# Erwin Schrödinger

gigatos | February 18, 2022

## Summary

Erwin Rudolf Joseph Alexander Schrödinger (August 12, 1887 – January 4, 1961, Vienna) – Austrian theoretical physicist and one of the founders of quantum mechanics. Winner of the Nobel Prize in Physics (1933). Member of the Austrian Academy of Sciences (1956), as well as a number of academies of sciences in the world, including a foreign member of the USSR Academy of Sciences (1934).

Schrödinger had a number of fundamental results in the field of quantum theory, which formed the basis of wave mechanics: he formulated the wave equations (stationary and time-dependent Schrödinger equation), showed the identity of the formalism he developed and matrix mechanics, developed the wave-mechanical theory of perturbations, obtained solutions for several specific problems. Schrödinger proposed an original interpretation of the physical meaning of the wave function; in later years he repeatedly criticized the generally accepted Copenhagen interpretation of quantum mechanics (Schrödinger”s cat paradox, etc.). He is also the author of many works in various fields of physics: statistical mechanics and thermodynamics, dielectric physics, color theory, electrodynamics, general relativity theory and cosmology; he has made several attempts to construct a unified field theory. In “What is Life?” Schrödinger addressed the problems of genetics, looking at the phenomenon of life from the perspective of physics. He paid much attention to philosophical aspects of science, ancient and oriental philosophical concepts, questions of ethics and religion.

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## Origins and Education (1887-1910)

Erwin Schrödinger was the only child of a wealthy and cultured Viennese family. His father, Rudolf Schrödinger, the prosperous owner of a linoleum and oilcloth factory, was interested in science and served for a long time as vice president of the Vienna Botanical and Zoological Society. Erwin”s mother, Georgina Emilie Brenda, was the daughter of the chemist Alexander Bauer, whose lectures Rudolf Schrödinger attended while studying at the Imperial and Royal Vienna University of Technology (k. k. Technischen Hochschule). The family environment and the company of highly educated parents helped to shape the diverse interests of the young Erwin. Until the age of eleven he received a home education, and in 1898 he entered the prestigious Academic Gymnasium (Öffentliches Academisches Gymnasium), where he studied mainly humanities. Schrödinger studied easily, becoming the best pupil in each class. Much time was devoted to reading, learning foreign languages. His maternal grandmother was English, so he mastered this language from an early age. He loved going to the theater; he especially liked Franz Grilparzer”s plays, which were staged at the Burgtheater.

Having passed the school examinations with flying colors, Erwin entered the University of Vienna in the fall of 1906, where he chose to study mathematics and physics. Franz Exner had a great influence on Schrödinger”s formation as a scientist, lecturing in physics and attaching particular importance to methodological and philosophical issues of science. His interest in theoretical problems of physics arose in Erwin after his acquaintance with Friedrich Hasenörl, Ludwig Boltzmann”s successor in the Department of Theoretical Physics. It was from Hasenöhrl that the future scientist learned about current scientific problems and the difficulties that arise in classical physics when trying to solve them. During his studies at the university Schrödinger mastered perfectly the mathematical methods of physics, but his dissertation work was experimental. It was devoted to the study of the effect of air humidity on the electrical properties of a number of insulating materials (glass, ebonite, amber) and was carried out under the supervision of Egon Schweidler in the Exner laboratory. On May 20, 1910, after defending the thesis and passing the oral examinations, Schrödinger was awarded the degree of Doctor of Philosophy.

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## Beginning of a scientific career (1911-1921)

In October 1911, after a year”s service in the Austrian army, Schrödinger returned to the Second Institute of Physics at Vienna University as Exner”s assistant. He taught a physics workshop and also participated in the experimental research carried out in Exner”s laboratory. In 1913 Schrödinger applied for the title of privat-docent, and after going through the proper procedures (submitting a scientific paper, giving a “test lecture” and so on) in early 1914 the ministry approved him of this rank (habilitations). The First World War delayed the beginning of Schrödinger”s active teaching activities for several years. The young physicist was drafted into the army and served in the artillery on the relatively quiet sections of the Austrian South-Western Front: in Raibl (Raibl), Komarom, then in Prosecco (Prosecco) and in the area of Trieste. In 1917 he was appointed to teach meteorology at the officers” school in Wiener Neustadt. This mode of service left him enough time to read special literature and work on scientific problems.

