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Category Archives: Quantum mechanics

Intermezzo: The Dirac Equation

After you’ve been Pharyngulated a couple times, you develop a protective strategy to deal with the aftermath. “How,” you ask yourself, “can I get rid of the extra readers whom I’ve probably picked up?” The answer, for me at least, is clear:

Math!

RECAP

Science After Sunclipse has been presenting an introduction to supersymmetric quantum mechanics. This area of inquiry stemmed from attempts to understand the complicated implications of supersymmetry in a simpler setting than quantum field theory; just as supersymmetry began in string theory and developed into its own “thing,” so too has this offshoot become interesting in its own right. In a five-part series, we’ve seen how the ideas of “SUSY QM” can be applied to practical ends, such as understanding the quantum properties of the hydrogen atom. I have attempted to make these essays accessible to undergraduate physics students in their first or possibly second term of quantum theory. Having undergraduates solve the hydrogen atom in this fashion is rather unorthodox, but this is a safe kind of iconoclasm, as it was endorsed by three of my professors.

The posts in this series to date are as follows:

Having solved the “Coulomb problem,” we have attained a plateau and can move in several directions. The solution technique of shape-invariant partner potentials is broadly applicable; virtually all potentials for which introductory quantum classes solve the Schrödinger Equation can be brought into this framework. We can also move into new conceptual territory, connecting these ideas from quantum physics to statistical mechanics, for example, or moving from the non-relativistic regime we’ve studied so far into the territory of relativity. Today, we’ll take the latter route.

We’re going to step aside for a brief interlude on the Dirac Equation. Using some intuition about special relativity, we’re going to betray our Vulcan heritage and take a guess — an inspired guess, as it happens — one sufficiently inspired that I strongly doubt I could make it myself. Fortunately, Dirac made it for us. After reliving this great moment in TwenCen physics, we’ll be in an excellent position to explore another aspect of SUSY QM.

REFRESHER ON RELATIVITY

Let’s ground ourselves with the basic principles of special relativity. (Recently, Skulls in the Stars covered the history of the subject.) First, we have that the laws of physics will appear the same in all inertial frames: if Joe and Moe are floating past each other in deep space, Joe can do experiments with springs and whirligigs and beams of light to deduce physical laws, and Moe — who Joe thinks is moving past with constant velocity — will deduce the same physical laws. Thus, neither Joe nor Moe can determine who is “really moving” and who is “really standing still.”

Second, all observers will measure the same speed of light. In terms of a space-time diagram, where time is conventionally drawn as the vertical axis and space as the horizontal, Joe and Moe will both represent the progress of a light flash as a diagonal line with the same slope. (This video has some spiffy CG renditions of the concept.) To make life easy on ourselves, we say that this line has a slope of 1, and is thus drawn at a 45-degree angle from the horizontal. This means we’re measuring distance and time in the same units, a meter of time being how long it takes light to travel one meter.
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Science Illiteracy of the Day

BPSDBToday’s installment of “We’re so ignorant about basic science you couldn’t make up the crap we say if you tried” comes from Y-Origins Connection, a magazine which uses “dramatic photos and contemporary graphics” to explain “both sides of the intelligent design debate,” namely the creationist side and the creationists’ view of the scientists’ side. This comes from their website, right up top:

Quantum mechanics has revealed that our material world is based upon an invisible world of subatomic particles that is totally non-material. And over 95% of our universe consists of dark matter and energy that is beyond scientific observation. Also, scientists are openly discussing dimensions beyond ours where walking through walls and teleportation could be realities. The dilemma for materialists is that these areas are beyond the purview of science.

They managed to pack at least one kind of wrong in each sentence. I’m impressed. The overall theme seems to be taking discoveries of science and claiming them to be beyond science. When that well of inspiration runs dry, they take bits of overheard science jargon (hep talk like “extra dimensions” or “quantum teleportation,” let’s say) and throw them together without regard to their meaning. Truly they are strong in the art of nonsense-fu.

