guide:D40ccd2cbc

[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Getting now to some truly exciting applications of light and spectroscopy, let us discuss the beginnings of the atomic theory. There is a long story here, involving many discoveries, around 1890-1900, focusing on hydrogen [math]{\rm H}[/math]. We will present here things a bit retrospectively. First on our list is the following discovery, by Lyman in 1906: \begin{fact}[Lyman] The hydrogen atom has spectral lines given by the formula

[[math]] \frac{1}{\lambda}=R\left(1-\frac{1}{n^2}\right) [[/math]]

where [math]R\simeq 1.097\times 10^7[/math] and [math]n\geq2[/math], which are as follows,

[[math]] \begin{matrix} n&{\rm Name}&{\rm Wavelength}&{\rm Color}\\ &-&-\\ 2&\alpha&121.567&{\rm UV}\\ 3&\beta&102.572&{\rm UV}\\ 4&\gamma&97.254&{\rm UV}\\ \vdots&\vdots&\vdots&\vdots\\ \infty&{\rm limit}&91.175&{\rm UV} \end{matrix} [[/math]]

called Lyman series of the hydrogen atom. \end{fact} Observe that all the Lyman series lies in UV, which is invisible to the naked eye. Due to this fact, this series, while theoretically being the most important, was discovered only second. The first discovery, which was the big one, and the breakthrough, was by Balmer, the founding father of all this, back in 1885, in the visible range, as follows: \begin{fact}[Balmer] The hydrogen atom has spectral lines given by the formula

[[math]] \frac{1}{\lambda}=R\left(\frac{1}{4}-\frac{1}{n^2}\right) [[/math]]

where [math]R\simeq 1.097\times 10^7[/math] and [math]n\geq3[/math], which are as follows,

[[math]] \begin{matrix} n&{\rm Name}&{\rm Wavelength}&{\rm Color}\\ &-&-\\ 3&\alpha&656.279&{\rm red}\\ 4&\beta&486.135&{\rm aqua}\\ 5&\gamma&434.047&{\rm blue}\\ 6&\delta&410.173&{\rm violet}\\ 7&\varepsilon&397.007&{\rm UV}\\ \vdots&\vdots&\vdots&\vdots\\ \infty&{\rm limit}&346.600&{\rm UV} \end{matrix} [[/math]]

called Balmer series of the hydrogen atom. \end{fact} So, this was Balmer's original result, which started everything. As a third main result now, this time in IR, due to Paschen in 1908, we have: \begin{fact}[Paschen] The hydrogen atom has spectral lines given by the formula

[[math]] \frac{1}{\lambda}=R\left(\frac{1}{9}-\frac{1}{n^2}\right) [[/math]]

where [math]R\simeq 1.097\times 10^7[/math] and [math]n\geq4[/math], which are as follows,

[[math]] \begin{matrix} n&{\rm Name}&{\rm Wavelength}&{\rm Color}\\ &-&-\\ 4&\alpha&1875&{\rm IR}\\ 5&\beta&1282&{\rm IR}\\ 6&\gamma&1094&{\rm IR}\\ \vdots&\vdots&\vdots&\vdots\\ \infty&{\rm limit}&820.4&{\rm IR} \end{matrix} [[/math]]

called Paschen series of the hydrogen atom. \end{fact} Observe the striking similarity between the above three results. In fact, we have here the following fundamental, grand result, due to Rydberg in 1888, based on the Balmer series, and with later contributions by Ritz in 1908, using the Lyman series as well: \begin{conclusion}[Rydberg, Ritz] The spectral lines of the hydrogen atom are given by the Rydberg formula, depending on integer parameters [math]n_1 \lt n_2[/math],

[[math]] \frac{1}{\lambda_{n_1n_2}}=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) [[/math]]

with [math]R[/math] being the Rydberg constant for hydrogen, which is as follows:

[[math]] R\simeq1.096\ 775\ 83\times 10^7 [[/math]]

These spectral lines combine according to the Ritz-Rydberg principle, as follows:

[[math]] \frac{1}{\lambda_{n_1n_2}}+\frac{1}{\lambda_{n_2n_3}}=\frac{1}{\lambda_{n_1n_3}} [[/math]]

Similar formulae hold for other atoms, with suitable fine-tunings of [math]R[/math]. \end{conclusion} Here the first part, the Rydberg formula, generalizes the results of Lyman, Balmer, Paschen, which appear at [math]n_1=1,2,3[/math], at least retrospectively. The Rydberg formula predicts further spectral lines, appearing at [math]n_1=4,5,6,\ldots\,[/math], and these were discovered later, by Brackett in 1922, Pfund in 1924, Humphreys in 1953, and others afterwards, with all these extra lines being in far IR. The simplified complete table is as follows:

[[math]] \begin{matrix} n_1&n_2&{\rm Series\ name}&{\rm Wavelength}\ n_2=\infty&{\rm Color}\ n_2=\infty\\ &&-&-\\ 1&2-\infty&{\rm Lyman}&91.13\ {\rm nm}&{\rm UV}\\ 2&3-\infty&{\rm Balmer}&364.51\ {\rm nm}&{\rm UV}\\ 3&4-\infty&{\rm Paschen}&820.14\ {\rm nm}&{\rm IR}\\ &&-&-\\ 4&5-\infty&{\rm Brackett}&1458.03\ {\rm nm}&{\rm far\ IR}\\ 5&6-\infty&{\rm Pfund}&2278.17\ {\rm nm}&{\rm far\ IR}\\ 6&7-\infty&{\rm Humphreys}&3280.56\ {\rm nm}&{\rm far\ IR}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \end{matrix} [[/math]]

Regarding the last assertion, concerning other elements, this was something conjectured and partly verified by Ritz, and fully verified and clarified later, via many experiments, the fine-tuning of [math]R[/math] being basically [math]R\to RZ^2[/math], where [math]Z[/math] is the atomic number.

