nature of matter [1]. (Justly enough, an idealisation of this experiment has become the de facto standard presentation of the quantum mechanical world!.) The funny thing is that this romantic picture has nearly nothing to do with what really happened. As it comes, D&G were looking for evidence of the atomic structure of metals and knew nothing about de Broglie. After the experiment had been going on for a somewhat sterile period, one of their widgets broke and overheated the nickel plaits, which crystalised and made (when used again for scattering electrons) the interference patterns apparent. The experimenters were all but bewildered, and only after Davisson discussed his results with other colleagues during a holiday in England, did they realize the importance of their discovery.

That's serendipity at its best. And, of course, it was not the first nor the last time that serendipity gave physicists a helping hand. The Oxford Dictionary gives a precise definition of this beautiful word:

serendipity noun the occurrence and development of events by chance in a happy or beneficial way.

or, even better, this one from Julius H. Comroe (as quoted by Simon Singh)

Serendipity is looking for a needle in a haystack and finding the Farmer's Daughter.

and its all too apt etymology:

ORIGIN 1754: coined by Horace Walpole, suggested by The Three Princes of Serendip, the title of a fairy tale in which the heroes “were always making discoveries, by accidents and sagacity, of things they were not in quest of.”

which in my opinion captures extremely well the kind of discoveries we're discussing. They were by chance, that's true, but not just chance: one needs to be in the quest of something, to begin with.

Another famous (and probably better known) example of serendipity at work is Penzias and Wilson's discovery of the cosmic microwave background. As explained by Ivan Kaminow,

He [Ivan] joked that Penzias was an unusually lucky guy. "Arno Penzias and Bob Wilson were trying to find the source of excess noise in their antenna, where pigeons were roosting," he said. "They spent hours searching for and removing the pigeon dung. Still the noise remained, and was later identified with the Big Bang."He laughed, "Thus, they looked for dung but found gold, which is just opposite of the experience of most of us."

The experiment was being conducted at Bell's Labs and its aim was to tune an ultra-sensitive microwave receiving system to study radio emissions from the Milky Way. It was only after Penzias talked with Robert H. Dicke (see also this nice memorial (PDF) for more on Dicke) that the misterious radiation was recognized as the relics of the Big Bang hypothesized by George Gamow some time before. I read the whole story for the first time in Weinberg's marvelous book (required reading), and I've always found a bit unfair that the Nobel prize went only to Penzias and Wilson.

My third serendipitous example comes also from the skies. In summer of 1974, Russell Hulse was a 23-year-old graduate student compiling data from the Arecibo Observatory radio telescope in Puerto Rico. The job was a little bit tedious: he was trying to detect periodic radio sources that could be interpreted as a pulsar [2]. One of the pulsar's earmarks is its extraordinary regularity (a few nanoseconds deviation per year for a period of about a second). Around 100 pulsars were known back then, all with a stable period with a extremely slow tendency to increase. At the end of the day, the data obtained by the telescope was processed by a computer program written by Russell, which selected candidate signals based on the stability of their period. Those were correlated with later or former observations of the same sky zone, to rule out earth-based, spurious sources. One night, Russell boringly noticed a very weak candidate, so weak that, had it been a mere 4 percent fainter, it would have passed unnoticed. On top of that, its period was too short (about 0,06 seconds) and, even worst, it was variable. Russell was on the verge of discarding it more than once during the following weeks, but eventually he persevered and, helped by his supervisor, Joe Taylor (a.k.a. K1JT), correctly interpreted the observation as a binary pulsar. The rest is history, and a Nobel prize [3] one. Russell tells the amazing story in his delicious Nobel lecture (PDF), which starts with these telling words:

I would like to take you along on a scientific adventure, a story of intense preparation, long hours, serendipity, and a certain level of compulsive behavior that tries to make sense out of everything that one observes.

I specially like this instance of serendipity, for it shows that, many a time, lucky strikes befall on those who work hard enough to get hit.

