Category Archives: Quantum Mechanics

Relativity becomes less comprehensible

“To get Hawking radiation we have to give up on the idea that spacetime always had 3 space dimensions and one time dimension to get a quantum theory of the big bang.”  I’ve been studying relativity for some years now in the hopes of saying something intelligent to the author (Jim Hartle), if we’re both lucky enough to make it to our 60th college reunion in 2 years.  Hartle majored in physics under John Wheeler who essentially revived relativity from obscurity during the years when quantum mechanics was all the rage. Jim worked with Hawking for years, spoke at his funeral and wrote this in an appreciation of Hawking’s work [ Proc.Natl. Acad. Sci. vol. 115 pp. 5309 – 5310 ’18 ].

I find the above incomprehensible.  Could anyone out there enlighten me?  Just write a comment.  I’m not going to bother Hartle

Addendum 25 May

From a retired math professor friend —

I’ve never studied this stuff, but here is one way to get more actual dimensions without increasing the number of apparent dimensions:
Start with a 1-dimensional line, R^1 and now consider a 2-dimensional cylinder S^1 x R^1.  (S^1 is the circle, of course.)  If the radius of the circle is small, then the cylinder looks like a narrow tube.  Make the radius even smaller–lsay, ess than the radius of an atomic nucleus.  Then the actual 2-dimensional cylinder appears to be a 1-dimensional line.
The next step is to rethink S^1 as a line interval with ends identified (but not actually glued together.  Then S^1 x R^1 looks like a long ribbon with its two edges identified.  If the width of the ribbon–the length of the line interval–is less, say, than the radius of an atom, the actual 2-dimensional “ribbon with edges identified” appears to be just a 1-dimensional line.
Okay, now we can carry all these notions to R^2.  Take S^1 X R^2, and treat S^1 as a line interval with ends identified.  Then S^1 x R^2 looks like a (3-dimensional) stack of planes with the top plane identified, point by point, with the bottom plane.  (This is the analog of the ribbon.)  If the length of the line interval is less, say, than the radius of an atom, then the actual 3-dimensional s! x R^2 appears to be a 2-dimensional plane.
That’s it.  In general, the actual n+1-dimensional S^1 x R^n appears to be just n-space R^n when the radius of S^1 is sufficiently small.
All this can be done with a sphere S^2, S^3, … of any dimension, so that the actual k+n-dimensional manifold S^k x R^n appears to be just the n-space R^n when the radius of S^k is sufficiently small.  Moreover, if M^k is any compact manifold whose physical size is sufficiently small, then the actual k+n-dimensional manifold M^k x R^n appears to be just the n-plane R^n.
That’s one way to get “hidden” dimensions, I think. “

Nicastrin the gatekeeper of gamma secretase

Once a year some hapless trucker from out of town gets stuck trying to drive under a nearby railroad bridge with a low clearance. This is exactly the function of nicastrin in the gamma secretase complex which produces the main component of the senile plaque, the aBeta peptide.

Gamma secretase is a 4 protein complex which functions as an enzyme which can cut the transmembrane segment of proteins embedded in the cell membrane. This was not understood for years, as cutting a protein here means hydrolyzing the amide bond of the protein, (e.g. adding water) and there is precious little water in the cell membrane which is nearly all lipid.

Big pharma has been attacking gamma secretase for years, as inhibiting it should stop production of the Abeta peptide and (hopefully) help Alzheimer’s disease. However the paper to be discussed [ Proc. Natl. Acad. Sci. vol. 113 p.n E509 – E518 ’16 ] notes that gamma secretase processes ‘scores’ of cell membrane proteins, so blanket inhibition might be dangerous.

The idea that Nicastrin is the gatekeeper for gamma secretase is at least a decade old [ Cell vol. 122 pp. 318 – 320 ’05 ], but back then people were looking for specific binding of nicastrin to gamma secretase targets.

The new paper provides a much simpler explanation. It won’t let any transmembrane segment of a protein near the active site of gamma secretase unless the extracellular part is lopped off. The answer is simple mechanics. Nicastrin is large (709 amino acids) but with just one transmembrane domain. Most of it is extracellular forming a blob extending out 25 Angstroms from the membrane, directly over the substrate binding pocket of gamma secretase. Only substrates with small portions outside the membrane (ectodomains) can pass through it. It’s the railroad bridge mentioned above. Take a look at the picture —

This is why a preliminary cleavage of the Amyloid Precursor Peptide (APP) is required for gamma secretase to work.