In November 1918 Schrödinger returned to Vienna, and around that time he was offered the position of Extraordinary Professor of Theoretical Physics at the University of Chernivtsi. However, after the collapse of the Austro-Hungarian Empire, this city was in another country, so the opportunity was lost. The difficult economic situation in the country, low wages and the bankruptcy of the family business forced him to look for a new place of work, including abroad. A suitable opportunity presented itself in the fall of 1919, when Max Wien, who headed the Institute of Physics at the University of Jena, invited Schrödinger to take over as his assistant and associate professor in the department of theoretical physics. The Austrian gladly accepted the offer, and in April 1920 he moved to Jena (this happened immediately after his wedding). Schrödinger only stayed in Jena for four months: he soon moved to Stuttgart as an extraordinary professor at the local Technical University (now the University of Stuttgart). An important factor in the context of rising inflation was the substantial increase in salary. However, very soon even better conditions and the post of professor of theoretical physics began to be offered by other institutions – the universities of Breslau, Kiel, Hamburg and Vienna. Schrödinger chose the first and left Stuttgart after just one semester. In Breslau, he lectured during the summer term, and at the end of that term, he changed jobs again, heading the prestigious Department of Theoretical Physics at the University of Zurich.

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## Zurich to Berlin (1921-1933)

Schrödinger moved to Zurich in the summer of 1921. Life here was more financially stable, the neighboring mountains provided the scientist, who loved mountaineering and skiing, with comfortable opportunities for recreation, and the company of famous colleagues Peter Debye, Paul Scherrer and Hermann Weil, who worked at the nearby Zurich Polytechnic, created the necessary atmosphere for scientific creativity. His time in Zurich was overshadowed in 1921-1922 by a serious illness; Schrödinger was diagnosed with pulmonary tuberculosis, so he had to spend nine months in the spa town of Arosa in the Swiss Alps. In terms of creativity, the Zurich years were most fruitful for Schrödinger, who wrote his classic works on wave mechanics here. It is known that Weil helped him a lot in overcoming mathematical difficulties.

The fame that Schrödinger”s pioneering work brought made him one of the main candidates for the prestigious post of professor of theoretical physics at the University of Berlin, vacated by Max Planck”s resignation. After Arnold Sommerfeld”s rejection and overcoming doubts about whether to leave his beloved Zurich, Schrödinger accepted the offer and took up his new duties on October 1, 1927. In Berlin, the Austrian physicist found friends and associates in Max Planck, Albert Einstein, and Max von Laue, who shared his conservative views on quantum mechanics and did not recognize its Copenhagen interpretation. At the university Schrödinger lectured on various sections of physics, led seminars, led the physics colloquium, participated in organizational activities, but in general he stood apart, as evidenced by the lack of students. As Viktor Weisskopf, who at one time worked as Schrödinger”s assistant, noted, the latter “played the role of an outsider at the university.

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## Oxford-Graz-Ghent (1933-1939)

The time spent in Berlin was described by Schroedinger as “the beautiful years when I studied and learned. That time came to an end in 1933, after Hitler came to power. In the summer of that year, the now-aged scientist, no longer wishing to remain under the rule of the new regime, decided once again to change his surroundings. It is worth noting that despite his negative attitude to Nazism, he never openly expressed it and did not want to interfere in politics, and it was almost impossible to maintain his apolitical nature in Germany at the time. Schrödinger himself, explaining the reasons for his departure, said, “I can”t stand to be pestered by politics. The British physicist Frederick Lindeman (later Lord Cherwell), who was visiting Germany at the time, invited Schrödinger to Oxford University. Having left for a summer holiday in the South Tyrol, the scientist did not return to Berlin and in October 1933, together with his wife, arrived in Oxford. Shortly after his arrival, he learned that he had been awarded the Nobel Prize in Physics (jointly with Paul Dirac) “for the discovery of new and fruitful forms of atomic theory. In an autobiography written on this occasion, Schrödinger gave the following assessment of his style of thinking:

In my scientific work, as well as in life in general, I have never adhered to any general line, nor have I followed a guiding program designed for a long time. Although I am very bad at teamwork, including, unfortunately, with students, nevertheless my work has never been completely independent, because my interest in a question always depends on the interest shown in the same question by others. I seldom say the first word, but often the second, as the impetus for it is usually a desire to object or correct…

At Oxford, Schrödinger became a member of Magdalen College, with no teaching responsibilities and, along with other émigrés, receiving funding from the Imperial Chemical Industry. However, he was never able to settle into the specific environment of one of the oldest universities in England. One reason for this was the lack of interest in modern theoretical physics at Oxford, which was focused mainly on teaching traditional humanities and theology, which made the scholar feel undeserving of his high position and large salary, which he sometimes called a kind of alms. Another aspect of Schroedinger”s discomfort at Oxford University was the peculiarities of social life, full of conventions and formalities, which he acknowledged constrained his freedom. The situation was complicated by the unusual nature of his private and family life, which caused quite a scandal in clerical circles in Oxford. In particular, Schroedinger came into sharp conflict with Clive Lewis, professor of English language and literature. All these problems, as well as the winding down of the immigrant scholarship program in early 1936, forced Schrödinger to consider options for pursuing a career outside Oxford. After a visit to Edinburgh, in the fall of 1936 he accepted an offer to return home and take up a post as professor of theoretical physics at the University of Graz.

Schrödinger”s stay in Austria did not last long: already in March 1938, the country was annexed to Nazi Germany. On the advice of the rector of the university, Schroedinger wrote a letter of reconciliation with the new government, which was published on March 30 in the Graz newspaper Tagespost and provoked a negative reaction from his emigrated colleagues. These measures, however, did not help: the scientist was dismissed from his post on grounds of political unreliability; he received official notification in August 1938. Realizing that leaving the country would soon be impossible, Schrödinger hastily left Austria for Rome (Fascist Italy was the only country that did not require a visa at the time). By this time he had established a connection with the Irish prime minister, Eamon de Valera, a mathematician by training, who planned to set up in Dublin an equivalent of the Princeton Institute for Higher Studies. De Valera, then in Geneva as president of the Assembly of the League of Nations, secured a transit visa for Schroedinger and his wife to travel through Europe. In the fall of 1938, after a brief stopover in Switzerland, they arrived in Oxford. While the Dublin institute was being organized, the scientist agreed to take a temporary position in Belgian Ghent, paid for by the Fondation Francqui. It was here that the outbreak of World War II caught up with him. Thanks to the intervention of de Valera Schrödinger, who after the Anschluss was considered a citizen of Germany (and therefore an enemy state), was able to travel through England and arrived in the Irish capital on 7 October 1939.

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## Dublin to Vienna (1939-1961)

The legislation for the Dublin Institute for Advanced Studies was passed by the Irish Parliament in June 1940. Schrödinger, who became the first professor in one of the Institute”s two original departments, the School of Theoretical Physics, was also appointed its first chairman. Later on, other members of the institute, including the well-known scientists Walter Geitler, Lajos Janosz and Cornelius Lanzos, as well as many young physicists, were able to devote their full attention to research. Schrödinger organized a permanent seminar, gave lectures at Dublin University, and initiated annual summer schools at the Institute attended by leading European physicists. During his years in Ireland, his main research interests were the theory of gravitation and questions at the interface between physics and biology. He served as director of the Department of Theoretical Physics from 1940-1945 and from 1949 to 1956, when he decided to return home.

Although after the war Schrödinger repeatedly received offers to move to Austria or Germany, he declined these invitations, not wanting to leave the place he had settled. Only after the signing of the Austrian State Treaty and the withdrawal of Allied troops did he agree to return to his homeland. At the beginning of 1956, the president of Austria approved a decree granting the scientist a personal post of professor of theoretical physics at the University of Vienna. In April of that year Schrödinger returned to Vienna and inaugurated his appointment by giving a lecture in the presence of several celebrities, including the president of the republic. He was grateful to the Austrian government, which had arranged for his return to the place where his career had begun. Two years later, the often ill scholar finally left the university, resigning. The last years of his life were spent mainly in the Tyrolean village of Alpbach. Schrödinger died as a result of acute tuberculosis in a Viennese hospital on 4 January 1961 and was buried in Alpbach.

