Discovery of the Relativistic Schrödinger equation

There are many great reminiscences and reviews of remarkable events of the early days of quantum mechanics. Among those are [2], [4], [14], [17], [18], [26], [30], [36], [37], [38], [39], [41], [42], [46], [56], [60], [61] (and references therein), as well as great biographies of the ‘pioneers’ of the quantum revolution, namely, Werner Heisenberg [6], [7], Erwin Schrödinger [25], [43], and Paul Dirac [19].

A traditional story of the discovery of Schrödinger’s wave equation – “rumor has it”– goes this way [2], [25], [43]: In early November 1925, Peter Debye, at the ETH,¹¹1Swiss Federal Institute of Technology in Zürich. asked Schrödinger to prepare a talk for the Zürich physicists about de Broglie’s work just published in Annales de Physique. The date on which this particular colloquium took place has not been recorded, but it must have been in late November or early December, before the end of the academic term [25], [39]. According to Felix Bloch [2],²²2An alternative recollection is due to Kapitza [28]. at the end of Schrödinger’s talk, Debye casually remarked that (de Broglie’s) way of talking was rather childish and someone had to think about an equation for these (de Broglie’s) ‘matter waves’. As a student of Sommerfeld, Debye had learned that to deal properly with waves, one had to have a wave equation. Indeed, at his second talk, just a few weeks later, Schrödinger mentioned: “My colleague Debye suggested that one should have a wave equation; well, I have found one!”

For a sequence of those remarkable events, a direct quote is quite appropriate [25]: “Schrödinger’s first thought was to attempt to find a wave equation, moving on from de Broglie’s work, that would describe the behavior of an electron in the simplest atom, hydrogen. He naturally included in his calculations allowance for the effects described by the special theory of relativity, deriving, probably early in December 1925, what became known as the relativistic Schrödinger (not hydrogen—our correction!) equation. Unfortunately, it didn’t work. The predictions of the relativistic equation did not match up with observations of real atoms. We now know that this is because Schrödinger did not allow for the quantum spin of the electron, which is hardly surprising, since the idea of spin had not been introduced into quantum mechanics at that time. But it is particularly worth taking note of this false start, since it highlights the deep and muddy waters in which quantum physicists go swimming—for you need to take account of spin, a property usually associated with particles, in order to derive a wave equation for the electron!³³3The concept of electron spin was introduced by G. E. Uhlenbeck and S. Goudsmit in a letter published in Die Naturwissenschaften; the issue of 20 November 1925; see [35] for more details.

With the Christmas break coming up, he had an opportunity to get away from Zürich and think things over in the clean air of Arosa … [25] .”(See also [39] and [43] for more details⁴⁴4Arosa, an Alpine Kurort at about 1800 m altitude, not far from ski-resort Davos, and overlooked by the great peak of the Weisshorn. For a related video, see: https://www.news.uzh.ch/en/articles/2017/Schroedinger.html . Here, among other things, a female physicist meets the grandson of Dr. Herwig and he shows the entry of the payment in a guest book, done by Schrödinger. .)

Whereas the honorable historians of science are arguing about the sequence and details of some events [26], [30], [39], [61], our remarks are dedicated to a somewhat missing, modern mathematical, part of this story, namely, the discovery of a relativistic (spin-zero) wave equation by Erwin Schrödinger (and why he did report and publish only the centennial nonrelativistic version for the hydrogen atom).

How Schrödinger derived his celebrated equation and subsequently applied it to the hydrogen atom? According to his own testimony [49], [54], de Broglie’s seminal work on a wave theory of matter of 1923–24 and Einstein’s work on ideal Bose gases of 1924–25 laid the ground for the discovery of wave mechanics. The crucial phase is dated approximately from late November 1925 to late January 1926 (see, for instance, [30] and [39]).

Let us start our discussion by following the original Schrödinger’s idea — “rumor has it”! For a classical relativistic particle of charge $e$ in the electromagnetic field, one has to replace

by

In relativistic quantum mechanics, we replace the four-vector of energy $E$ and linear momentum $\mathbf{p}$ with the following operators

As a result, we arrive at the relativistic Schrödinger equation [12], [47]:

In the special case of Coulomb field,

one can separate the time and spatial variables

Finally, the stationary relativistic Schrödinger equation takes the form

(As one can see, it is NOT a traditional eigenvalue problem!) The standard rumor says that Schrödinger invented this equation sometime in late 1925, found the corresponding relativistic energy levels, and derived the fine structure formula, different from Sommerfeld’s one [13]. This is why he had to abandon this equation and to consider a nonrelativistic approximation. Moreover, they say that he had to withdraw the ‘relativistically framed’ article, whose draft never been found, from a journal and start all over with his centennial article on the nonrelativistic Schrödinger equation [49]. Our goal here is to follow the original ‘relativistic idea’. Our analysis, nonetheless, will suggest that it is hardly possible to believe that a complete relativistic solution (for a charged spin-zero particle) had been found at that time (see, nonetheless, [39] for an opposite opinion and a remarkable article of Vladimir A. Fock [20] for the first published solution of this problem among other things; see also [21]).

