The Particle in a Box I: the Schrödinger Equation in One-dimension

In 1926, the Austrian physicist Erwin Schrödinger (1887-1961) made a fundamental mathematical discovery that had a profound impact on the study of the molecular world (in 1933, Schrödinger was awarded with the Nobel prize in Physics just 7 years later his breaktrough discovery). He discovered that a state of a quantum system composed by particles (such as electrons and nucleons) can be described by postulating the existence of a function of the particle coordinates and time, called state function or wave function (\Psi, psi function). This function are solution of a wave equation: the so-called the Schrödinger equation (SE). Although the SE equation can be solved analytically only for relatively simple cases, the development of computer and numerical methods has made possible the application of SE to study complex molecular. 

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Buon Natale e Felice Anno Nuovo

Gentilissimi/e Lettori e Lettrici, Dear Reader, Sehr geehrte Leserinnen und Leser,

grazie mille per aver fatto tappa durante le vostre peregrinazioni cibernautiche nel mio sito web e per aver dedicato un po’ del vostro tempo nel leggere i miei ariticoli. Spero che li avete trovati tanto interessanti e utili da continuate a tornare a leggermi.
Voglio anticipare alcune delle prossime publicazioni.
Tra breve usciranno nuovi titoli:

  • The Logistic Map and the Feigenbaum Constants: a Retro Programming Inspired Excursion
  • L’integrazione numerica di equazioni differenziali, parte II: 50 anni fa l’uomo ha messo piede sulla Luna
  • Retro Programming: Acid-base Titration
  • Retro Programming: Plant evolution

Per il momento auguro a tutti voi di trascorrere con le vostri cari un felice Natale e di avere un nuovo anno pieno di buone notizie.

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A personal tribute to the founder of MD simulation of biological molecules: Prof Herman J.C. Berendsen (1934-2019)

On the 7 October 2019, Prof Dr Herman Johan Christiaan Berendsen passed away just shortly after his 85 birthday. Prof Berendsen is considered the founder of the molecular dynamics simulation of biological system: the area of theoretical research that also shaped my scientific career. He was working at the University of Groningen in the pictoresque northen part of the Netherlands. It was there that I meet him the first time as it allowed me to conduct research in his lab during the last year of my doctorate researches training at the University of Rome “La Sapienza”. After I completed my doctorate, Herman gave me the opportunity to continue working in his group with a postdoc position within the “Protein Folding” EU Training network. This happen just two years before his retirement and therefore I was also one of his last postdocs. After retirement, Herman dedicated himself to write two books that distillate all his experience in the area of molecular simulation [1] and in the education [2]. He stated in a project on the social scientific network Researchgate that “I am retired and work occasionally on methods for multiscale simulations.”

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The First 150 Years of the Periodic Table of the Elements

That the nobility of man, acquired in a hundred centuries of trial and error, lay in making himself the conquerer of matter, and that I had enrolled in chemistry because I wanted to remain faithful to this nobility. That conquering matter is to understand it, and understanding matter is necessary to understanding the universe and ourselves: and that therefore Mendeleev’s Periodic Table, which just during those weeks we were laboriously learning to unravel, was poetry, loftier and more solemn than all the poetry we had swallowed down in liceo; and come to think of it, it even rhymed!

Primo Levi, The Periodic Table.

This year marks the 150th anniversary of the periodic table of the elements (TPE) which currently has 118 entries, the latest arrival (the Tennessium) was discovered 10 years ago (2009), and I feel obliged as a chemist to give some a small informative contribution to celebrate this important event.

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Retro programming nostalgia III: the MSX Microcomputer and the Orbit of the Planets in the Solar System