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SUSY QM 5: The Hydrogen Atom

Last time, we found that the problem of the hydrogen atom could be split into a radial part and an angular part. Thanks to spherical symmetry, the angular part could be studied using angular momentum operators and spherical harmonics. We found that the 3D behavior of the electron could be reinterpreted as a 1D wavefunction of a particle in an effective potential which was the two-body interaction potential plus a “barrier” term which depended upon the angular momentum quantum number. Today, we’re going to solve the radial part of the problem and thereby find the eigenstates and eigenenergies of the hydrogen atom.

The technique we’ll employ has a certain charm, because we solved the first part, the angular dependence, using commutator relations, while as we shall see, the radial dependence can be solved with anticommutator relations.
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An Item of Prime Importance

I was busy with something or other, so I didn’t get to see the event Dennis Overbye describes in the New York Times, where the director and the star of the new film Jumper chatted with MIT professors Ed Farhi and Max Tegmark before a live audience in lecture hall 26-100. Not having been there, I don’t have very much to say, but I do feel the need to quote one paragraph of Overbye’s article and add just a tiny bit of emphasis:

The real lure, [Farhi] said, is not transportation, but secure communication. If anybody eavesdrops on the teleportation signal, the whole thing doesn’t work, Dr. Farhi said. Another use is in quantum computing, which would exploit the ability of quantum bits of information to have different values, both one and zero, at the same time to perform certain calculations, like factoring large prime numbers, much faster than ordinary computers.

Ahem.

In any other circumstance, I’d probably pontificate on how exponential parallelism is not the source of quantum computing’s calculation-fu, but I think we have a few other points to address first. . . .

Attentive readers will recall that Dr. Farhi was the guy who signed my paperwork when I was an undergraduate. For an amusing story from those days, see my post of last November, “Pay No Attention to the Man.”

(Tip o’ the fedora to Dave Bacon.)

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SUSY QM 4: Separation of Variables

Today, on a very special episode of Science After Sunclipse, Mary Sue discovers that for a quantum-mechanical system with a central potential, the eigenfunctions of the Hamiltonian can be separated into radial and angular factors, and the angular dependence can be understood using angular momentum operators.
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The Name is Planck, Max Planck

The algorithm informs me that the title for the next James Bond movie, the sequel to the franchise reboot Casino Royale (2006), will be titled Quantum of Solace. This title comes from a short story in Ian Fleming’s collection For Your Eyes Only (1960); the movie of that name used elements from two stories in the book, a third story became part of Licence to Kill (1989), and the title of a fourth story was affixed to the film A View To A Kill (1985).

The title “Quantum of Solace” appears in the story as the smallest possible unit of human compassion. The following paragraph appears in both the Everything2 article on the story (dated 6 March 2001) and today’s Telegraph piece about the movie:

The crux of the story is the emotional phenomenon the Governor calls the Quantum of Solace, the smallest unit of human compassion that two people can have. As long as that compassion exists, people can survive, but when it is gone, when your partner no longer cares about your essential humanity, the relationship is over.

Well, OK, Lucy Cockcroft’s story in the Telegraph doesn’t have the words “of the story” following “crux.” Eit!

Popularizers of physics are going to have a field day with this one. Perhaps, just perhaps, we’ll finally have an example of quantum meaning small!

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SUSY QM 3: The Two-Body Problem

Our goal in this series is the solution of the hydrogen atom using the methods of supersymmetric quantum mechanics. Last time, we constructed the following picture of the procedure:

If the potential we wish to study satisfies a certain criterion, which we called “shape invariance,” we can construct a hierarchy of Hamiltonians, each missing the lowest-energy eigenstate of the last, and find the complete spectrum of the original Hamiltonian by “working leftward” in the state diagram. We shall see that with the hydrogen atom, each state in the diagram corresponds to a physical eigenstate of the system, but in order to get there, we have to turn the three-dimensional Coulomb potential of the hydrogen atom into the kind of problem we can study with the SUSY QM machinery we’ve built up so far. Two steps will be necessary to do this: first, moving to the center-of-mass reference frame, and second, separating the radial and angular dependencies. In this post, we’ll tackle the first of those two tasks.