We follow [44] with somewhat different details (see also [11], [20], [47] for a different approach) and, first of all, separate the variables in spherical coordinates, $\chi\left(r,\theta,\varphi\right)=R(r)Y_{lm}(\theta,\varphi),$ where $Y_{lm}(\theta,\varphi)$ are the spherical harmonics with familiar properties [59]. As a result,

(see Figure 3 for the original version⁵⁵5Schrödinger’s notebooks are reproduced in the Archive for the History of Quantum Physics (AHQP); for more details, see [26], [39]. ). In the dimensionless quantities,

for the new radial function,

one gets

Given the identity,

we finally obtain

This is the generalized equation of hypergeometric form (A.1) below, when

(see also Table 2 provided in Appendix A below).

In view of the normalization condition,

it’s useful to start from asymptotic as $r\rightarrow 0$ $($or $x\rightarrow 0).$ In this limit, from (3.6), one gets Euler’s equation:

which can be solved by the standard substitution $u=x^{\alpha},$ resulting in the following quadratic equation:

Thus the general solution of Euler’s equation has the form

which is bounded as $x\rightarrow 0$ with $C_{2}=0:$ $\ u\simeq C_{1}x^{\nu+1}\/.$⁶⁶6The substitution $u=x^{\nu+1}U$ in (3.6) results in Laplace’s equation of the form: $$xU^{\prime\prime}+2(\nu+1)U^{\prime}+\left(2\varepsilon\mu-a^{2}x\right)U=0\/\quad\left(a=\sqrt{1-\varepsilon^{2}}\right)\/,$$ which can be solved by a complex integration [33], [48]; see also Appendix C for some details. According to [39], Schrödinger did analyze these integral solutions in his three page Notebook $N1$! (As the first step towards solution of the nonrelativistic problem in Arosa?) It was not realized at that time that a simple substitution $U=e^{-ax}y$ results in the Sturm–Liouville eigenvalue problem for the Laguerre polynomials! (Thus the quantization rule is related to the number of zeros of the radial wave functions.)

We continue our analysis of Schrödinger’s equation (3.6), applying Nikiforov–Uvarov’s technique, and transform (A.1) into the simpler form (A.3) by using $u\left(x\right)=\varphi\left(x\right)y\left(x\right),$ where

or, in this particular case,

Here, one has to choose the second possibility, when $\tau^{\prime}=2\pi^{\prime}<0,$ namely$:$

Thus

For example,

(All these data are collected in Table 2, Appendix A, for the reader’s convenience.)

In Nikiforov–Uvarov’s method, the energy levels can be obtained from the quantization rule [44]:

which gives Schrödinger’s fine structure formula (for a charged spin-zero particle in the Coulomb field) [20]:

Here,

The corresponding eigenfunctions are given by the Rodrigues-type formula

Up to a constant, they are Laguerre polynomials [44]. By (3.8),

The corresponding integral is given by [58]:

As a result,

The normalized radial eigenfunctions, corresponding to the relativistic energy levels (3.20), are explicitly given by

where

Our consideration here, or somewhat similar to that in [11], [20], [47], outlines the way which Schrödinger had to complete (during his Christmas vacation in 1925–26?) to discover the above solution of the relativistic spin-zero charged particle in the central Coulomb field. One can see that it’s a bit more involved than the corresponding nonrelativistic case originally published in Ref. [49] (see also [1], [33], [44], [58] for modern treatments).

In his letter to Wilhelm Wien from Arosa ([40], pp. 162–165), Schrödinger admits that working on “a new atomic theory” he has “… to learn mathematics to handle the vibration problem …” (see also [37], p. 775). In the first ‘quantization’ article [49], the Laplace method based on complex integration was utilized to solve the radial equation (for a nonrelativistic hydrogen atom). At that time, Schrödinger refers to book [48], well-known to him from his student days, and acknowledges some guidance from Hermann Weyl. Nowadays, we may say that the corresponding quantization rule for those complex integrals is not that straightforward and a bit complicated — it has hardly ever been utilized in the aforementioned form after that and now is completely forgotten.