So all you had to do was write down the wavefunction for Nicastrin (all 709 amino acids) and solve it (assuming you even write it down) and you’d have the same answer — NOT. Only the totally macroscopic world explanation (railroad bridge) is of any use. What keeps proteins from moving through each other? Van der Waals forces. What help explain them. The Pauli exclusion principle, as pure quantum mechanics as it gets.

A book recommendation, not a review

My first encounter with a topology textbook was not a happy one. I was in grad school knowing I’d leave in a few months to start med school and with plenty of time on my hands and enough money to do what I wanted. I’d always liked math and had taken calculus, including advanced and differential equations in college. Grad school and quantum mechanics meant more differential equations, series solutions of same, matrices, eigenvectors and eigenvalues, etc. etc. I liked the stuff. So I’d heard topology was cool — Mobius strips, Klein bottles, wormholes (from John Wheeler) etc. etc.

So I opened a topology book to find on page 1

A topology is a set with certain selected subsets called open sets satisfying two conditions
l. The union of any number of open sets is an open set
2. The intersection of a finite number of open sets is an open set

Say what?

In an effort to help, on page two the book provided another definition

A topology is a set with certain selected subsets called closed sets satisfying two conditions
l. The union of a finite number number of closed sets is a closed set
2. The intersection of any number of closed sets is a closed set

Ghastly. No motivation. No idea where the definitions came from or how they could be applied.

Which brings me to ‘An Introduction to Algebraic Topology” by Andrew H. Wallace. I recommend it highly, even though algebraic topology is just a branch of topology and fairly specialized at that.


Because in a wonderful, leisurely and discursive fashion, he starts out with the intuitive concept of nearness, applying it to to classic analytic geometry of the plane. He then moves on to continuous functions from one plane to another explaining why they must preserve nearness. Then he abstracts what nearness must mean in terms of the classic pythagorean distance function. Topological spaces are first defined in terms of nearness and neighborhoods, and only after 18 pages does he define open sets in terms of neighborhoods. It’s a wonderful exposition, explaining why open sets must have the properties they have. He doesn’t even get to algebraic topology until p. 62, explaining point set topological notions such as connectedness, compactness, homeomorphisms etc. etc. along the way.

This is a recommendation not a review because, I’ve not read the whole thing. But it’s a great explanation for why the definitions in topology must be the way they are.

It won’t set you back much — I paid. $12.95 for the Dover edition (not sure when).

More Quantum Weirdness

If you put 3 pigeons in two pigeonholes, two pigeons must be in one of them. Not so says quantum mechanics in a new paper [ Proc. Natl. Acad. Sci. vol. 113 pp. 532 – 535 ’16 ]. I can’t claim to understand the paper, despite auditing a course in QM in the past decade, but at least I do understand the terms they are throwing about. I plan to print it out and really give it a workout

The paper does involve something called a beam splitter, which splits photon waves into two parts. I’ve never understood how this works on a mechanistic level. Perhaps no such understanding is possible. Another thing I don’t understand is what happens when a photon wave (or particle) is reflected from a mirror. Perhaps no such understanding is possible. Another thing I don’t understand is how a photon is diffracted when it passes through a slit. It must in some way sense the edge of the slit. I’ve certainly watched waves being diffracted by a breakwater — just go up to the bar at the top of the Sears Tower in Chicago, get an estimate on a beer and look out at lake Michigan.

Can anyone out there help?

Time to get busy

Well I asked for it (the answer sheets to my classmate’s book on general relativity). It came today all 347 pages of it + a small appendix “Light Orbits in the Schwarzschild Geometry”. It’s one of the few times the old school tie has actually been of some use. The real advantages of going to an elite school are (1) the education you can get if you want (2) the people you meet back then or subsequently. WRT #1 — the late 50s was the era of the “Gentleman’s C”.

It should be fun. The book is the exact opposite of the one I’d been working on which put the math front and center. This one puts the physics first and the math later on. I’m glad I’m reading it second because as an undergraduate and graduate student I became adept at mouthing mathematical incantations without really understanding what was going on. I think most of my math now is reasonably solid. I did make a lot of detours I probably didn’t need to make — manifold theory,some serious topology — but that was fun as well.

When you’re out there away from University studying on your own, you assume everything you don’t understand is due to your stupidity. This isn’t always the case (although it usually is), and I’ve found errors in just about every book I’ve studied hard, and my name features on errata web pages of most of them. For one example see

How formal tensor mathematics and the postulates of quantum mechanics give rise to entanglement

Tensors continue to amaze. I never thought I’d get a simple mathematical explanation of entanglement, but here it is. Explanation is probably too strong a word, because it relies on the postulates of quantum mechanics, which are extremely simple but which lead to extremely bizarre consequences (such as entanglement). As Feynman famously said ‘no one understands quantum mechanics’. Despite that it’s never made a prediction not confirmed by experiments, so the theory is correct even if we don’t understand ‘how it can be like that’. 100 years of correct prediction of experimentation are not to be sneezed at.