While the SUSY part isn’t widely taught, these preliminary steps are more familiar. This brief note is based on Chapter VII of Cohen-Tannoudji, Diu and Laloë.
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Irony, Thy Name is Quantum

I wonder how many instances of the phrase “quantum leap” Scott Bakula is personally responsible for? There is, as many others have observed before me, a hefty dose of irony in calling a major transition a “quantum leap” or “quantum jump,” as the original leaps to gain the name were the transitions of electrons between energy levels. We’re talking about an electron’s “orbit” changing its diameter from one zillionth of a centimeter to four zillionths of a centimeter. But science never stands in the way of evocative, quasi-scientific jargon!

It’s old irony, but it’s still worth a chuckle, as when India’s prime minister declares, “We need a quantum jump in science education and research.” Start at the top, my friend, start at the top. OK, points for effort:
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SUSY QM 2: Shape Invariance

Blogging on Peer-Reviewed ResearchAfter a while, you just get tired. An honest science blogger can only handle so much science jargon thrown around without meaning, only a limited amount of Choprawoo and quantum flapdoodle. How long can anyone with integrity, curiosity and a dose of genuine knowledge endure the trumpeting that, say, the brain’s limited ability to recover after injury is evidence for some quantum spirit? The brain is living flesh, made of living cells: by the same so-called logic, the scabbed knees of childhood are all evidence of quantum skin.

After a while, a deep reserve of psyche cries out, “Enough! If my freedom means aught, I must stop responding to these charlatans and move beyond. I must send a message of my own, a message which is not a reaction but an expression unto itself. I must sing the quantum genuine!”

In my case, this means another post on supersymmetric quantum mechanics.

RECAP

Last time, we deduced some interesting properties of Hamiltonians which can be factored into operators and adjoints:

[tex]H_1 = A^\dag A,\ H_2 = AA^\dag.[/tex]

We observed that [tex]H_1[/tex] and [tex]H_2[/tex] are isospectral. That is, while the forms of their eigenfunctions may be different, the eigenvalues associated with those functions are the same; or, in physical terms, the wavefunctions have different shapes, but the energies match. The only exception is the ground state: if [tex]H_1[/tex] has a zero-energy ground state, then [tex]H_2[/tex] will not. Furthermore, the operator [tex]A[/tex] maps eigenstates of [tex]H_1[/tex] into those of [tex]H_2[/tex], and the operator [tex]A^\dag[/tex] maps eigenstates in the reverse direction:


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Shorter Sal Cordova

BPSDBSal Cordova, famous for calling Charles Darwin a puppy-killer, has attempted to reply to Tyler DiPietro’s demonstration that he, Cordova, is blitheringly ignorant of quantum physics. (Such ignorance would not be a crime, of course, except that Cordova is hell-bent on using quantum physics to prop up his “Advanced Creation Science.” See here, here, here, here and here if you’re morbidly curious.) Cordova’s latest reply compresses down rather nicely; once you do him the favor of cutting out the prevarications and the contradictions, the result resembles the following.

Stringing words I don’t understand together in a row is just as good as a logical argument, thank you very much, and if the laws of physics impose any limits on the “Ultimate Observer”, well then Jesus will come down and make everything better.

Click away if you want the gory details. . . .

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The Unbinding Problem

We human folk are pretty good at taking information from different sensory channels and combining it into unified impressions. When I watch a video recording of Richard Feynman, for example, my optic nerves carry a flood of data pulses, signals which somehow arrive in my brain and sync up with the voice which sounds like a Brooklyn gangster (but which actually came from Far Rockaway, of course). A unified whole then emerges, without my conscious effort.

