## March Meeting 2019 Boston

I’m currently in Boston for the 2019 March Meeting, which is as exciting, overwhelming, and exhausting as usual!

Circling Boston to approach for landing. Keen viewers can see the Boston Commons (just on the far side of the river), the MIT campus (on the near side of the river, toward the right), and even the convention center (beyond the Commons)

You may remember last March Meeting, we were in LA, which was naturally nice and warm.  Boston welcomed the March Meeting with one of the first big snow storms of the season — about 8 inches of very wet, heavy snow — right in time for Monday morning.  Fortunately, I flew in on Sunday so had no trouble, but I heard of a local presenter who missed their invited 8 am talk because they couldn’t drive in.  The roads were definitely sloppy, but it was a beautiful snow.

Monday morning, outside the Parker House hotel.

Snowy statue of Ben Franklin outside Old City Hall, just across from my hotel

I ended up walking into the convention center (about a mile) which was actually quite nice.  But I wished that I had brought boots.  I was impressed at how quickly the downtown sidewalks were being cleared.  The trouble was really at the intersections with the piles from the snow plows.

Snow on the Bass River, crossing toward the Convention Center

Once I arrived at the meeting, there were lots of interesting talks, of course.  I spent the morning in a session on Geophysical Applications of Granular Flows, with a series of really interesting invited talks. I learned a lot more about erosion and incipient flow — when does a fluid flowing over a granular material make those grains also flow?  When do grains that have been picked up and carried in the fluid drop down and settle?  It’s fluid dynamics and turbulence and granular materials all in one complicated problem!

More updates to come — but I need to get to the convention center this morning for our last day at the meeting!

## Relativistic Colors

Metallium, Inc. is attempting to manufacture coins made from as many different metals (and elements) as possible, typically 99 to 99.9% pure. My Metallium coin collection currently includes aluminum, titanium, iron, nickel, copper, zinc, silver, tin, and gold coins.

Most metals are silvery gray because they absorb ultraviolet light and reflect visible light. However, relativistic effects contract some of the atomic orbitals of copper and gold so they absorb some visible light and reflect the complementary colors. (Heuristically, in large atoms some electrons move at near light speed and appear more massive.) These colors are a striking example of relativistic physics in everyday life — and of the Dirac equation corrections to the Schrödinger equation.

My Metallium coin collection includes Al, Ti, Fe, Ni, Cu, Zn, Ag, Sn, Au coins. All are silvery gray except Cu and Au, whose colors result from relativistic changes to atomic orbitals.

## 720° untangles 360° tangles

Despite growing up in three dimensions, as a kid I did not recognize one of 3D’s deep and subtle properties: full rotations tangle, but double rotations untangle!

Following physicist Paul Dirac, twist a belt one full turn about its length. The 360° single twist cannot be undone without changing the belt buckles’ orientations, although the twist can be changed from clockwise to counterclockwise. Now twist the belt two full turns about its length. Amazingly, the 720° double twist can be undone without changing the belt buckles’ orientations. The double twist is the true identity.

The animations below show belts twisting concretely in 3D space and abstractly in a 3D projection of the 4D quaternion unit sphere. Points on the sphere represent all 3D rotations (twice). Each blue dot represents a belt cross section rotation and is located by a radius vector. Suppressing rotations about the 1‑direction, the radius vector’s projection onto the 2-3-plane is the section’s rotation axis and twice its co-latitude is the section’s rotation angle.

The sphere’s north and south poles represent the same orientation, 0° ≡ 720° and 360°, but different orientation-entanglements. Identifying the north and south poles as the same orientation allows closed loops on the quaternion sphere to represent both 360° and 720° belt twists, but only the latter can smoothly contract to the north pole identity rotation. This multiple connectivity is reminiscent of a torus (or donut with hole), where toroidal loops (around-the-hole) are contractible but poloidal (through-the-hole) loops are not, rather than the simple connectivity of a sphere, where all loops are contractible.

Elastic belt with a 360° twist. Blue dots on quaternion sphere projection represent belt cross section rotations. Blue curve connecting dots can not be smoothly contracted to the untwisted state represented by the north pole, but without changing the orientation of the belt’s ends, the twist can be changed from clockwise to counterclockwise as indicated.

Elastic belt with a 720° twist. Blue dots on quaternion sphere projection represent belt cross section rotations. Blue curve connecting dots can be smoothly contracted to the untwisted state represented by the north pole, so without changing the orientation of the belt’s ends, the twist can be be undone as indicated.