If you’re a bit foggy on just what entanglement is — have a look at Even better; read the book by Zeilinger referred to in the link (if you have the time).

Actually you don’t even need all the postulates for quantum mechanics (as given in the book “Quantum Computation and Quantum Information by Nielsen and Chuang). No differential equations. No Schrodinger equation. No operators. No eigenvalues. What could be nicer for those thirsting for knowledge? Such a deal ! ! ! Just 2 postulates and a little formal mathematics.

Postulate #1 “Associated to any isolated physical system, is a complex vector space with inner product (that is a Hilbert space) known as the state space of the system. The system is completely described by its state vector which is a unit vector in the system’s state space”. If this is unsatisfying, see an explication of this on p. 80 of Nielson and Chuang (where the postulate appears)

Because the linear algebra underlying quantum mechanics seemed to be largely ignored in the course I audited, I wrote a series of posts called Linear Algebra Survival Guide for Quantum Mechanics. The first should be all you need. but there are several more.

Even though I wrote a post on tensors, showing how they were a way of describing an object independently of the coordinates used to describe it, I did’t even discuss another aspect of tensors — multi linearity — which is crucial here. The post itself can be viewed at

Start by thinking of a simple tensor as a vector in a vector space. The tensor product is just a way of combining vectors in vector spaces to get another (and larger) vector space. So the tensor product isn’t a product in the sense that multiplication of two objects (real numbers, complex numbers, square matrices) produces another object of the exactly same kind.

So mathematicians use a special symbol for the tensor product — a circle with an x inside. I’m going to use something similar ‘®’ because I can’t figure out how to produce the actual symbol. So let V and W be the quantum mechanical state spaces of two systems.

Their tensor product is just V ® W. Mathematicians can define things any way they want. A crucial aspect of the tensor product is that is multilinear. So if v and v’ are elements of V, then v + v’ is also an element of V (because two vectors in a given vector space can always be added). Similarly w + w’ is an element of W if w an w’ are. Adding to the confusion trying to learn this stuff is the fact that all vectors are themselves tensors.

Multilinearity of the tensor product is what you’d think

(v + v’) ® (w + w’) = v ® (w + w’ ) + v’ ® (w + w’)

= v ® w + v ® w’ + v’ ® w + v’ ® w’

You get all 4 tensor products in this case.

This brings us to Postulate #2 (actually #4 on the book on p. 94 — we don’t need the other two — I told you this was fairly simple)

Postulate #2 “The state space of a composite physical system is the tensor product of the state spaces of the component physical systems.”

Where does entanglement come in? Patience, we’re nearly done. One now must distinguish simple and non-simple tensors. Each of the 4 tensors products in the sum on the last line is simple being the tensor product of two vectors.

What about v ® w’ + v’ ® w ?? It isn’t simple because there is no way to get this by itself as simple_tensor1 ® simple_tensor2 So it’s called a compound tensor. (v + v’) ® (w + w’) is a simple tensor because v + v’ is just another single element of V (call it v”) and w + w’ is just another single element of W (call it w”).

So the tensor product of (v + v’) ® (w + w’) — the elements of the two state spaces can be understood as though V has state v” and W has state w”.

v ® w’ + v’ ® w can’t be understood this way. The full system can’t be understood by considering V and W in isolation, e.g. the two subsystems V and W are ENTANGLED.

Yup, that’s all there is to entanglement (mathematically at least). The paradoxes entanglement including Einstein’s ‘creepy action at a distance’ are left for you to explore — again Zeilinger’s book is a great source.

But how can it be like that you ask? Feynman said not to start thinking these thoughts, and if he didn’t know you expect a retired neurologist to tell you? Please.

Watching electrons being pushed

Would any organic chemist like to watch electrons moving around in a molecule? Is the Pope Catholic? Attosecond laser pulses permit this [ Science vol. 346 pp. 336 – 339 ’14 ]. An attosecond is 10^-18 seconds. The characteristic vibrational motion of atoms in chemical bonds occurs at the femtosecond scale (10^-15 seconds). An electron takes 150 attoseconds to orbit a hydrogen atom [ Nature vol. 449 p. 997 ’07 ]. Of course this is macroscopic thinking at the quantum level, a particular type of doublethink indulged in by chemists all the time —

The technique involves something called pump probe spectroscopy. Here was the state of play 15 years ago — [ Science vol. 283 pp. 1467 – 1468 ’99 ] Using lasers it is possible to blast in a short duration (picoseconds 10^-12 to femtoseconds 10^-15) pulse of energy (pump pulse ) at one frequency (usually ultraviolet so one type of bond can be excited) and then to measure absorption at another frequency (usually infrared) a short duration later (to measure vibrational energy). This allows you to monitor the formation and decay of reactive intermediates produced by the pump (as the time between pump and probe is varied systematically).