Later on, Schrödinger always uses (and develops) methods from the ‘magnificent classical mathematics’ book [8] (in Erwin’s own words [39], pp. 582–583) first published in 1924. It’s almost certain that this book was not available to him in Arosa. Only in the second article on wave mechanics, Schrödinger thanks his assistant E. Fues for pointing out a connection with the Hermite polynomials for the harmonic oscillator problem and acknowledges the relation of his wave function in the ‘Kepler problem’ with the ‘polynomial of Laguerre’ [50].

According to Dirac [13], [14] (see also [2], [25], and [39]), Schrödinger was able to solve his relativistic wave equation (and submitted an article to a journal, which original manuscript had never been found?). But later he had to abandon the whole idea after discovering an inconsistency of his fine structure formula with Sommerfeld’s one, which was in agreement with the available experimental data [57]. During his Christmas 1925-vacation at Arosa, Schrödinger introduced, eventually, his famous nonrelativistic stationary wave equation and solved it for the hydrogen atom [49] (thus ignoring the fine structure). The relativistic Schrödinger equation for a free spin-zero particle was later rediscovered by Klein, Fock, and Gordon.⁷⁷7According to [30], Louis de Broglie discovered the relativistic wave equation for a spin-zero free particle on February 1925 [5], which did not attract any interest at that time. The relativistic wave equation for a free electron, spin-$1/2$ particle, was discovered by Dirac only in 1928 [12].⁸⁸8For a definition of the spin, in terms of representations of the Poincaré group, see [31]. Its solution in the central Coulomb field, when Sommerfeld–Dirac’s fine structure formula had been finally obtained from relativistic quantum mechanics, was later found by Darwin [10] and Gordon [23]. As a result, the relativistic energy levels of an electron in the central Coulomb field are given by

In Dirac’s theory,

where $j=1/2,3/2,5/2,\ ...\$ is the total angular momentum including the spin of the relativistic electron. A detailed solution of this problem, including the nonrelativistic limit, can be found in [44], [58] (following Nikiforov–Uvarov’s paradigm), or in classical sources [1], [11], [22], [47].

Let us analyze a nonrelativistic limit of Schrödinger’s fine structure formula (3.20)–(3.21) and compare with the corresponding results for Sommerfeld–Dirac’s one (4.1)–(4.2). For the relativistic Schrödinger equation, in the limit $c\rightarrow\infty$ (or $\mu\rightarrow 0),$ one gets [11], [47]:

which can be derived by a direct Taylor’s expansion and/or verified by a computer algebra system. Here, $n=n_{r}+l+1$ is the corresponding nonrelativistic principal quantum number. The first term in this expansion is simply the rest mass energy $E_{0}=mc^{2}$ of the charged spin-zero particle, the second term coincides with the energy eigenvalue in the nonrelativistic Schrödinger theory and the third term gives the so-called fine structure of the energy levels, which removes the degeneracy between states of the same $n$ and different $l\/.$ Unfortunately, Schrödinger’s relativistic approach didn’t account correctly for the fine structure of hydrogenlike atoms, such as hydrogen, ionized helium, doubly-ionized lithium, etc. For example, the total splitting of the fine-structure levels for the principal quantum number $n=2$ became $8/3$ times as large as the one obtained from Sommerfeld’s theory, which was in perfect agreement with experiments, notably for ionized helium [11], [39], [47], [57].⁹⁹9The maximum spreads of the fine-structure levels occur when $l=0\/,$ $l=n-1$ and $j=1/2\/,$ $j=n-1/2$ for (4.3) and (4.4), respectively [11], [47]. Therefore, for the quotient, one gets $$\frac{\Delta E_{\text{Schr\"{o}dinger}}}{\Delta E_{\text{Sommerfeld}}}=\frac{4n}{2n-1}\qquad(n=2,3,\,...)\/.$$ This is why, it is almost certain that Schrödinger, who always insisted on a very good fit of experimental data by the theoretical description, would never consider publishing a ‘wrong theory’ of the relativistic hydrogen atom!