Posted in Mathematics, Physics | 1 Comment

## Wooster physics reunion in Eugene, Oregon!

I recently returned from a refreshing and productive leave at the University of Oregon in Eugene.  I received my own Ph D in the field of quantum optics at Oregon, and my leave was a great opportunity to continue work with prior colleagues.  However, I am excited to be back and interacting daily with Wooster students again!

Five Wooster alumni are currently enrolled in U of Oregon’s Physics Ph D program.  This summer I managed to gather four of these physicists together in one place–one of my favorite old haunts from my graduate school days, McMenamins Cafe.

Pictured here from left to right are Deepika Sundaraman ’14, myself, Tzula Propp ’15, Nicu Istrate ’15, and Andrew Blaikie ’13.  Not present–but with us in spirit–is Amanda Steinhebel, who had recently left Oregon to spend a year on site at the world’s largest particle accelerator (the Large Hadron Collider) at CERN in Switzerland.  All attendees (plus Amanda) received this year’s Wooster physics club t-shirt!

Later in the summer, I attended a the fall conference of the Oregon Center for Optical, Molecular, and Quantum Science, where Andrew Blaikie ’13 presented his fascinating research on a nanoscale device employing graphene to make extremely sensitive measurements (see Andrew in action below).

As for the research I worked on in Oregon while on leave–that is a different blog post for a different day!

## Luna’s Convex Orbit

Luna orbits Earth and Earth orbits Sol (where Luna is Earth’s moon and Sol is Earth’s star, the sun). As a kid, I thought Luna’s solar orbit formed a loopy spirograph pattern. Instead, Luna’s orbit is convex!

Neglecting the eccentricity and tilts of the orbits and the incommensurability of the periods, Luna’s counterclockwise path around Sol is roughly a dodecagon, as in the idealized animation, where the distance between Earth and Luna varies from cartoon large to proportionally correct. Colors code the path’s curvature, red for counterclockwise-rotating velocity vectors and cyan for clockwise-rotating velocity vectors, with magnitudes proportional to the saturation, so white is straight. Due to the convexity of both Earth and Luna’s orbits, some planetary scientists consider Earth and Luna to be a double planet.

Luna’s solar orbit is almost a convex rounded dodecagon. Red-white-cyan colors code curvature.

## 1+2+3+… = -1/12?

In quantum electrodynamics, the bare charge of an electron is infinite, but the renormalized dressed charge is finite. The bare electron shields itself by polarizing the virtual electron-positron pairs of the nearby quantum vacuum to reduce its coupling at large distances to

$$\infty - \infty \stackrel{I}{=} \sqrt{\frac{1}{137}}$$

in natural units, where the “I” decorating the equals sign denotes an informal relationship. Renormalization techniques help physicists make sense of divergent sums and integrals used to calculate quantities like running coupling “constants” that vary with interaction energy and set the strengths of the fundamental forces.

Corresponding sum and integral diverge together

For a simple example, the sum of natural numbers

$$\sum_{n=0}^N n = 1+2+3+\cdots+N= \frac{N}{2}(N+1)$$

diverges as $N\rightarrow \infty$. Similarly, the analogous integral of positive real numbers

$$\int_{0}^X \hspace{-0.8em}dx\ x =\frac{X^2}{2}$$

diverges as $X\rightarrow \infty$. To cancel these corresponding discrete and continuous divergences, introduce an exponential convergence factor and formally subtract the integral from the sum

$$\sum_{n=0}^\infty n -\int_{0}^\infty \hspace{-0.8em} dx\, x =\color{red}\lim_{\lambda \rightarrow 0}\color{black}\left( \sum_{n=0}^\infty n\, \color{red}e^{-\lambda n}\color{black}-\int_{0}^\infty \hspace{-0.8em} dx\, x\, \color{red}e^{-\lambda x} \color{black}\right).$$

To evaluate the cancellation, differentiate the integral with respect to the convergence parameter $\lambda>0$ to get

$$\mathcal{I}=\int_{0}^\infty \hspace{-0.8em}dx\ x\, \color{red}e^{-\lambda x}\color{black} =-\frac{d}{d\lambda}\int_{0}^\infty \hspace{-0.8em}dx\ e^{-\lambda x} =-\frac{d}{d\lambda}\left(\frac{e^{-\lambda x}}{-\lambda}\right)\bigg|_{0}^\infty=-\frac{d}{d\lambda}\left(\frac{1}{\lambda}\right)=\frac{1}{\lambda^2}.$$