Time has marched on and we now have lasers capable of producing attosecond pulses of electromagnetic energy (e.g. light).

A single optical cycle of visible light of 6000 Angstrom wavelength lasts 2 femtoseconds. To see this just multiply the reciprocal of the speed of light (3 * 10^8 meters/second) by the wavelength (6 * 10^3 *10^-10). To get down to the attosecond range you must use light of a shorter wavelength (e.g. the ultraviolet or vacuum ultraviolet).

The paper didn’t play around with toy molecules like hydrogen. They blasted phenylalanine with UV light. Here’s what they said “Here, we present experimental evidence of ultrafast charge dynamics in the amino acid phenylalanine after prompt ionization induced by isolated attosecond pulses. A probe pulse then produced a doubly charged molecular fragment by ejection of a second electron, and charge migration manifested itself as a sub-4.5-fs oscillation in the yield of this fragment as a function of pump-probe delay. Numerical simulations of the temporal evolution of the electronic wave packet created by the attosecond pulse strongly support the interpretation of the experimental data in terms of charge migration resulting from ultrafast electron dynamics preceding nuclear rearrangement.”

OK, they didn’t actually see the electron dynamics but calculated it to explain their results. It’s the Born Oppenheimer approximation writ large.

You are unlikely to be able to try this at home. It’s more physics than I know, but here’s the experimental setup. ” In our experiments, we used a two-color, pump-probe technique. Charge dynamics were initiated by isolated XUV sub-300-as pulses, with photon energy in the spectral range between 15 and 35 eV and probed by 4-fs, waveform-controlled visible/near infrared (VIS/NIR, central photon energy of 1.77 eV) pulses (see supplementary materials).”

Physics to the rescue

It’s enough to drive a medicinal chemist nuts. General anesthetics are an extremely wide ranging class of chemicals, ranging from Xenon (which has essentially no chemistry) to a steroid alfaxalone which has 56 carbons. How can they possibly have a similar mechanism of action? It’s long been noted that anesthetic potency is proportional to lipid solubility, so that’s at least something. Other work has noted that enantiomers of some anesthetics vary in potency implying that they are interacting with something optically active (like proteins). However, you should note sphingosine which is part of many cell membrane lipids (gangliosides, sulfatides etc. etc.) contains two optically active carbons.

A great paper [ Proc. Natl. Acad. Sci. vol. 111 pp. E3524 – E3533 ’14 ] notes that although Xenon has no chemistry it does have physics. It facilitates electron transfer between conductors. The present work does some quantum mechanical calculations purporting to show that
Xenon can extend the highest occupied molecular orbital (HOMO) of an alpha helix so as to bridge the gap to another helix.

This paper shows that Xe, SF6, NO and chloroform cause rapid increases in the electron spin content of Drosophila. The changes are reversible. Anesthetic resistant mutant strains (in what protein) show a different pattern of spin responses to anesthetic.

So they think general anesthetics might work by perturbing the electronic structure of proteins. It’s certainly a fresh idea.

What is carrying the anesthetic induced increase in spin? Speculations are bruited about. They don’t think the spin changes are due to free radicals. They favor changes in the redox state of metals. Could it be due to electrons in melanin (the prevalent stable free radical in flies). Could it be changes in spin polarization? Electrons traversing chiral materials can become spin polarized.

Why this should affect neurons isn’t known, and further speculations are given (1) electron currents in mitochondria, (2) redox reactions where electrons are used to break a disulfide bond.

The article notes that spin changes due to general anesthetics differ in anesthesia resistant fly mutants.

Fascinating paper, and Mark Twain said it the best “There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”

Are Van der Waals interactions holding asteroids together?

A recent post of Derek’s concerned the very weak (high kD) but very important interactions of proteins within our cells.

Most of this interaction is due to Van der Waals forces — Shape shape complementarity (e.g. steric factors) and dipole dipole interactions are also important.

Although important, Van der Waals interactions have always seemed like a lot of hand waving to me.

Well guess what, they are now hypothesized to be what is holding an asteroid together. Why are people interested in asteroids in the first place? [ Science vol. 338 p. 1521 ’12 ] “Asteroids and comets .. reflect the original chemical makeup of the solar system when it formed roughly 4.5 billion years ago.”