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## Personal life and hobbies

From the spring of 1920, Schrödinger was married to Annemarie Bertel of Salzburg, whom he met in the summer of 1913 at Seeham while conducting experiments on atmospheric electricity. This marriage lasted until the end of the scientist”s life, despite the spouses” regular affairs “on the side”. For example, among Annemarie”s lovers were her husband”s colleagues Paul Ewald and Hermann Weil. Schrödinger, for his part, had numerous affairs with young women, two of whom were still teenagers (with one of them he spent the winter of 1925 on vacation in Arosa, during which he worked intensively on the creation of wave mechanics). Although Erwin and Annemarie had no children, Schrödinger is known to have several children out of wedlock. The mother of one of them, Hilde March, wife of Arthur March, one of the Austrian friends of the scientist, became Schrödinger”s “second wife. In 1933, while leaving Germany, he was able to arrange financing at Oxford not only for himself but also for the Marchs; in the spring of 1934 Hilde gave birth to a daughter, Ruth Georgine March, by Schrödinger. The following year the Marches returned to Innsbruck. Such a free lifestyle shocked the puritanical inhabitants of Oxford, which was one reason for Schrödinger”s discomfort there. Two more children out of wedlock were born to him during his time in Dublin. Beginning in the 1940s, Annemarie was regularly hospitalized for bouts of depression.

Biographers and contemporaries have repeatedly noted Schroedinger”s versatile interests, his profound knowledge of philosophy and history. He spoke six foreign languages (in addition to “gymnasium” ancient Greek and Latin, they are English, French, Spanish and Italian), read classical works in the original and translated them, wrote poetry (in 1949 was published collection), was fond of sculpture.

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## Early and Experimental Works

At the beginning of his scientific career Schrödinger did a lot of theoretical and experimental research, which was in line with the interests of his teacher Franz Exner – electrical engineering, atmospheric electricity and radioactivity, the study of the properties of dielectrics. At the same time the young scientist actively studied purely theoretical issues of classical mechanics, the theory of oscillations, the theory of Brownian motion, and mathematical statistics. In 1912, at the request of the compilers of the Handbook of Electricity and Magnetism (Handbuch der Elektrizität und des Magnetismus) he wrote a major review article “Dielectrics,” which was evidence of the recognition of his work in the scientific world. In the same year Schrödinger gave a theoretical estimate of the probable altitude distribution of radioactive substances, which is required to explain the observed radioactivity of the atmosphere, and in August 1913 at Seeham he carried out corresponding experimental measurements, confirming some conclusions of Victor Franz Hess about the insufficient value of the concentration of decay products to explain the measured ionization of the atmosphere. For this work Schrödinger was awarded the Haitinger-Preis of the Austrian Academy of Sciences in 1920. Other experimental studies carried out by the young scientist in 1914 were checking the formula for the capillary pressure in gas bubbles and studying the properties of soft beta-rays, which appear when gamma-rays fall on a metal surface. He carried out the latter work together with his experimental friend Karl Wilhelm Friedrich Kohlrausch. In 1919 Schrödinger performed his last physical experiment (studying the coherence of rays emitted at a large angle to each other) and subsequently concentrated on theoretical research.

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## The Doctrine of Color

Exner”s laboratory paid special attention to the science of color, continuing and developing the work of Thomas Jung, James Clerk Maxwell, and Hermann Helmholtz in this field. Schrödinger dealt with the theoretical side of the question, making important contributions to colorometry. The results of his work were presented in a large article published in Annalen der Physik in 1920. As the basis the scientist took not a flat color triangle, but a three-dimensional color space, the basic vectors of which were the three primary colors. Pure spectral colors settle down on a surface of some figure (a color cone) whereas its volume is occupied with mixed colors (for example, white). To each concrete color there corresponds the radius-vector in this color space. The next step in the direction of so-called higher colorometry was a strict definition of a number of quantitative characteristics (such as brightness) to be able to compare objectively their relative values for different colors. For this purpose Schrödinger, following Helmholtz”s idea, introduced into three-dimensional color space the laws of Riemannian geometry, and the shortest distance between two given points of such space (on a geodesic line) should serve as quantitative value of difference of two colors. Further he has offered concrete metrics of color space which allowed to calculate brightness of colors in conformity with Weber-Fechner”s law.