Update: I've just found an excellent article by Alan Lightman, Wheels of Fortune, which gives some very nice examples of serendipitous discoveries, as well as a nice discussion. After reading Michael post on serendipity in HEP, i was wondering about non-experimental lucky strikes, and Lightman gives an excellent example: Steve Weinberg's electroweak theory:

Serendipitous discovery strikes not only in the photographic plates, test tubes, and petri dishes of the laboratory. It also can strike in the pencil-and-paper world of theoretical scientists. In the fall of 1967, theoretical physicist Steven Weinberg was working out a new theory of the so-called “weak force,” one of the four fundamental forces of nature, when he discovered, to his surprise, that his new theory was actually two theories in one. Weinberg was approaching the weak force with the seminal idea that pairs of particles it acted upon, electrons and neutrinos for example, might be identical as far as the force is concerned, just as yellow and white tennis balls are identical as far as the game of tennis goes. When he cast this idea into the mathematical language of quantum physics, Weinberg found that his theory necessarily included the electromagnetic force as well as the weak force. The mathematics required the union of the two forces. As he later remarked, “I found in doing this, although it had not been my idea at all to start with, that it turned out to be a theory not only of the weak forces, based on an analogy with electromagnetism; it turned out to be a unified theory of the weak and electromagnetic forces.” 


[1] As an aside, i find the constant chatter about matter being some sort of schizophrenic mix between particles and waves misleading, if not outright wrong. As stressed (to no avail, it seems) by Feynman (see and hear him on this and much more in his Vega Lectures, for instance), electrons (and photons, for that matter) are particles. You never detect half an electron, or a pi-fold-photon. There's always ticks in a detector (a photo-multiplier, a photographic plate, or trails in a Wilson chamber, for instance). The wave function is not real (neither in the physical nor in the mathematical sense of real), and it 'oscillates' in an imaginary space which is not even 3-dimensional when more than a particle is described. The interference patterns observed (which arise from the addition of complex amplitudes which are squared afterwards) are not associated with single electrons, the only thing wavelike (with a twist) about them being the statistics of their hits on the wall. Even if you believe in Bohm's pilot waves, the particles are still particles! Of course, there's ample room for analogy, but i still find the typical discussions misleading.

[2] The discovery of pulsars had also its share of serendipity. They were found, also unexpectedly, by Jocelyn Bell and Anthony Hewish while they were looking studying scintillating radio signals from compact sources. Jocelyn has written a lively report of their discovery, including the funny story of how they were on the verge on attributing the signals to extraterrestrials, and jokingly use monikers starting with the prefix LGM (for little green men) to name the misterious radio sources. There's also a good review of the tale over at the Hitchhiker's Guide to the Galaxy funny website.

[3] The pulsar discovery also won a Nobel in 1974. But, curiously enough, the undergraduate hero of the story (Jocelyn Bell) was not awarded this time. One wonders.

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The cyclist team

May 29, 2006

 Fieldtheory Figs FeynmancovGeorgia Tehch’s Pedrag Cvitanovic’ and friends write physics and maths books under the nome de guerre of The cyclist team. These books are interesting in many ways. First of all, they are comprehensive and of excellent quality, although, fortunately, these are not extremely rare in the field. What is not so common is the dose of humour and engaging wittiness distilled in their pages. And, besides, they’re being written over a span of many years in a totally public manner: you can view and download them in PDF of Postscript and are frequently updated. As they explain in their webbook rationale, they don’t even plan to ever publish them:

The relevant parts of a good text will be printed and perused, no less than a good electronic preprint. A bad text should be junked anyway. If a student in Buenos Aires or Salamanca reads a chapter and is wiser for it, that is all it takes to make us happy. The webbook has done something to further little piece of wisdom that we know and love.

The oldest book is Pedrag’s (Quantum) Field Theory, and its companion (and more modern) lecture notes: Quantum Field Theory, a cyclist tour. In Pedrag’s own words:

Relax by reading Classics Illustrated, diagrammatic, Predragian vision of field theory. The exposition assumes no prior knowledge of anything (other than Taylor expansion of an exponential, taking derivatives, and inate knack for doodling). The techniques covered apply to QFT, Stat Mech and stochastic processes.

As is a norm, the book’s site contains many other bits of additional information, including the delicious fable of Quefithe.

Next comes the Group Theory Book, which, under the subtitle of Birdtracks, Lie’s and Exceptional Groups and spanning almost 300 pages, will tell you all you’ll ever need about Lie Groups and Algebras. This nice PDF presentation makes for a good summary of its contents, or, as Pedrag says, of “most of the Webbook at a cyclist pace, in 50 overheads” (see also here for more short intros). In case you’re wondering, birdtracks are to Lie Groups what Feynman’s diagrams to QFT, and then more. As you can see, Pedrag loves diagrams and pictures, in a way that reminds me of Penrose’s fondness for geometrical descriptions (actually, birdtracks have many a point in common with Penrose’s diagrammatic tensor notation, who even wrote a letter claiming his precedence on it). And, again, don’t miss the book’s site for lots of additional goodies.

