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Category Archives: Quantum mechanics

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Via Imaginary Potential comes Sidney Coleman’s lecture on how quantum mechanics differs from classical and what that whole “collapsing the wave function” business is all about. The lecture is geared to those who have a working familiarity with first-term quantum physics: the Schrödinger Equation, spin operators and such.

The video quality is not always quite good enough to capture what’s written on the transparencies, but the audio makes up for it.

EDIT TO ADD: I don’t actually agree with the final thesis of Coleman’s lecture (I’ve gone too far in my reading of Appleby, Barnum, Caves, Fuchs, Kent, Leifer, Peres, Schack, Spekkens, Unruh, Zeilinger and so on to make that retreat). However, I would say that (a) the GHZ story is easier to remember than the Bell story, and (b) “vernacular” quantum mechanics is a good term to have on hand, as the mishmash we get from several generations of skipping-past-the-weird-bits shouldn’t necessarily be called a “school of thought” in its own right.

Today’s prize for best use of scientific terminology for humorous effect goes to Jeff Medkeff:

I did notice in the course of my reading that the density of baloney in this story is very high — maybe even degenerate.

Well, I laughed.

What can the LHC tell us, and how long will we have to wait to find out?

Over at Symmetry Breaking, David Harris has a timeline for when the amount of data gathered at the LHC will be large enough to detect particular exciting bits of physics which we expect might be lurking in wait, at high-energy realms we can’t currently reach. (The figures come from Abe Seiden’s presentation at the April 2008 meeting of the American Physical Society.) Assuming the superconducting cables — all 7000 kilometers of them! — get chilled down to their operating temperatures by mid-June and particles start whirling around the ring on schedule after that, then we could hope to spot the Higgs boson as early as 2009.
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So there I was, quietly standing in Lobby 10, queuing to buy myself and a few friends advance tickets to Neil Gaiman’s forthcoming speech at MIT, when a strange odor proturbed onto my awareness. “That’s odd,” thought I, “it smells like backstage at my high school’s auditorium. [snif snif] Or the bathroom at Quiz Bowl state finals. . . And it’s not even 4:20. Something very unusual is going on, here on this university campus.”

I became aware of a, well, perhaps a presence would be the best way to describe it: the sort of feeling which people report when their temporal and parietal lobes are stimulated by magnetic fields. Something tall and imposing was standing. . . just. . . over. . . my. . . right. . . shoulder! But when I turned to see, I saw nothing there.

Feeling a little perturbed, I bought my tickets and tried to shrug it off. Not wanting to deal with the wet and yucky weather currently sticking down upon Cambridge, I descended the nearest staircase and began to work my way eastward through MIT’s tunnel system, progressing through the “zeroth floors” of the classroom and laboratory buildings, heading for Kendall Square and the T station. Putting my unusual experience in the ticket queue out of my mind, I returned to contemplating the junction of physics and neuroscience:

“So, based on the power-law behavior of cortical avalanches, we’d guess that the cortex is positioned at a phase transition, a critical point between, well, let’s call them quiescence and epileptic madness. This would allow the cortex to sustain a wide variety of functional patterns. . . but at a critical point, the Wilson-Cowan equations should yield a conformal field theory in two spatial dimensions. . . .

“But if you reinterpret the classical partition function as a quantum path integral, a field theory in 2D becomes a quantum field theory in one spatial and one temporal dimension. And the central charge of the quantum conformal field theory is equal to the normalized entropy density. . . so we should be able to apply gauge/gravity duality and model the cortex as a black hole in anti-de Sitter spacetime —”

Suddenly, a tentacle wrapped around my chest, and constricted, and pulled, and lifted — not up, but in a direction I had never moved before. Like a square knocked out of Flatland, I had been displaced.
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Ben Allen is now on the arXivotubes, with a category-theoretic arithmetic of information.

The concept of information has found application across the sciences. However, the conventional measures of information are not appropriate for all situations, and a more general mathematical concept is needed. In this work we give axioms that characterize the arithmetic of information, i.e. the way that pieces of information combine with each other. These axioms allow for a general notion of information functions, which quantify the information transmitted by a communication system. In our formalism, communication systems are represented as category-theoretic morphisms between information sources and destinations. Our framework encompasses discrete, continuous, and quantum information measures, as well as familiar mathematical functions that are not usually associated with information. We discuss these examples and prove basic results on the general behavior of information.

It looks like a discussion about this is starting over at the n-Category Café. If I didn’t have to spend today cutting down a 12-page paper to eight pages for an overpriced book of conference proceedings which nobody will read, I’d totally be writing more about it!

Whew! We spent a considerable amount of wordage developing the Dirac Equation. Now, it’s time to tie this development back to the supersymmetry material we studied earlier in the non-relativistic context. The result will be a surprising mapping between relativistic and non-relativistic quantum mechanics. Today, we’ll just get the gist of it, and to get started, we’ll begin with the final equation we had before,

[tex](i\displaystyle{\not} \partial – m)\psi = 0.[/tex]

Recalling Feynman’s notation of slashed quantities,

[tex]\displaystyle{\not} a = \gamma^\mu a_\mu,[/tex]

we can unpack this a little to

[tex]\left(i\gamma^\mu\partial_\mu – m\right) \psi = 0,[/tex]

which we can elaborate to include an electromagnetic field as follows:

[tex]{\left[i\gamma^\mu(\partial_\mu + iA_\mu) - m\right] \psi = 0.[/tex]

The Dirac Hamiltonian [tex]H_D[/tex] has a rich SUSY structure, of which we can catch a glimpse even having pared the problem down to its barest essentials. To take the simplest possible case, consider a Dirac particle living in one spatial dimension, on which there also lives a scalar potential [tex]\phi(x^1)[/tex]. (We could call this a “1+1-dimensional” system, to remind ourselves of the difference between time and space.) The SUSY structure can be seen most clearly when we look at the limit of a massless particle; this eliminates the [tex]m[/tex] term we had before.
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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!

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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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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.

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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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.)

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 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!

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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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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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.

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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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