In the following years, Schrödinger devoted several papers to physiological features of vision (in particular, the color of stars observed at night) and wrote a large survey on visual perception for another edition of the popular Müller-Pouillet Lehrbuch der Physik textbook (Müller-Pouillet). In another article he examined the evolution of color vision, trying to relate the sensitivity of the eye to light of different wavelengths to the spectral composition of solar radiation. He believed that color insensitive rods (retinal receptors responsible for night vision) emerged at much earlier stages of evolution (probably in ancient creatures that led underwater life) than cones. These evolutionary changes, he claims, can be traced in the structure of the eye. Thanks to his work, by the mid-1920s Schrödinger had gained a reputation as one of the leading specialists in color theory, but from that time on his attention was completely absorbed by other problems, and in the following years he did not return to this subject.

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## Statistical Physics

Schrödinger, educated at the University of Vienna, was greatly influenced by his famous compatriot Ludwig Boltzmann and his work and methods. Already in one of his first articles (1912) he applied the methods of kinetic theory to describe the diamagnetic properties of metals. Although these results were of limited success and in general could not be correct in the absence of correct quantum statistics for electrons, Schrödinger soon decided to apply the Boltzmann approach to a more difficult problem: the kinetic theory of solids and, in particular, the description of crystallization and melting. Based on the recent results of Peter Debye, the Austrian physicist generalized the equation of state for liquids and interpreted its parameter (critical temperature) as the melting temperature. After the discovery of X-ray diffraction in 1912, the problem arose of theoretical description of this phenomenon and, in particular, taking into account the influence of thermal motion of atoms on the structure of the observed interference patterns. In a paper published in 1914, Schrödinger (independently of Debye) considered this problem within the framework of Born-Von Karman dynamic lattice model and obtained the temperature dependence for X-ray intensity distribution by angles. This dependence was soon confirmed experimentally. These and other early works of Schrödinger were of interest for him also from the point of view of approval of atomistic structure of matter and further development of kinetic theory, which, in his opinion, was to finally displace continuous media models in the future.

During his military service Schrödinger studied the problem of thermodynamic fluctuations and related phenomena, paying special attention to the works of Marian Smoluchowski. After the war, statistical physics became one of the main themes in Schrödinger”s work, and he devoted to it most of the works he wrote in the first half of the 1920s. For example, in 1921 he argued for the difference between isotopes of the same element from the thermodynamic point of view (the so-called Gibbs paradox), although they might be virtually indistinguishable chemically. In a number of papers Schrödinger specified or clarified specific results obtained by his colleagues on various questions of statistical physics (specific heat capacity of solids, thermal equilibrium between light and sound waves, and so on). Some of these papers used considerations of a quantum nature, such as the article on the specific heat capacity of molecular hydrogen or the publications on the quantum theory of ideal (degenerate) gas. These works preceded the emergence in the summer of 1924 of the works of Chatyendranath Bose and Albert Einstein, who laid the foundations of the new quantum statistics (Bose-Einstein statistics) and applied it to the development of the quantum theory of the ideal one-atom gas. Schrödinger joined the study of the details of this new theory, discussing in its light the question of determining the entropy of gas. In the fall of 1925, using Max Planck”s new definition of entropy, he derived expressions for quantized energy levels of gas as a whole rather than its individual molecules. Work on this subject, communication with Planck and Einstein, as well as acquaintance with the new idea of Louis de Broglie on the wave properties of matter were the prerequisites for further research, which led to the creation of wave mechanics. In the immediately preceding paper “Toward Einstein”s Theory of Gas” Schrödinger showed the importance of de Broglie”s concept for understanding Bose-Einstein”s statistics.

