The Birth of Quantum Mechanics Tutorial
Creator of Tutorial: Michele_Laino

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$$\qquad \qquad \qquad \qquad{\mathbf{The\; Days\; of \;Quantum \;Mechanics}}$$ $$\qquad \qquad \qquad \qquad \qquad \qquad \qquad {\mathbf{Abstract}}$$ This brief article is a summary about how the famous scientist Karl Werner Heisenberg, came to formulate his mechanics of matrices. Between the ligatures << >>, I reported the original comments of Heisenberg, furthermore, at the end of the present article, I give my reference publication, on which, these notes are based. $$\qquad \qquad \qquad \qquad {\mathbf{On\; the\; year\; of\; 1925: \;The\; birth\; of\; a \;new \;theory}}$$ On the April of 1925 Heisenberg completed his article about the anomalous Zeeman effect, and after that he left Copenhagen for G\"ottingen. At this point, his objective was <>. In order to do that, he turned his attention to his paper, written some time before, with the collaboration of H. A. Kramers about the spectral lines of hydrogen atom, nevertheless such starting point of his research was not fruitful <>. Therefore, Heisenberg turned his interest to simpler mechanical systems. In his letter dated 5th of June, he provided the first clues of what he was searching for. Nevertheless in that period of time he didn't feel good: <>. So he decided to searching for a better air, and on the 7th June he departed for the Helgoland island in the North Sea. <>. There, at Helgoland Heisenberg slept very little, he spent his time inventing the quantum mechanics, by climbing out on many rocks of that island, and learning by heart the poems from the western-eastern divan by Goethe. It was at Helgoland that Heisenberg made the decisive step. <>. That was <>. On the 19th of June Heisenberg went back to G\"ottingen. On the 24th of June he sent to W. Pauli a summary of his own results, noting that: <>. On the 9th of July he wrote to Pauli which his point of view became, day by day, increasingly more radical, and his attempts were directed to eliminate the concept of orbit. The reaction of Pauli was definitely positive. On the 25th of July the magazine Zeitschrift f\"ur Physik received the paper of Heisenberg which announced the discovery of quantum mechanics. $$\qquad \qquad \qquad \qquad{\mathbf{On\; the\; Heisenberg\; paper\; of\; 1925}}$$ From the abstract we can read: <> (...) <> (...) Then he starts in this way: <>. Heisenberg then terminates his paper, highlighting the necessity <>. Here are the essential aspects of the reasoning of Heisenberg. What is, in one dimension, a classical orbit? It is a geometrical shape described by one coordinate, $x$, which varies, continuously as a function of time $t$, so we will use the notation $x(t)$. At this point Heisenberg gets the inspiration from his preceding work, written in collaboration with Kramers. In that case, the problem was how to determine the amplitude $A(\nu)$ for scattering of a radiation with frequency $\nu$ from the hydrogen atom. Such amplitude had to depend on the transitions between atomic states $n$ to $m$, as noted by the symbol $A_{mn}(\nu)$ (such symbol was not used in the paper with Kramers). At this point, Heisenberg try to apply such reasoning to the orbit $x(t)$, so we represent such orbit by the quantum symbol $x_{mn}(t)$, where, the indexes $m,\;n$ refer to quantum states of a harmonic oscillator. There are two possibilities: the first is that indexes $m,\;n$ are equal each to other, therefore $x_{mn}(t)$ will represent the coordinate of $x$ at time $t$ when the system is in the state $n$; or it is in the state $m$; the second possibility is when $m$ and $n$ are different each from other, in such case $x_{mn}(t)$ will represent what we could call a transition coordinate. ***ogously the classical speed, $v(t)$ will be substituted by $v_{mn}(t)$. All these quantities satisfy the Heisenberg criterion, namely, to be in principle \emph{observables}. Classically, the continuous orbit $x(t)$ obeys to an equation which tells us how the particle moves from an initial position and speed, to another one. Heisenberg takes the guess that every quantity $x_{nn}(t)$ obeys to the same equation. Then he asks to himself: what is the energy of the particle which is in the state $n$? He also takes in consideration, for energy, the classical expression, as a function of $x(t)$ and $v(t)$. In classical mechanics the procedure consists in determining the solutions of the equation of motion and to substitute them into the expression for energy, getting so the corresponding values of the latter. Heisenberg proceeds in the same manner in order to attempt to determine the quantized energies $E_n$. Nevertheless in that case he have to make a drastic choice. The energy of an oscillator (as in our example) depends on the square of $x(t)$ and $v(t)$. If we represent $x(t)$ with $x_{mn}(t)$, what becomes $x^2(t)$? <>: $$x_{mn}^2\left( t \right) = \sum\limits_k {\left( {{x_{mk}}\left( t \right)} \right) \cdot } \left( {{x_{kn}}\left( t \right)} \right)$$ where $\sum\limits_k {}$ means <>. Since 1925 onwards, such hypothesis has proven to be perfectly guessed. Before to go on, Heisenberg notes that <>, however, a significant difficulty when we consider two quantities $x(t)$ and $y(t)$, and we want to express their product $x(t)y(t)$. Based on what we have wrote before for $x^2(t)$ we have to suppose, more in general, that it is: $${\left[ {x\left( t \right)y\left( t \right)} \right]_{mn}} = \sum\limits_k {\left( {{x_{mk}}\left( t \right)} \right) \cdot } \left( {{y_{kn}}\left( t \right)} \right)$$ Heisenberg was upset by this unusual property of multiplication: <>. Turning on the question of energy, Heisenberg asks to himself: <> The classical energy was substituted by the set $H_{mn}$: it was necessary to show that the energy $H_{nn}=E_n$ doesn't depend on time $t$, that $H_{mn}$ is equal to zero when $m$ is different from $n$ (energy doesn't make jumps), and that the values of $E_n$ are quantized (discrete values). While Heisenberg was at Helgoland, he was able to show all those properties, which afterward he called <>. The 19th of June Heisenberg came back to G\"ottingen, and on 9th of July he finished transcribing his results: He carried his work to M. Born, and asked to him to submit it for publishing on some magazine, if, of course, it also liked to him. Subsequently, Heisenberg left for Leida and Cambridge, for a holiday; he came back again to G\"ottingen, only at the end of august. Shortly after, he wrote to Pauli, speaking about on <>. In deliver his own work to Born, Heisenberg had said (as the same Born remembered) <>. Finally <>. (The preceding ligatures enclose a comment of M. Born). At this point Born starts, with the collaboration of his student Pascual Jordan, to transcribe and to extend systematically the work of Heisenberg using the language of matrices. When Heisenberg, who was still in holidays, received a copy of their manuscript, he was <>. Mathematical entities, like $x_{mn}$ and $v_{mn}$ are called <>; it is better to write them as a square table wherein the element $x_{12}$, for example, is located at the intersection between the second row with the third column. A diagonal matrix, is a matrix whose non-zero elements are the ones for which $m=n$. Born and Jordan showed that the conditions put by Heisenberg for the matrix of energy, namely to have non-zero elements and which are independent from time, only along the main diagonal, can be satisfied, generally, from all one-dimensional mechanical systems. The main relationship of their work regards the matrix $p_{mn}$ of momentum (mass multiplied by velocity), and the matrix $q_{mn}$ of the coordinate of position for any one-dimensional system: $$\sum\limits_k {\left( {{p_{nk}} \cdot {q_{km}} - {q_{nk}} \cdot {p_{km}}} \right)} = \left\{ {\begin{array}{*{20}{c}} {h/2\pi i\quad {\text{if}}\;n = m} \ {0\quad {\text{ if}}\;n \ne m} \end{array}} \right.\;.$$ Shortly after the completion of this little paper, Born received a copy of a work by the British physicist P. A. M. Dirac, containing many of the results just found by him and Jordan. <>. Dirac had found the same \emph{``relationships of commutation"} between $p_{mn}$ and $q_{mn}$ and he had written them in the more compact form: $$pq - qp = \frac{h}{{2\pi i}}{\mathbf{I}}$$ where $\mathbf{I}$ is the identity matrix, whose elements are worth $1$ along the main diagonal and zero off diagonal. At that time Dirac invented different notations which are remained in the language used by physicists. The numbers $a,b,c,...$ which they commute with any other quantity $x: (ax=xa)$ are called $c-$numbers, where $c$ stands for classic, or commuting. The numbers which don't commute are called $q-$numbers, where $q$ stands for quantum, or queer. When Heisenberg wrote his first article on quantum mechanics, he didn't know the matrix calculus: <>. He put himself at equal, rapidly, and he started at soon to collaborate with Born and Jordan. The resultant paper completed the first stage of the formulation of the bases of matrix mechanics, which was making Heisenberg exasperated, since he considered such calculus type too mathematical. The derivation of the formula of Balmer, which was obtained by the same Pauli in the autumn of 1925, remains one of the most difficult computations in the scope of matrices methods. It also was the first case wherein the new techniques were applied to a solution of a real atomic problem, ***ogously as made by Bohr, for the hydrogen atom in the scope of the old quantum theory. At the beginning of November, Heisenberg expressed his own admiration for the skill of Pauli. $$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\mathbf{References}}$$ Pais A., Il Danese Tranquillo. Niels Bohr un Fisico e il suo Tempo 1885-1962, Bollati-Boringhieri, Torino (Italy), 1993. Original Title: \textit{Niels Bohr's Times. In physics, Philosophy and Polity}, Oxford University Press, 1981.