In Dirac’s theory of the relativistic electron, the corresponding limit has the form [1], [11], [47], [58]:

where $n=n_{r}+j+1/2$ is the principal quantum number of the nonrelativistic hydrogenlike atom. Once again, the first term in this expansion is the rest mass energy of the relativistic electron, the second term coincides with the energy eigenvalue in the nonrelativistic Schrödinger theory and the third term gives the so-called fine structure of the energy levels — the correction obtained for the energy in the Pauli approximation which includes the interaction of the spin of the electron with its orbital angular momentum. The total spread in the energy of the fine-structure levels is in agreement with experiments (see [1], [11], [22], [47], [57] for further discussion of the hydrogenlike energy levels including comparison with the experimental data). For an alternative discussion of the discovery of Sommerfeld’s formula, see [24], [45].

In a handwritten supplement to the February 20th, 1926 letter to Sommerfeld ([40], p. 181–182), Schrödinger had analyzed introduction of magnetic field into his relativistic wave equation (see also [11] for the nonrelatiistic limit).

Let us also consider a nonrelativistic limit of the radial wave functions (3.26)–(3.27) of the relativistic Schrödinger equation. From (3.21), $\nu\rightarrow l$ as $c\rightarrow\infty\/.$ Moreover,

and, therefore,

where $a_{0}=\hbar^{2}/me^{2}$ is Bohr’s radius. As a result, from (3.26)–(3.27) one gets the nonrelativistic radial wavefunctions of the hydrogenlike atom [1], [33], [49], [58]:

with

and

In Table 1, provided in Appendix A, we present the corresponding data for the nonrelativistic hydrogenlike atom for the reader’s convenience. (More details can be found in [44], [58].)

Most likely that Schrödinger was unaware of de Broglie’s relativistic wave equation of February 1925, which according to [30] was never mentioned by any physicist in 1925–27. Moreover, it’s almost certain that this equation doesn’t seem to have had any impact on his route towards the nonrelativistic eigenvalue problem (stationary Schrödinger equation; see also [9]). In a letter to Hermann Weyl written on April 1st, 1931 (see [40], pp. 483–485), regarding the second German edition of his book on group theory and quantum mechanics [62], Schrödinger describes his own contribution versus de Broglie’s as follows: “Now one more thing, please do not resent me. You have titled the §§2 and 3 in Chapter II with the names of de Broglie and Schrödinger. It looks as if the first of them knew about the assignment of the operators and the wave equation in field-free space, but not about the curvature of the rays in the potential field; and as if only the latter had taken this influence of the field into account by introducing the potentials into the wave equation. I don’t think that’s consistent. If you are of the opinion that de Broglie’s considerations about beam path and wavelength are actually a description of the wave equation (even if he does not use them in algebraic symbols written down), then you should put the wave equation including potentials under ‘de Broglie’. Then he was not one iota less aware of the behavior in the field than of the behavior in the field-free space. Otherwise, however, it is really a bit annoying to read, under the de Broglie name, for example, the operator mapping ($p_{1}\to{\partial}/{\partial q_{1}}$), which as far as I know is actually announced for the first time in my paper from March 18th, 1926 (see [52], p. 47), and also the scalar relativistic equation, which I described on the first page of my original paper [49] (of course only in words) and the solutions of which I published in §3, section 2 of the same paper. I never protested the fact that this equation is now commonly known under the name of Gordon, because this is convenient for distinguishing it [from non-relativistic one]. But for obvious reasons, I don’t like to see it under the name de Broglie (for example, on p. 211 of your book) [page 188 in the second German edition].”

Our translation of the entire letter from German to English, that seems is missing in the available literature, can be found in Appendix D. Here, Schrödinger’s somewhat emotional response may be related to a certain Nobel Prize controversy. Louis de Broglie was awarded the prize in 1929 after discovery of electron-diffraction picture as an experimental confirmation of his idea of the wave nature of electrons. At that time, the Nobel committee was very reluctant to give the prize for a pure theoretical work because, due to the will of Alfred Nobel, the prize should be given to the person whose ‘discovery or invention’ shall have ‘conferred the greatest benefit on mankind’. In the next two years the prizes in physics were not awarded for the discovery of quantum mechanics although both Heisenberg and Schrödinger were nominated multiple times. (The 1930 prize went to Raman ‘for his work on the scattering of light and for the discovery of the effect named after him’ and no prize was awarded in 1931 after vigorous debates in the committee and the international physics community.) Only after experimental discovery of the positron in cosmic rays, The Nobel Prize in Physics 1932 was given to Heisenberg ‘for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen’ and The Nobel Prize for 1933 was divided between Schrödinger and Dirac ‘for the discovery of new productive forms of atomic theory’ [43].