Finally, there’s the Chaos Book, probably my favourite. Again, the authors introduce it far better than i would:

Quite a few excellent mathematics monographs on nonlinear dynamics and ergodic theories have been published in last three decades. On the whole, they are unreadable for non-mathematicians, and they give no hint that the theory is applicable to problems of physics, chemistry and other sciences.
By now, there are also many physics textbooks on “chaos”. Most lack depth, and many of them are plain bad, emphasizing pictorial and computer-graphics aspects of dynamics and short changing the student on the theory. That’s a pity, as the subject in its beauty and intellectual depth ranks alongside statistical mechanics and quantum field theory, with which it shares many fundamental techniques. The book represents authors’ attempt to formulate the subject as one of the basic cornerstones of the advanced graduate physics curriculum of future.

The amount of additional information for this book is almost overwhelming, including computer programs, additional exercises (the book itself contains many) and a long list of projects written by students. I won’t try to summarize the wide range of themes covered by the book (here you have the table of contents of its three volumes–classical chaos, quantum chaos and appendices), but a very good way of getting a glimpse of its scope and fun style is reading its Overture (PS). An amazing way to become acquainted with an amazing subject!

The cyclist team

Quantum probability

May 22, 2006

I just stumbled upon John Baez’s page on Bayesian probability and Quantum Mechanics, which nicely summarizes one of the first difficulties i had with the latter: Born’s interpretation of the wave function as a probability. The problem hinged on my naive (frequentist) interpretation of probabilities, and the conclusion that QM describes only ensembles, not individual systems. For to compute the probability of an experiment’s outcome, i reasoned, you need to repeat the experiment a large number of times. Then, counting the number of times your outcome happens and dividing by the total number of repetitions one obtains the sought for probability. Problem is, what is large? Well, nothing short of infinite, it seems. Because, with this frequentist definition of probability, nothing prevents your tossing a coin a hundred times and getting a hundred tails. And such a situation may still be compatible with a half and half probability for heads and tails! My unsettling conclusion was that QM predicts nothing at all about individual systems! Come to think of it, it doesn’t even predict anything about finite ensembles.

One way out of this conundrum is Everett’s many worlds interpretation: since all possible outcomes really happen, frequentist probabilities are well-defined. I still remember being genuinely surprised when i learnt that there existed serious attempts at making sense of such an idea. I still am. John gives an excellent argument to be done with this peculiar interpretation:

Here is a sample conversation between two Everettistas, who have fallen from a plane and are hurtling towards the ground without parachutes:

Mike: What do you think our chances of survival are?

Ron: Don’t worry, they’re really good. In the vast majority of
possible worlds, we didn’t even take this plane trip.

A second way out is revising our definition of probability. We forget (initially) about frequencies, and take a Bayesian stance. In a nutshell, Bayesian probability is not measured from scratch because it is defined as a degree of belief on a given outcome. One starts with an a priori value for such a belief, and revises it (if needed) according to experiment. The gist of it is that Bayes’ theorem lets you calculate the likelihood of future outcomes based solely on your a priori probabilities. So, the tale goes, when a wave function collapses as a result of a measurement, there’s nothing real out there undergoing a physical collapse; it’s only that we have improved our knowledge of the system and must update our a priori likelihood assignments accordingly. This view mixes well, by the way, with the orthodox Copenhagen interpretation of QM, which also denies an objective reality of the wave function.

The so called relational interpretations have, i think, a clear Bayesian substrate. Probably the best known relational theory nowadays is Rovelli‘s, whose recent paper Relation EPR (nicely reviewed in Alejandro’s blog) has been widely discussed elsewhere.

While i have nothing against Bayesian probability for describing our knowledge of any system, considering it as a final interpretation of QM makes me feel uneasy. I’d rather have a theory which describes something out there, some kind of (possibly inter-subjective) reality. Atoms, stars and the whole universe seem to care little about our knowledge of them, and the quantum mechanics rules look a bit too simple to explain, out of the blue, our way of acquiring information about the world. I would rather put my money on some sort of objective, physical reduction of the state vector, maybe along the lines of some non-linear modification of Schrödinger’s equation (and probably not as fancy as Penrose’s objective reduction, but who knows!). Call me a (perhaps non-local) realist.