From a theoretical physics viewpoint, the main result remains the middle assertion, called Ritz-Rydberg combination principle. This is something at the same time extremely simple, and completely puzzling, the informal conclusion being as follows: \begin{thought} The simplest observables of the hydrogen atom, combining via

[[math]] \frac{1}{\lambda_{n_1n_2}}+\frac{1}{\lambda_{n_2n_3}}=\frac{1}{\lambda_{n_1n_3}} [[/math]]

look like quite weird quantities. Why wouldn't they just sum normally. \end{thought} Fortunately, mathematics comes to the rescue. Indeed, the Ritz-Rydberg combination principle reminds the formula [math]e_{n_1n_2}e_{n_2n_3}=e_{n_1n_3}[/math] for the usual matrix units [math]e_{ij}:e_j\to e_i[/math]. In short, we are in familiar territory here, and we can start dreaming of: \begin{thought} Observables in quantum mechanics should be some sort of infinite matrices, generalizing the Lyman, Balmer, Paschen lines of the hydrogen atom, and multiplying between them as the matrices do, as to produce further observables. \end{thought} Good news, time to put everything together. As a main problem that we would like to solve, we have the understanding the intimate structure of matter, at the atomic level. There is of course a long story here, regarding the intimate structure of matter, going back centuries and even millennia ago, and our presentation here will be quite simplified. As a starting point, since we need a starting point, let us agree on: \begin{claim} Ordinary matter is made of small particles called atoms, with each atom appearing as a mix of even smaller particles, namely protons [math]+[/math], neutrons [math]0[/math] and electrons [math]-[/math], with the same number of protons [math]+[/math] and electrons [math]-[/math]. \end{claim} As a first observation, this is something which does not look obvious at all, with probably lots of work, by many people, being involved, as to lead to this claim. And so it is. The story goes back to the discovery of charges and electricity, which were attributed to a small particle, the electron [math]-[/math]. Now since matter is by default neutral, this naturally leads to the consideration to the proton [math]+[/math], having the same charge as the electron.

But, as a natural question, why should be these electrons [math]-[/math] and protons [math]+[/math] that small? And also, what about the neutron 0? These are not easy questions, and the fact that it is so came from several clever experiments. Let us first recall that careful experiments with tiny particles are practically impossible. However, all sorts of brutal experiments, such as bombarding matter with other pieces of matter, accelerated to the extremes, or submitting it to huge electric and magnetic fields, do work. And it is such kind of experiments, due to Thomson, Rutherford and others, “peeling off” protons [math]+[/math], neutrons [math]0[/math] and electrons [math]-[/math] from matter, and observing them, that led to the conclusion that these small beasts [math]+,0,-[/math] exist indeed, in agreement with Claim 8.27.

Of particular importance here was as well the radioactivity theory of Becquerel and Pierre and Marie Curie, involving this time such small beasts, or perhaps some related radiation, peeling off by themselves, in heavy elements such as uranium [math]\!\!{\ }_{92}{\rm U}[/math], polonium [math]\!\!{\ }_{84}{\rm Po}[/math] and radium [math]\!\!{\ }_{88}{\rm Ra}[/math]. And there was also Einstein's work on the photoelectric effect, light interacting with matter, suggesting that even light itself might have associated to it some kind of particle, called photon. All this goes of course beyond Claim 8.27, with further particles involved, and more on this later, but as a general idea, all this deluge of small particle findings, all coming around 1900-1910, further solidified Claim 8.27.

So, taking now Claim 8.27 for granted, how are then the atoms organized, as mixtures of protons [math]+[/math], neutrons [math]0[/math] and electrons [math]-[/math]? The answer here lies again in the above-mentioned “brutal” experiments of Thomson, Rutherford and others, which not only proved Claim 8.27, but led to an improved version of it, as follows:

\begin{claim} The atoms are formed by a core of protons [math]+[/math] and neutrons [math]0[/math], surrounded by a cloud of electrons [math]-[/math], gravitating around the core. \end{claim} This is a considerable advance, because we are now into familiar territory, namely some kind of mechanics. And with this in mind, all the pieces of our puzzle start fitting together, and we are led to the following grand conclusion: \begin{claim}[Bohr and others] The atoms are formed by a core of protons and neutrons, surrounded by a cloud of electrons, basically obeying to a modified version of electromagnetism. And with a fine mechanism involved, as follows:

• The electrons are free to move only on certain specified elliptic orbits, labeled [math]1,2,3,\ldots\,[/math], situated at certain specific heights.
• The electrons can jump or fall between orbits [math]n_1 \lt n_2[/math], absorbing or emitting light and heat, that is, electromagnetic waves, as accelerating charges.
• The energy of such a wave, coming from [math]n_1\to n_2[/math] or [math]n_2\to n_1[/math], is given, via the Planck viewpoint, by the Rydberg formula, applied with [math]n_1 \lt n_2[/math].
• The simplest such jumps are those observed by Lyman, Balmer, Paschen. And multiple jumps explain the Ritz-Rydberg formula.

\end{claim} And isn't this beautiful. Moreover, some further claims, also by Bohr and others, are that the theory can be further extended and fine-tuned as to explain many other phenomena, such as the above-mentioned findings of Einstein, and of Becquerel and Pierre and Marie Curie, and generally speaking, all the physics and chemistry known.

And the story is not over here. Following now Heisenberg, the next claim is that the underlying mathematics in all the above can lead to a beautiful axiomatization of quantum mechanics, as a “matrix mechanics”, along the lines of Thought 8.26. We will be back to all this in chapter 16 below, at the end of the present book.

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].