Phenomenological quantization rules of ‘old’ quantum mechanics [57] are derived in modern physics from the corresponding wave equations in the so-called semiclassical approximation [3], [11], [33] (WKB-method). In this approximation, for a particle in the central field, one can use a generic radial equation of the form [34], [44]:

where $x^{2}\,q(x)$ is continuous together with its first and second derivatives for $0\leq x\leq b<\infty\/.$ As is well-known, the traditional semiclassical approximation cannot be used in a neighborhood of $x=0\/.$ However, the change of variables $x=e^{z}\/$, $u=e^{z/2}\,v(z)$ transforms the equation into a new form,

where

(Langer’s transformation [34]). As $z\to-\infty$ (or $x\to 0$), the new function $q_{1}(z)$ is changing slowly near the constant $-1/4+\lim_{x\to 0}x^{2}q(x)\/$ and $\lim_{z\to-\infty}q_{1}^{(k)}(z)=0\/$ ($k=1,2$). Hence the function $q_{1}(z)$ and its derivatives are changing slowly for large negative $z\/.$ The WKB-method can be applied to (5.2) and, as a result, in the original equation (5.1) one should replace $q(x)$ with $q(x)-1/(4x^{2})\/$ (see [34], [44] for more details).

For example, in the well-known case of nonrelativistic Coulomb’s problem, one gets

(in dimensional units, see Table 1 or [58]). The Bohr–Sommerfeld quantization rule takes the form

with

(see, for instance, [44] for more details).

For all Coulomb problems under consideration, we utilize the following generic integral, originally evaluated by Sommerfeld [57] performing complex integration: If

Then

provided $p(r_{1})=p(r_{2})=0\/.$ (See Appendix B for an elementary evaluation of this integral.)

In (5.4), $A=-2\varepsilon_{0}\/,$ $B=2Z\/,$ and $C=(l+1/2)^{2}.\/$ In view of the quantization rule (5.5) ,

As a result, we obtain exact energy levels for the nonrelativistic hydrogenlike problem:

Here, $n=n_{r}+l+1$ is known as the principal quantum number.

Our main goal is to analyze the corresponding relativistic problems. In the case of the relativistic Schrödinger equation (3.6)–(3.7), one should write

and

Here, $A=1-{\varepsilon}^{2}\/,$ $B=2\mu\varepsilon\/,$ and $C=(l+1/2)^{2}-{\mu}^{2}\/.$ The quantization rule (5.5) implies

and the exact formula for the relativistic energy levels (3.20)–(3.21) follows once again.¹⁰¹⁰10This is, almost certain, how Schrödinger was able to derive his fine structure formula so quickly, from ‘old’ quantum mechanics, and verify the result with the help of his new ‘complex’ quantization rule. But a true reason why this algebraic equation, obtained here in semiclassical approximation, coincides with the exact one in relativistic quantum mechanics still remains a mystery!

Sommerfeld’s fine structure formula (4.1)–(4.2), for the relativistic Coulomb problem, can be thought of as the main achievement of the ‘old’ quantum mechanics [24], [41], [57]. Here, we will derive this result in a semiclassical approximation for the radial Dirac equations (separation of variables in spherical coordinates is discussed in detail in Refs. [44] and [58]). In the dimensionless units, one of these second-order differential equations has the form

and the second equation can be obtained from the first one by replacing $\nu\to-\nu\/$ (see Eqs. (3.81)–(3.82) in Ref. [58]). By Langer’s transformation, we obtain

Hence, for the Dirac equation, $A=1-{\varepsilon}^{2}\/,$ $B=2\mu\varepsilon\/,$ and $C=(\nu+1/2)^{2}.\/$ The quantization rule (5.5) implies (5.13). (One has to replace $n_{r}\to n_{r}-1\/$, see [58] for more details.) Thus the fine structure formula for the energy levels (4.1)–(4.2) has been derived here in semiclassical approximation for the corresponding radial Dirac equations. This explains the quantization rules of the ‘old’ quantum mechanics of Bohr and Sommerfeld [57] (see also [24]). The integral (5.7)–(5.8) was well-known to Schrödinger — it is mentioned in his late January 1926 letter to Sommerfeld when he reported on the first success of the wave mechanics (see [39], p. 462 or [40], p. 172).