Like they’ve done with all the other odd aspects of brain function, people have invoked quantum mechanics to “explain” this: the mind is mysterious, quantum physics is mysterious, and so the two must be related. Only your blind devotion to linear, Western, patriarchal science prevents you from seeing the Truth! (Oddly enough, the Oprah-Chopra-woo crowd never seem to remark on how the original developers of quantum physics were, almost all of them, White Males who are now Dead.) The “binding problem” is no exception.

However, as both Max Tegmark and Ray F. Streater have pointed out, the proponents of what we might call “quantum binding” — Henry Stapp in particular — fail to consider that correlations are perfectly possible in classical physics. The invocation of quantum physics is only a maladroit solution to a non-problem. In Tegmark’s words,

For instance, oscillations in a guitar string are local in Fourier space, not in real space, so in this case the “binding problem” can be solved by a simple change of variables. As Eddington remarked [77], when observing the ocean we perceive the moving waves as objects in their own right because they display a certain permanence, even though the water itself is only bobbing up and down. Similarly, thoughts are presumably highly non-local excitation patterns in the neural network of our brain, except of a non-linear and much more complex nature.

Now, a patient has turned up in whose brain binding does not occur. G. Lee and H. B. Coslett write of the patient “K.E.,”

He was unable to report more than one attribute of a single object. For example, he was unable to name the color of the ink in which words were written despite naming the word correctly. Several experiments demonstrated, however, that perceptual attributes that he was unable to report influenced his performance.

K.E. had suffered stroke damage to his parietal lobes on both sides of his brain, or in technical terms, “bilateral posterior parietal infarcts.” In addition to his binding difficulties, he experienced simultanagnosia, an inability to see more than one object out of several presented to his view at once.

It may be a “dog bites man” report by now, but I think it’s worth noting that finding patients with neural dysfunctions is still a more productive way of gaining new neuroscientific knowledge than woolly-headed speculation about quantum mechanics, speculation which incidentally ignores big facts about physics.

(Tip o’ the fedora to Mind Hacks, and to Warren Ellis, whose “4am” provides my current background score.)

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Teh Burning Stupid: Relativity Edition

To put the “moral” at the beginning, let’s summarize. If you want to raise my blood pressure, one good way to do it is to write a completely wrong, back-to-front absurd tirade against all of twentieth-century physics. Anyone can slip a few errors into an essay, or even a few “fundamental” errors, but if you want the brass ring, you need at the very least to misrepresent the special theory of relativity, the general theory of relativity, quantum mechanics, the use of mathematics in physics and the scientific method. Bonus points if you confuse general relativity with quantum physics; a woop-woop-woop special prize for taking a non-true assertion and calling it a “fundamental premise” of quantum mechanics; and an extra cherry on top if you take three famous observations which support general relativity, lie about two of them and forget the third.

The story so far:

I might write an actual, non-linkfesty post about this. . . but then again, other corners of the Network are calling to me and reminding me of overdue obligations, so I might leave it to my colleagues.

UPDATE (6 December): gg now has Part 2 out on the blogonets.

UPDATE (16:47 o’clock): Tyler DiPietro dons the asbestos and joins the fun, followed quickly by Mark Chu-Carroll.

UPDATE (9 December): Flavin of the St. Louis Skeptical Society offers an essay.

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Yawn: More Abuse of the Quantum

Binocular rivalry is a phenomenon which occurs when conflicting information is presented to each of our two eyes, and the brain has to cope with the contradiction. Instead of seeing a superimposition or “average” of the two, our perceptual machinery entertains both possibilities in turn, randomly flickering from one to the other. This presents an interesting way to stress-test our visual system and see how vision works. Unfortunately, talk of “perception” leads to talk of “consciousness,” and once “consciousness” has been raised, an invocation of quantum mechanics can’t be too far behind.

