How to Find Papers

A site named “ResearchGate” has become one of my tools for locating papers. The site is a bit annoying at times. Sort of like Facebook, for scholarly work. There is a continual stream of “helpful” emails, and popups asking one to consider a job posting, or add a publication, etc etc. Rather like a hyperactive and gossipy personal assistant. Also, there is a lack of confidentiality — if one “follows” a particular paper, to be apprised of future citations of it, all of one’s contacts are advised that one is following that paper. So if one wants to work under the radar, then ResearchGate is not the tool for you. Be advised!

However, on the plus side: ResearchGate is excellent as a way of finding out what a person has published, who has cited it, what that person has published, and so on. I recently was looking at a paper by Neil Turok, “On quantum tunneling in real time”, and wanted to find out if anyone had cited it. Web of Science / Knowledge said not. But Google Scholar reported some related papers, and one of them led me to a paper by Carl Bender and Daniel Hook which cites the Turok paper. And that, in turn, via Hook’s ResearchGate publications list, led me to the very interesting and stimulating paper by Bender and Hook, Arxiv 1011.0121, “Quantum tunneling as a classical anomaly”.

That enables me to return to a subject I’ve been interested in for some time, whether the complex plane tangencies of the Lambert W lines with the strength contours of a quantum well, represent sensitivities which have a physical implication — for example, whether one can devise a sensor which uses that tangency. The QWIP, quantum well infrared photodetector, which is found in night vision apparatus, is an example of such a sensor. In general, a sensor can be made by conditioning a quantum well device at or near a context which changes the number of bound states or a tunneling probability, and then allowing the environment to stress the sensor — changing the energy, changing a dimension, changing temperature, and so on. Finding the Bender and Hook papers, and a couple of Turok papers, offers the possibility of a new look at that topic.

Best wishes,
Ken Roberts
31-Jan-2016

Finite Square Well

Finally the paper my colleague and I have been working on, about a description of the exact solutions of the finite square well problem in quantum mechanics, is complete and on the Arxiv database as number 1403.6685. I want to talk about those ideas a bit here. I’ll leave the details of the math in the paper, and here try to outline the problem and the method we devised to deal with it.

The finite square well problem refers to the possible energy levels of a particle which is in a bound state in a potential well. It is a staple of the introductory quantum mechanics course. If you’ve taken such a course, you may recall that it is customary to hear some remark about the FSW problem not being exactly solvable, and then be advised to try a solution on the computer. That’s not really correct — there are exact solutions of the FSW, but the solutions involve contour integration, which is a mathematical technique not usually covered at the time one is taking an introductory QM course.

My colleague and I have devised a simple way to describe the exact solutions of the FSW problem. It turns out that the solutions can be described via either of two copies of the complex plane, let’s call them the w-plane and the z-plane, and they are related by the complex variable w in the w-plane mapping to the complex variable z = w*exp(w) in the z-plane. Simple geometric shapes — lines and circles — mapped back and forth between those two planes, are sufficient to construct the exact solutions. Look at the two diagrams below, which show the w-plane and the z-plane.

First the w-plane:

This has a circle, in this example drawn with radius R = 5. The value R is called the strength parameter of the FSW problem, and is a pure number (unitless) based upon the dimensions of the square well — its spatial dimensions, and its potential depth. In material design, it can be altered based upon the choice of materials and structures. Here I’m using 5 to illustrate the solution. You can see some curvy lines on the w-plane diagram which intersect the circle — 14 intersections marked with dots. I’ll talk in a minute about those curvy lines. Those intersections correspond to the solutions. If you’ve looked at the FSW problem solution in a QM textbook, you probably recognize the first quadrant of this w-plane diagram as very similar to the one in your textbook, but flipped, the axes interchanged.

Now let’s look at the z-plane:

The circle has mapped, under the w –> z = w*exp(w) map, to a multi-loop closed curve which wraps several times around the origin. Wherever that curve intersects one of the axial rays — the four rays which go outward from the origin, along the positive or negative real axial rays, or the positive or negative imaginary axial rays, there is a solution. There are 14 such intersections, counting multiples where the looping curve crosses an axial ray along two different trajectories. You cannot see all the intersections in that diagram, because lots of them are close to the origin. The paper on Arxiv has magnifications which show the structure of the curve close to the origin. Or you can play with the curve on your computer — that is probably going to show you the curve’s behavior in a more memorable manner. And you may wish to see what happens to the curve as you adjust the value of R. Try R = 4.603 for a fun time!

So, what happens to the axial rays under the inverse of the w –> z map? They go into the curvy lines on the w-plane diagram. Those are called Lambert W lines, as the inverse of the w –> z = w*exp(w) map is the Lambert W function. It has multiple branches. Only the branches called (by convention) branches -1, 0 and 1 are shown in the w-plane diagram above, as it is only those branches whose Lambert lines have intersections with the R=5 circle.

So there you have it — an exact solution of the FSW problem, achieved by a simple mapping from one complex domain to another, a circle in the w-plane mapping to a multi-loop curve in the z-plane, and axial rays in the z-plane mapping to Lambert lines in the w-plane. The intersections are the solutions which give the energy levels which a particle bound in a quantum finite square well can possess. You can work in either the w-plane or the z-plane whichever is convenient for the rest of your task related to an FSW model.

Is there an application? Yes there is. This is a nice result, regardless of application. Looking at these diagrams in the whole complex plane, instead of only the first quadrant, shows that the R=4.603 value produces a tangency of the R-circle to the Lambert line. (As do several other R-values — you can explore the details.) That tangency occurs in the second and third w-quadrants, not the first w-quadrant which is the one usually considered. (If one worked only in the first quadrant, the R value of interest might be determined to be R=4.712, close but maybe not quite on.) Tangency in the w-plane corresponds to tangency in the z-plane, because the mapping is conformal — that is, it preserves angles of intersection. Two curves in the w-plane which intersect at some angle (or are tangent, ie just graze one another), when transformed to the z-plane, will there intersect at the same angle (or graze) — and vice versa.

There is a device known as a QWIP — Quantum Well Infrared Photodetector — which is commonly used in infrared imaging applications. QWIPs, at their most simple description, are based upon materials which have quantum wells structured so that a single photon, at infrared energy, will raise an electron within the well from its ground state to an excited state, which has energy just about the same as the depth of the well. That is, the electron when excited is really very loosely bound. A bit of bias and one gets a current whenever an infrared photon arrives. Tangency corresponds to sensitivity. Real QWIPs are not as simple as the FSW. There are many complexities and practicalities. Still, the fact that R=4.603 (and other radii) is a tangency strength parameter may be of some use in improving QWIP devices.

The paper is at http://arxiv.org/abs/1403.6685 if you are interested in the details.

If you are interested in discussing the scientific aspects, please drop me an email.

Oh — by the way — the quadratix of Hippias is also related to these FSW solutions.

Best wishes,
Ken Roberts
28-Mar-2014