Category Archives: Quantum mechanics

Calloo! Callay!

My copy of Quantum Mechanics and Path Integrals by Feynman and Hibbs just arrived! If, say, David Griffiths’ textbook epitomizes the ordinary “vernacular” treatment of quantum mechanics, QMaPI is a classic unorthodox approach. Intended for students who already have a bit of background in the subject, it builds up the Lagrangian alternative to the Hamiltonian method, a highly useful idea when one goes on to study field theory, string theory or advanced statistical physics.

For years, this book was only available in beat-up old library copies and illegal DJVU files from Lithuania, but now, Dover has brought forth a new edition. I’m not certain on this, but it appears as if the book was so heavily pre-ordered that sold out of it the day it became available for purchase.

EDIT TO ADD: One erratum — on p. 364, Thorber should be Thornber.

Monday BPSDB: Null Physics

BPSDBA fellow named Terry Witt has been advertising his self-published book, Our Undiscovered Universe, in places like Discover magazine and Scientific American. Unfortunately, the ad pages aren’t exactly peer-reviewed, or even cross-checked with a nearby grad student; being businesses, magazines naturally care about revenue. Upon examination, Our Undiscovered Universe turns out to be brimming over with crank physics and general nonsense. Ben Monreal, who was one of the intimidatingly smart people in the lab where I did my undergrad thesis, has weighed Witt’s “Null Physics” and found it wanting; his review of Our Undiscovered Universe is quite a good read.

Witt’s book starts with pseudomathematics before moving on to pseudophysics. As Ben explains,

Chapter 1 is where Witt lays out a series of “proofs” derived from what he calls the “Null Axiom”. That axiom is: “Existence sums to nonexistence” (pg. 28)—something that Witt calls self-evident after a page of invalid set theory. The central mistake, if I had to identify one, is the claim that “X does not exist” is the same as “everything except X exists”. This is utter baloney, whether in formal logic or in set theory or in daily experience.

Actually, as the book unfolds, Witt doesn’t appear to use this dead-in-the-water non-axiom for anything. He does, however, pile on more pseudomathematics:

Chapter 3 contains such gems as Theorem 3.1: “The Existence of Any Half of the Universe is Equal to the Nonexistence of the Other Half” (pg. 66) and Theorem 3.9: “The Time Required for Light to Traverse the Universe is Eternity, infinity/c” (pg. 72). I am not making this up. Witt throws around “infinity” as though it were an ordinary real number; he multiplies and divides by it, etc., with normal algebraic cancellation. This is complete nonsense; there are two centuries of mathematical thought figuring out the mathematical properties of infinity, and Witt’s approach is valid in exactly none of them. (Witt later explained on his online forum—currently disabled—that he’s reinvented all of the mathematics associated with “infinity”. His reasoning, if that’s what you call it, was that his new definition jibed with a grand idea about math being dependent on nature; it was an argument from incredulity.)

When Witt does finally get around to physics, five chapters into the book, he doesn’t do any better.
Continue reading Monday BPSDB: Null Physics

Quantum Woo, Part N

BPSDBTime for a little BPSDB! The redoubtable Ben Goldacre has the dirt on Bill Nelson’s “QXCI machine,” a device for “bioenergetic health auditing,” a medical procedure well-known among specialists as an essential step in the surgical removal of cash from wallets. Best of all, though, is what QXCI stands for: Quantum Xrroid Consciousness Interface. Now, quantum physics has jack to do with consciousness, but more importantly, “quantum xrroid” just sounds. . . painful. Like a blood boil growing inside your X, if you know what I mean.

Maybe a “quantum xrroid” means that your X is in a superposition of inflamed and not inflamed and only settles on one or the other option when your doctor examines it.

(Incidentally, I met the redoubtable Ben Goldacre in Vegas a few weeks ago — and thereby would hang a tale, if he weren’t still hoarding the photo evidence.)

Quantum One

Michael Nielsen’s recent essay “Why the world needs quantum mechanics,” about the quintessential weirdness of quantum phenomena, provoked Dave Bacon to ask if there’s a better way to teach introductory courses in quantum physics. This question strikes a chord with me, since my first semester of college quantum — the class known as “8.04” — was rather remarkably dreadful.

It began with some fluff about early models of the atom, leaving out most of the ideas actually proposed in favor of a “textbook cardboard” version of the discoveries made in early TwenCen. If we can’t teach history well, why teach it at all? We’re certainly not promoting a genuine understanding of how science works if we only present a caricature of it. I doubt one could even instil an appreciation for the problems which Bohr, Heisenberg, Schrödinger and company solved in the years leading up to 1927: sophomore physics students can’t follow in their footsteps, because sophomore physics students don’t know as much classical physics as professional physicists of the 1920s did. To understand their starting point and the steps they took requires, oddly enough, subject material which even MIT undergrads don’t learn until later.
Continue reading Quantum One

Quantum Mechanics in Your Face

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.

What Can the LHC Tell Us?

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.
Continue reading What Can the LHC Tell Us?

An Unusual Occurrence

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.
Continue reading An Unusual Occurrence

Categorical Information Theory

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!

SUSY QM: The 1D Dirac Hamiltonian

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.
Continue reading SUSY QM: The 1D Dirac Hamiltonian

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:



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.


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.
Continue reading Intermezzo: The Dirac Equation

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.

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.
Continue reading SUSY QM 5: The Hydrogen Atom

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.


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.)