In saved correspondence and numerous working notes, Schrödinger never gave a chronological pathway to the birth of wave mechanics that made him one of the classics of science. Schrödinger’s personal diaries for the years 1925–26 are not available. This is why we may never know for sure all the details of his revolutionary discovery. Nonetheless, our analysis of the relativistic Schrödinger equation allows suggesting that he, probably, didn’t have a chance to solve this equation for Coulomb potential, say from the viewpoint of the later standards, when eigenvalues and the normalized eigenfunctions should be explicitly presented? (The first solution was published by V. A. Fock just a few months later [20]). Instead, he seems found the corresponding fine structure formula for a charged spin-zero particle by using the Bohr–Sommerfeld quantization rule and/or by the Laplace method (correcting a mistake in the original work [24], [26]?) and observed discrepancies with the experimental data [39]. Analyzing this ‘disaster’, he switched to the nonrelativistic case and introduced the stationary Schrödinger equation, which he was able to solve around December-January of 1925–26 (by the Laplace method, consulting with Hermann Weyl?). His main legacy, namely, the time-dependent Schrödinger equation (Figure 4), de facto required in the article dedicated to the coherent states [51],¹¹¹¹11The coherent states were introduced as a result of correspondence with Hendrik Lorentz [29], [43]; see also [32] for an extension to the minimum-uncertainty squeezed states. was published only about six months later [53]. His review [54] gives an account of the new form of quantum theory. That article was submitted on September 3, 1926. At the end, addressing the controversy with the relativistic hydrogen atom, Schrödinger concludes: “The deficiency must be intimately connected with Uhlenbeck–Goudsmit’s theory of the spinning electron. But in what way the electron spin has to be taken into account in the present theory is yet unknown”. Less than two years later, Paul Dirac would derive a first-order wave equation for a four-component spinor field that describes relativistic spin-1/2 particles like electrons [12].

The year 1926 revolutionary changed the world of physics forever — the Schrödinger equation turned out to have an enormously wide range of applications [42], from the quantum theory of atoms and molecules to solid state physics, quantum crystals, superfluidity, and superconductivity. In a historical perspective, the ‘golden years’ following the foundation of quantum mechanics have paved the way to quantum field theory and the physics of elementary particles. Just in a few months, Erwin Schrödinger turned the matter wave hypothesis of Louis de Broglie into a detailed, perfectly operating technique, which eventually eliminated many of ‘old’ quantum physics postulates, that had been employed in the previous decades, as unnecessary. However, at that time, a ‘true nature’ of quantum motion, including the probabilistic interpretation of the wave function, the uncertainty principle, and the wave-particle duality, was not yet well-understood — it came later as a result of vigorous debates in the newly (re)born ‘quantum community’ [39]. Some of these debates are not completed even 60 years after Schrödinger! In mathematics, the time-dependent Schrödinger equation gave rise to a new class of partial differential equations [16].

In November 1926, Schrödinger wrote in the preface to a collected edition of his first papers on wave mechanics [55]: “Referring to these six papers (the present reprint of which is solely due to the great demand for separate copies), a young lady friend¹²¹²12The young friend was fourteen-year old Itha Junger, whom Schrödinger was tutoring in mathematics [25], [43]. recently remarked to the author: ‘When you began this work you had no idea that anything so clever would come out of it, had you?’ This remark, with which I wholeheartedly agreed (with due qualification of the flattering adjective), may serve to call attention to the fact that the papers now combined in one volume were originally written one by one at different times. The results of the later sections were largely unknown to the writer of the earlier ones.” No alterations of the originals were allowed. In the end, Schrödinger concludes: “On the contrary, this comparatively cheap method of issue seemed advisable on account of the impossibility at the present stage of giving a fresh exposition which would be really satisfactory or conclusive.” Since then his classical works combined in one volume and translated to English remain a monumental ‘snapshot’ in the history of science. After Schrödinger’s review [54] (immediately translated into Russian by Soviet Physics Uspekhi, 1927) and the subsequent blast of research, the first ‘really satisfactory and conclusive’ expositions of the newly emerged quantum theory were given by Werner Heisenberg [27] (in German, 1930), Paul Dirac [15] (in English, 1930), and Vladimir A. Fock [22] (in Russian, 1931). A detailed list of early sources can be found in [18] and [62].

In a way, we have had somewhat similar ‘feelings’ (with due qualification, of course!), originally writing these notes mostly for our colleagues, friends, and students and thoroughly checking the resulting interpretations with the available bibliography (letters, notebooks, and articles). We hope that this consideration, despite potential imperfections, will encourage the readers to study quantum physics at one of the crucial moments of its creation and draw their own conclusions.