Next differentiate the sum with respect to $\lambda$

$$\mathcal{S}=\sum_{n=0}^\infty n\,\color{red} e^{-\lambda n}\color{black} = -\frac{d}{d\lambda} \sum_{n=0}^\infty e^{-\lambda n} = -\frac{d}{d\lambda} \sum_{n=0}^\infty \left(e^{-\lambda}\right)^n$$

and sum the resulting geometric series to find

$$\mathcal{S}= -\frac{d}{d\lambda} \left( \frac{1}{1-e^{-\lambda}} \right)= \frac{e^{-\lambda}}{\left(1-e^{-\lambda}\right)^2}\color{red}\frac{e^{2\lambda}} {e^{2\lambda}}\color{black}= \frac{e^{\lambda}}{\left(e^{\lambda}-1\right)^2}\color{red}= \frac{e^{\lambda}}{\left(1-e^{\lambda}\right)^2}\color{black},$$

which is thus an even function of $\lambda$. Replace the exponentials by their power series expansions

$$\mathcal{S}=\frac{1+\lambda+\lambda^2/2!+\lambda^3/3!+\cdots}{\left(1+\lambda+\lambda^2/2!+\lambda^3/3!+\cdots - 1 \right)^2}=\frac{1+\lambda+\lambda^2/2+\cdots}{\lambda^2\left(1+\lambda/2+\lambda^2/6+\cdots \right)^2},$$

expand the denominator square

$$\begin{array}{ccc}\left(1+\lambda/2+\lambda^2/6+\cdots\right)^2=&+1\hspace{1.2em}+\lambda/2\hspace{0.7em}+\lambda^2/6\hspace{0.2em}\phantom{+\cdots}\\ \rule{0pt}{1.3em}&+\lambda/2\hspace{0.43em}+\lambda^2/4\hspace{0.37em}\color{red}+\lambda^3/12\color{black}\phantom{+\cdots}\\ \rule{0pt}{1.3em}&+\lambda^2/6\color{red}+\lambda^3/12+\lambda^4/36+\cdots&\color{black} = 1+\lambda+7\lambda^2/12+\cdots\end{array}$$

and use long division

$$\begin{array}{r}1-\lambda^2/12+\cdots\\ 1+\lambda+7\lambda^2/12+\cdots \,{\overline{\smash{\big)}\,1+\lambda+\phantom{9}\lambda^2/2\phantom{9} +\cdots}}\\ \underline{1+\lambda+7\lambda^2/12+\cdots }\\0-\lambda^2/12+\cdots\\ \underline{-\lambda^2/12+\cdots}\\0+\cdots \end{array}$$

to show

$$\mathcal{S} = \frac{1}{\lambda^2} \left(1-\frac{\lambda^2}{12}+\mathcal{O}[\lambda^4]\right) =\frac{1}{\lambda^2}-\frac{1}{12}+\mathcal{O}[\lambda^2].$$

For finite $\lambda$, the difference

$$\mathcal{S}-\mathcal{I}=\sum_{n=0}^\infty n\, e^{-\lambda n}-\int_{0}^\infty \hspace{-0.8em} dx\, x\,e^{-\lambda x} = \frac{1}{\lambda^2}-\frac{1}{12}+\mathcal{O}[\lambda^2]-\frac{1}{\lambda^2}= -\frac{1}{12}+\mathcal{O}[\lambda^2],$$

and for vanishing $\lambda$

$$\color{red}\lim_{\lambda \rightarrow 0}\color{black}\left( \sum_{n=0}^\infty n\, \color{red}e^{-\lambda n}\color{black}-\int_{0}^\infty \hspace{-0.8em} dx\, x\, \color{red}e^{-\lambda x} \color{black}\right) = -\frac{1}{12}.$$

Thus, removing nonphysical infinity in a controlled way exposes a finite component

$$\infty - \infty \stackrel{I}{=} -\frac{1}{12}$$

of the divergent natural numbers sum. To celebrate this hidden “golden nugget”, write

$$1+2+3+\cdots \stackrel{R}{=} -\frac{1}{12},$$

where”R” denotes regularized, renormalized, and remainder. “R” also denotes Ramanujan, who discovered this association without any formal mathematical training, and Riemann, whose famous zeta function $\zeta[s]$ provides an alternate path to it.

In brief, use the convergence factor to tame infinity by isolating the diverging, remaining, and vanishing parts of the natural numbers sum

$$\sum_{n=1}^\infty n=\color{red}\lim_{\lambda \rightarrow 0}\color{black}\sum_{n=0}^\infty n\color{red}\, e^{-\lambda n}\color{black}=\color{red}\underbrace{\ +\frac{1}{\lambda^2}\ \ }_\text{Diverge}\color{black}\underbrace{\ -\frac{1\vphantom{\lambda^2}}{12}\ \ }_\text{Remain}\color{red} \underbrace{\vphantom{-\frac{1\vphantom{\lambda^2}}{12}}+\mathcal{O}[\lambda^2]}_\text{Vanish}\color{black}\ \stackrel{R}{=}-\frac{1}{12}.$$

Discard the first, retain the second, and let go the third.