One last thing. One of the best ways to learn about Bayesian theory is from “Probability Theory : The Logic of Science” (E. T. Jaynes). The good news is that a draft version of it is available online. (See also Matthew Leifer’s comment below recommending Bruno de Finetti‘s work.)

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The music of emergence

May 2, 2006

I just stumbled upon a beautiful site, The Music of the Quantum, whose (apparent) main theme is a peculiar composition by Jaz Coleman. It was commissioned by the Institute for Complex Adaptive Matter (ICAM) as a public outreach event, first performed in New York at Columbia University in 2003. The event was performed by the Sporcl quintet from Prague, and narrated by Robert Laughlin and Piers Coleman (yes, it was that 1998 Nobel laureate Laughlin, for his theory of the fractional quantum Hall effect). According to the site,

The piece was written to bring out, musically, some of the themes of the quantum emergent world. The melody of this unique piece is carried between a violin and an accordion, the idea being to capture the duality of quantum mechanics between these two contrasting instruments.

Besides hearing to the (pretty good, to my taste) music, you can see three nicely done video clips of its perfomance, and an interview with Coleman. But that’s not all.

As it happens, the site has a sort of double agenda, and is full of information on what one may call the emergent viewpoint of physics, championed (among many others) by Laughlin, the ICAM and friends. I first read more or less seriously about this viewpoint a few months ago, via Laughlin’s very interesting “A Different Universe: Reinventing Physics from the Bottom Down”, which was a bit of an eye opener to me. As a theoretical physicist, i’ve had a reductionist upbringing. When i was in high school, a dear maths prof of mine’s used to tell me that i was what Einstein called (in this classic article) a tamed metaphysicist:

I believe that every true theorist is a kind of tamed metaphysicist, no matter how pure a “positivist” he may fancy himself. The metaphysicist believes that the logically simple is also the real. The tamed metaphysicist believes that not all that is logically simple is embodied in experienced reality, but that the totality of all sensory experience can be “comprehended” on the basis of a conceptual system built on premises of great simplicity. The skeptic will say that this is a “miracle creed.”

… and i felt i really was (and probably still am) one of those beasts. From that stance to reductionism there’s just a tiny step: to me, physics was the pursue of ultimate causes, the art of reducing complex systems to its constituents and explaining everything in terms of the interactions between those constituents. A very naive philosophy, if you like, but one that is reinforced by many science books and academic curricula, and which is implicit in much of the research in fundamental physics even these days (for instance, Steven Weinberg’s “Dreams of a Final Theory” is a perfect exponent of this ideology). In my experience, there are still many theoretical physicists that look at colleagues in experimental physics, biology or chemistry over their shoulders, feeling like some sort of priesthood in search of the ultimate truth. But, hopefully, maybe i’m just overreacting, as usual.

Anyway, people like Laughlin have a very different worldview, and are all for explaining natural phenomena in terms of emergent behaviours, that is, properties that appear, as a consequence of organizational principles, when great numbers of, say, atoms are put together. Take, for instance, metals: according to this view, there’s nothing in a gold atom that explains its macroscopic qualities, which appear only when you put many of these little pieces together and let them interact. The laws according to which these swarms of subsystems organize themselves are not to be viewed as a direct consequence of their structure, and in fact the claim is that the behaviours come from organizational laws that are independent of the detailed structure of those constituents (or is, at least, compatible with many different ones). The website contains several (short) clips where these ICAM guys give a far better explanation of these and similar ideas. I find them very refreshing and a good antidote against narrow-mindedness. Which does not mean, by the way, that one has to accept all this emergent worldview uncritically. Laughlin likes to call reductionism an ideology, but, to be fair, reading his book now and then i felt he was close to making emergency an ideology too (in the sense that, according to him, everything seems to be explainable in terms of emergentism)! It must be the reductionist in me.

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A Quantum Game

April 27, 2006

More quantum non-local amusements, this time in the form of A Quantum Game based on Greenberger-Horne-Zeilinger three particle entanglements: try to find the (perhaps by now not so) surprising answers before following the links to them. Those links are, by the way, to a blog, koantum matters, written by Ulrich Mohrhoff, a physics professor with some definitely non-mainstream ideas about the interpretation of quantum mechanics. As a matter of fact, i was on the verge of classifying him as a crackpot, but after reading his review on the so-called Pondicherry interpretation of QM, i must say that he’s an honest physicist doing a remarkable and original job. Of course, it’s for you to decide whether his ideas hold any water.

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