[ Nature vol. 512 p. 118 ’14 ] The Rosetta spacecraft reached the comet 67P/Churyumov-Gerasimenko after a 10 year journey becoming the first spacecraft to rendezvous with a comet. It will take a lap around the sun with the comet and will watch as the comet heats up and releases ice in a halo of gas and dust. It is now flying triangles in front of the comet, staying 100 kiloMeters away. In a few weeks it will settle into a 30 kiloMeter orbit around he comet. It will attempt to place a lander (Philae) the size of a washing machine on its surface in November. The comet is 4 kiloMeters long.

[ Nature vol. 512 pp. 139 – 140, 174 – 176 ’14 ] A kiloMeter sized near Earth asteroid called (29075) 1950 DA (how did they get this name?) is covered with sandy regolith (heterogeneous material covering solid rock { on earth } it includes dust, soil, broken rock ). The asteroid rotates every 2+ hours, and it is so small that gravity alone can’t hold the regolith to its surface. An astronaut could scoop up a sample from its surface, but would have to hold on to the asteroid to avoid being flung off by the rotation. So the asteroid must have some degree of cohesive strength. The strength required is 64 pascals to hold the rubble together — about the pressure that a penny exerts on the palm of your hand. A Pascal is 1/101,325 of atmospheric pressure.

They think the strength comes from van der Waals interactions between small (1 – 10 micron) grains — making it fairy dust. It’s rather unsatisfying as no one has seen these particles.

The ultimate understanding of the large multi-protein and RNA machines (ribosome, spliceosome, RNA polymerase etc. etc. ) without which life would be impossible will involve the very weak interactions which hold them together. Along with permanent dipole dipole interactions, charge interactions and steric complementarity, the van der Waals interaction is high on anyone’s list.

Some include dipole dipole interactions as a type of van der Waals interaction. The really fascinating interaction is the London dispersion force. These are attractions seen between transient induced dipoles formed in the electron clouds surrounding each atomic nucleus.

It’s time to attempt the surmount the schizophrenia which comes from trying to see how quantum mechanics gives rise to the macroscopic interactions between molecules which our minds naturally bring to matters molecular (with a fair degree of success).

Steric interactions come to mind first — it’s clear that an electron cloud surrounding molecule 1 should repel another electron cloud surrounding molecule 2. Shape complementarity should allow two molecules to get closer to each other.

What about the London dispersion forces, which are where most of the van der Waals interaction is thought to be. We all know that quantum mechanical molecular orbitals are static distributions of electron probability. They don’t fluctuate (at least the ones I’ve read about). If something is ‘transiently inducing a dipole’ in a molecule, it must be changing the energy level of a molecule, somehow. All dipoles involve separation of charge, and this always requires energy. Where does it come from? The kinetic energy of the interacting molecules? Macroscopically it’s easy to see how a collision between two molecules could change the vibrational and/or rotation energy levels of a molecule. What does a collision between between molecules look like in terms of the wave functions of both. I’ve never seen this. It has to have been worked out for single particle physics in an accelerators, but that’s something I’ve never studied.

One molecule inducing a transient dipole in another, which then induces a complementary dipole in the first molecule, seems like a lot of handwaving to me. It also appears to be getting something for nothing contradicting the second law of thermodynamics.

Any thoughts from the physics mavens out there?

Keep on truckin’ Dr. Schleyer

My undergraduate advisor (Paul Schleyer) has a new paper out in the 15 July ’14 PNAS pp. 10067 – 10072 at age 84+. Bravo ! He upends what we were always taught about electrophilic aromatic addition of halogens. The Arenium ion is out (at least in this example). Anyone with a smattering of physical organic chemistry can easily follow his mechanistic arguments for a different mechanism.

However, I wonder if any but the hardiest computational chemistry jock can understand the following (which is how he got his results) and decide if the conclusions follow.

Our Gaussian 09 (54) computations used the 6-311+G(2d,2p) basis set (55, 56) with the B3LYP hybrid functional (57⇓–59) and the Perdew–Burke–Ernzerhof (PBE) functional (60, 61) augmented with Grimme et al.’s (62) density functional theory with added Grimme’s D3 dispersion corrections (DFT-D3). Single-point energies of all optimized structures were obtained with the B2-PLYP [double-hybrid density functional of Grimme (63)] and applying the D3 dispersion corrections.

This may be similar to what happened with functional MRI in neuroscience, where you never saw the raw data, just the end product of the manipulations on the data (e.g. how the matrix was inverted and what manipulations of the inverted matrix was required to produce the pretty pictures shown). At least here, you have the tools used laid out explicitly.

For some very interesting work he did last year please see