Acceptable Methods

What is an Acceptable Method of solution for a problem? Let’s say a physics problem, such as determining the bound state energy levels in a 1-dimensional finite square well potential. I wrote blog posts about that particular task a couple of months ago (March 28th and 31st). The quantum mechanics textbooks, ones from 1951 to 2006 (the latest date I’ve checked), state (correctly) that the FSW problem is a transcendental equation and (incorrectly) that it can only be solved using graphical or numerical methods. That is, there is no analytic solution.

The truth is, that there IS an analytic solution, known in effect since a paper of Burniston and Siewert, and known even more explicitly in a paper published by Siewert in 1978, where he states the fact of the solution in the title of his paper, and addresses the problem directly. Two later papers present variant methods of solution, one by Paul and Nkemzi in 2000, and one by Blumel in 2006. So there are three methods of exact solution of the FSW problem, all rather related but with variations.

The common feature of those three solution methods is that they use contour integration (usually taught at first year of graduate school in the math methods in physics course), and consider the exact FSW solution as a Riemann-Hilbert problem (2nd year grad school level, if it gets covered at all — not very likely). Whereas introductory quantum mechanics finite square well is a 2nd year undergraduate course topic. The students in the QM course are not expected to understand contour integration, etc. So the exact solution cannot reasonably be given time within the beginning QM course. It is much more important to get on with the FSW as an example of how to think about QM and how to do some practical calculation.

But, it is NOT RIGHT to outright mislead the students, by telling them “the FSW problem CANNOT be solved exactly”. Rather, what should be said is that “the FSW problem can be solved exactly by an advanced math method known as contour integration. We will however use simpler graphical and computational techniques here…”

So, one property of an Acceptable Method seems to be that it will suit the mathematical maturity of the audience.

As mentioned in the prior blog posts, the geometric-analytic technique devised by my colleague and myself is quite simple to describe, definitely accessible at the 2nd year undergraduate level. But is it an Acceptable Method? We had an interesting discussion of this question after a talk which I gave yesterday, to some grad students, detailing the solution technique. The solution is a method, in effect a sort of geometric construction (though beyond what Euclid would have used) using conformal maps of straight lines and circles. I’ll not go into the details here — see the earlier blog posts and our paper (Arxiv 1403.6685) if you want details.

Think about Newton. Newton had published his optics book (Optiks), written in English. He must have gotten some feedback, adverse, about having written in English instead of Latin, the acceptable language of science at the time. So when it came to his Principia (published about 327 years ago), he wrote in Latin. Also, despite having devised (as did Liebnitz) the differential and integral calculus and using that calculus for obtaining his deductions, Newton presented his demonstrations of those deductions in the Acceptable mathematical manner, using geometry.

Newton wanted his book Principia to be accepted, so he wrote it in Latin and using Geometry.

Nowadays, one has to write in English and use Algebra/Calculus. That is the Acceptable Method of our times.

What will be the acceptable method some 327 years in the future?

I read an anecdote (don’t recall the details) of a paper in particle physics, which presented some good and novel ideas, having been rejected by the reviewer, because the paper had not used Feynman diagrams. “I cannot evaluate this paper until the calculations are presented via Feynman diagrams.”

Incidentally, if you are looking for an enjoyment, check out “Feynman’s Lost Lecture”. It consists of an audio recording of a talk which Richard Feynman gave, in which he explained the details of one of Isaac Newton’s geometric demonstrations. Photographs had been made of the chalkboard, and so the lecture was reconstructed. The whole thing is a pleasurable read, and if you wish, you can hear Feynman speak while you read along with the text and following the diagrams.

I guess the term “Feynman diagram” has multiple meanings.

Best wishes,
Ken Roberts
29-May-2014