In a recent article, I have explained the Euler’s method for solving ordinary differential equations using as a motivation the fictionalized version in the film Hidden Figures of the scientific contribution of Katherine Goble and her two colleagues to the NASA space program. As an example of application, I have also shown a program written in the awk programming language for calculating the orbits of planets of the solar system. However, my interest in astrodynamics come back to my juvenile age, when still going to high school, my parents decided to gift me a more sophisticated microcomputer than my previous one (the celebrated Commodore VIC 20). So I became a programmer of a Philips MSX VG 8010 that I still jelously own in its original box. So, powered by the versatile Federico Faggin’s Zilog Z80 processor with a clock 3.58 MHz, with an impressive (for a previous owner of a VIC20 with a mere 3.583 kB!) memory of 32 kB RAM , 16kB of video RAM and a dedicated tape-record device as storage system, I started to write more sophisticated in MSX Basic. At that time, I was eagerly following the department “Ricreazioni al Computer” by the famous computer scientist A. K. Dewdney on the magazine “Le Scienze”, the Italian edition of Scientific American. The new microcomputer allowed me to experiment with the fascinating computational topics that Dewdney was offering every month. One of these topics was dedicated to the simulation of stars using the algorithm based on the Euler integration of the Newton equation. Following the instruction of Dewney, I managed to write a small program in MSX basic and this was the starting of my interest in computational astronomy.

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The Logistic Map and the Feigenbaum Constants: a Retro Programming Inspired Excursion

“… Mitchell Feigenbaum was an unusual case. He had exactly one published article to his name, and he was working on nothing that seemed to have any particular promise. His hair was a ragged mane, sweeping back from his wide brow in the style of busts of German composers… At the age of twenty he had already become a savant among savants, an ad hoc consultant [at Los Alamos National Laboratory, USA] whom scientist would go to see about any expecially intractable problem.”

James Gleick, Chaos: the amazing science of the unpredectable.

This year, on June 30th 2019, Mitchell J. Feigenbaum died at the age of 74. Feigenbaum was an American mathematician that became famous with the discovery of the universal constants that bear his name. In the middle of the ’80, reading Le Scienze magazine (the Italian edition of Scientific American) I got to know of the contribution to the chaos theory of this charismatic mathematician. In particular, I was delighted by reading the Douglas Hofstadter’s article in the rubric “Temi Metamagici” ( Methamagical themes) (Scientific American, November 1981). The article explained the emergence of the chaos in the iteration map of the logistic equation, the same equation deeply studied by Feigenbaum. The full story about the Mitchell Feigenbaum and his discovery of his universal constants is delightly narrated in the beautiful book Chaos:the amazing science of the unpredectable by J. Gleick [1]. Here it is just another small extract:

“… in the summer of 1975, at a gathering in Aspen, Colorado, he heard Steve Smale [another key mathematicial in the developing of the chaos theory, NDA] talk about some of the mathematical qualities of the same quadratic difference equation [the same studied by Robert May, NDA]. Smale seemed to think that there were some interesting open questions about the exact point at which the mapping changes from periodic to chaotic. As always, Smale had a sharp instinct for questions worth exploring. Feigenbaum decided to look into it once more.”[1]

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Numerical Integration of Differential Equations. Part I.: Katherine Goble and the Euler’s Method.

This article was inspired by the beautiful 2016 movie Hidden Figures (based on the book of the same name by M. L. Shetterley) which tell the dramatic story of three talented black women scientist that worked as “human computers” for NASA in 1961 for the Mercury project.

Figure 1: Official theatrical poster of the movie and the phFoto of the real protagonist. From left to right. Mary Jackson, Katherine Goble and Dorothy Vaughan. (source: wikipedia)

In the movie, the mathematician Katherine Goble (interpreted by Taraji P. Henson), had a brilliant intuition on how to numerically solve the complex problem to find the transfer trajectory for the reentry into the Earth atmosphere of the Friendship 7 capsule with the astronaut John Glenn on board. In the particular scene, she was standing together with other engineers and the director of the Langley Research Center (a fictional character interpreted by Kevin Coster) in front of the vast blackboard looking to graph and equations when she says that the solution might be in the “old math” and she runs to take an old book from a bookshelf with the description of the Euler method. The scene is also nicely described in the youtube video lesson by Prof. Alan Garfinkel of the UCLA. A detailed description of the numerical solution based on the original derivation of K. Globe is in the Wolfram blog website.

Katherine Globe was using for these complex calculation her brilliant brain with the support of a mechanical calculator (the Friden STW-10, in the movie, this machine is visible in different scenes). In a scene of the film, she revealed that her typical computing performance was of 10000 calculations per day and probably for calculations, she was not referring to single arithmetic operations! These exceptional mathematical skills have given a significative contribution at the beginning of the American space program, but it became insufficient to handle the more complex mathematics necessary to land the man on the Moon, and the other fantastic NASA achievements.

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