I’m late to join the critical party surrounding E. Manousakis’ paper, “Quantum theory, consciousness and temporal perception: Binocular rivalry,” recently uploaded to the arXiv and noticed by Mo at Neurophilosophy. Manousakis applies “quantum theory” (there’s a reason for those scare quotes) to the problem of binocular rivalry and from this hat pulls a grandiose claim that quantum physics is relevant for human consciousness.

A NOTE ON WIRES AND SLINKYS

First, we observe that there is a healthy literature on this phenomenon, work done by computational neuroscience people who aren’t invoking quantum mechanics in their explanations.

Second, one must carefully distinguish a model of a phenomenon which actually uses quantum physics from a model in which certain mathematical tools are applicable. Linear algebra is a mathematical tool used in quantum physics, but describing a system with linear algebra does not make it quantum-mechanical. Long division and the extraction of square roots can also appear in the solution of a quantum problem, but this does not make dividing 420 lollipops among 25 children a correlate of quantum physics.

Just because the same equation applies doesn’t mean the same physics is at work. An electrical circuit containing a capacitor, an inductor and a resistor obeys the same differential equation as a mass on a spring: capacitance corresponds to “springiness,” inductance to inertia and resistance to friction. This does not mean that an electrical circuit is the same thing as a rock glued to a slinky.

MIXING THE QUANTUM AND THE CLASSICAL

One interesting thing about this paper is that the hypothesis is really only half quantum, at best. In fact, three of the four numbers fed into Manousakis’ hypothesis pertain to a classical phenomenon, and here’s why:

Manousakis invokes the formalism of the quantum two-state system, saying that the perception of (say) the image seen by the left eye is one state and that from the right eye is the other. The upshot of this is that the probability of seeing the illusion one way — say, the left-eye version — oscillates over time as

[tex]P(t) = \cos^2(\omega t),[/tex]

where [tex]\omega[/tex] is some characteristic frequency of the perceptual machinery. The oscillation is always going, swaying back and forth, but every once in a while, it gets “observed,” which forces the brain into either the clockwise or the counter-clockwise state, from which the oscillation starts again.

The quantum two-state system just provides an oscillating probability of favoring one perception, one which goes as the square of [tex]\cos(\omega t)[/tex]. Three of the four parameters fed into the Monte Carlo simulation actually pertain to how often this two-state system is “observed” and “collapsed”. These parameters describe a completely classical pulse train — click, click, click, pause, click click click click, etc.

What’s more, the classical part is the higher-level one, the one which intrudes on the low-level processing. Crudely speaking, it’s like saying there’s a quantum two-state system back in the visual cortex, but all the processing up in the prefrontal lobes is purely classical.
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Cectic Speaks for Me

I don’t have this conversation in person, because I hang out with academics, engineers and martial artists whose idea of “ancient Eastern wisdom” runs to better ways of extracting an enemy’s eyeball from its socket during combat. However, it happens on the Internet often enough to be all too familiar:

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Coleman on Quantum Flapdoodle

If you mentioned quantum physics at a cocktail party ten years ago, no one would invite you back. More recently a woman approached me and said, “Isn’t quantum physics just what Eastern mystics have been saying for the last two thousand years?” I had to summon every ounce of dignity and told her, “No!”

— Sidney Coleman, theoretical physicist (1937–2007)

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Currently Reading

Oliver Johnson, Christophe Vignat (2006). Some results concerning maximum Renyi entropy distributions.

We consider the Student-t and Student-r distributions, which maximise Renyi entropy under a covariance condition. We show that they have information-theoretic properties which mirror those of the Gaussian distributions, which maximise Shannon entropy under the same condition. We introduce a convolution which preserves the Renyi maximising family, and show that the Renyi maximisers are the case of equality in a version of the Entropy Power Inequality. Further, we show that the Renyi maximisers satisfy a version of the heat equation, motivating the definition of a generalized Fisher information.

Luciano da F. Costa, Francisco A. Rodrigues, Gonzalo Travieso, P. R. Villas Boas (2006). Characterization of complex networks: A survey of measurements.
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