## A Better Table

The periodic table of the elements is almost as old as The College of Wooster, and I am a big fan. As we approach next year’s sesquicentennial of Dmitri Mendeleev‘s 1869 periodic table, I present a modest addition to the over 600 known periodic tables, which include $2D$, $3D$, and even $4D$ designs!

I wanted the table to reflect an element’s electron configuration, its energy and angular momentum quantum numbers $n$, $l$, $m_l$ , $m_s$, and its mass $M$ and nuclear charge $Z$. I also did not want to alienate viewers familiar with the classic twentieth-century short-form periodic table, but problems with the short form include gaps and jumps in the presentation and the way the lanthanide and actinide elements appear as footnotes to the main table.

I first created a large electron-configuration periodic table for the Taylor 111 physics lecture hall about 15 years ago. Recently, I learned that French amateur scientist Charles Janet (pronounced “sharl shuh nay”?) first created such designs circa 1930. I have a paper Janet table rolled into a spiral and a wooden one arranged like a stepped layer cake.

In my $2D$ version, rainbow colors code the principle quantum number $n$. The levels and blocks tilt slightly to emphasize that lower means larger mass $M$ and charge $Z$. Lutetium and lawrencium are directly below scandium and yttrium, as they should be, not appended to the end of the lanthanide-actinide footnote. The placement of helium is nontraditional, and if it makes you uncomfortable, I feel your pain, but chemistry literature exists to support it.

Black corner triangles tag elements with exceptional electron configurations, the lightest being chromium and copper. Grayed symbols indicate elements without stable isotopes: technetium, promethium, and all elements heavier than lead. (Bismuth and uranium are nearly stable having isotopes with half-lives much longer than the age of the universe and about the age of Earth.)

Click for a larger version of my latest periodic table design

Posted in Physics | 2 Comments

I just bought a new calculator. New to me, that is, but older than me.

Inspired by the 1600s Gottfried Leibniz stepped cylinder and the 1800s Thomas de Colmar arithmometer, the Curta mechanical calculator design was developed by Curt Herzstark while imprisoned in the Buchenwald concentration camp during World War II. Curta calculators were manufactured between 1947 to 1972 in Liechtenstein until electronic calculators became widely available.

A true number cruncher, the Curta has been nicknamed the pepper grinder because of its crank — or the math grenade due to its resemblance to some hand grenades. My Curta II has about 719 metal parts yet fits comfortably in my hand. I remember seeing advertisements for Curta calculators in Scientific American magazine. Now collector’s items, the one I just bought costs an order of magnitude more.

“The world’s first, last, and only hand-held mechanical calculator”, the Curta is an elegant marvel of mechanical engineering, the culmination of generations of work to mechanize arithmetic. And as I can now attest, it is a joy to hold and use.

A Liebniz wheel at the heart of the Curta computes 5 + 3 = 8 with 2 slides and 2 cranks

## Optical Tweezers

A focused light beam can trap a small particle, such as a micron-sized latex sphere (or biological cell). If the sphere is much larger than the light’s wavelength, ray optics suffices to explain the trapping.

Light bends as it passes through the sphere, as in Fig. A. The piconewton forces (red and blue arrows) on the sphere are equal and opposite to the rate of momenta transferred to the light rays: the rays deflect one way, and the sphere deflects the opposite way. The net force (black arrow) is longitudinal and pushes the bead along the beam. A beam that is less intense at its edge nudges a laterally offset sphere back towards its center, as in Fig. B. A focused beam pulls a sphere behind the focus toward the focus, as in Fig. C. The push in Fig. A stabilized by the nudge in Fig. B balances the pull in Fig. C to hold the sphere just behind the focus, as in Fig. D: a Nobel Prize hiding in plain sight.

Microsphere in a light beam: Fig. A push stabilized by Fig B nudge balances Fig. C pull to trap sphere

A single Nobel Prize cannot be awarded to more than three people and cannot be awarded posthumously. Arthur Ashkin developed optical tweezers in the 1970s and 1980s. Over 20 years ago his collaborator Steven Chu shared the Nobel Prize with Claude Cohen-Tannoudji and William Phillips for trapping atoms. This year at age 96 Ashkin receives the Nobel Prize for his optical tweezing pioneering.