In the following years Schrödinger regularly returned to the questions of statistical mechanics and thermodynamics in his writings. In the Dublin period of his life he wrote several papers on the foundations of probability theory, Boolean algebra, and the application of statistical methods to the analysis of cosmic ray detector readings. In Statistical Thermodynamics (1946), written on the basis of a course of lectures he had read, the scientist examined in detail some fundamental problems that were often given insufficient attention in ordinary textbooks (difficulties in determining entropy, bose condensation and degeneracy, zero-point energy in crystals and electromagnetic radiation, and so on). Schrödinger devoted several articles to the nature of the second principle of thermodynamics, the reversibility of physical laws in time, whose direction he associated with the increase of entropy (in his philosophical essays he pointed out that perhaps the sense of time is due to the very fact of human consciousness).

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## Quantum Mechanics

Already in the early years of his scientific career Schrödinger became familiar with the ideas of quantum theory, developed in the works of Max Planck, Albert Einstein, Niels Bohr, Arnold Sommerfeld and other scientists. This acquaintance was facilitated by his work on some problems of statistical physics, but the Austrian scientist at that time was not yet ready to part with the traditional methods of classical physics. Despite Schrödinger”s recognition of the success of quantum theory, his attitude to it was ambiguous, and he tried, if possible, not to use new approaches with all their ambiguities. Much later, after the creation of quantum mechanics, he said, remembering this time:

The old Vienna Institute of Ludwig Boltzmann … gave me the opportunity to be penetrated by the ideas of this mighty mind. The circle of these ideas became my first love of science, nothing else so captivated me and, perhaps, never will. I approached the modern theory of the atom very slowly. Its inner contradictions sound like shrill dissonances compared to the pure, relentlessly clear consistency of Boltzmann”s thought. There was a time when I was ready to flee, but prompted by Exner and Kohlrausch, I found salvation in the doctrine of color.

Schrödinger”s first publications on atomic and spectral theory did not begin to appear until the early 1920s, following his personal acquaintance with Arnold Sommerfeld and Wolfgang Pauli and his move to work in Germany, which was the center for the development of new physics. In January 1921 Schrödinger completed his first paper on the subject, examining within the framework of the Bohr-Sommerfeld theory the influence of electron interactions on certain features of the spectra of alkali metals. Of particular interest to him was the introduction of relativistic considerations into quantum theory. In the fall of 1922 he analyzed electron orbits in the atom from a geometrical point of view, using the methods of the famous mathematician Hermann Weil. This work, in which it was shown that quantum orbits can be compared with certain geometrical properties, was an important step that anticipated certain features of wave mechanics. Earlier in the same year Schrödinger obtained a formula for the relativistic Doppler effect for spectral lines, based on the hypothesis of light quanta and considerations of conservation of energy and momentum. However, he had great doubts about the validity of the latter considerations in the microcosm. He was close to his teacher Exner”s idea of the statistical nature of the laws of conservation, so he enthusiastically accepted the appearance in the spring of 1924 of an article by Bohr, Kramers and Slater, which suggested the possibility of violation of these laws in individual atomic processes (for example, in the emission processes of radiation). Although the experiments of Hans Geiger and Walter Bothe soon showed the incompatibility of this assumption with experience, the idea of energy as a statistical concept attracted Schrödinger throughout his life and was discussed by him in several reports and publications.

The immediate impetus for the beginning of the development of wave mechanics was Schrödinger”s acquaintance in early November 1925 with Louis de Broglie”s dissertation containing the idea of wave properties of matter, and also with Einstein”s article on the quantum theory of gases, which cited the work of the French scientist. The success of Schrödinger”s work in this direction was due to his mastery of the appropriate mathematical apparatus, in particular the technique of solving problems on eigenvalues. Schrödinger attempted to generalize de Broglie waves to the case of interacting particles, taking into account, like the French scientist, relativistic effects. After some time he managed to represent energy levels as eigenvalues of some operator. However, the test for the case of the simplest atom, the hydrogen atom, was disappointing: the calculation results did not coincide with experimental data. The reason was that in fact Schrödinger obtained the relativistic equation, now known as the Klein-Gordon equation, which is valid only for particles with zero spin (at that time spin was not yet known). After this failure the scientist left this work and returned to it only some time later, having found that his approach gives satisfactory results in the nonrelativistic approximation.

In the first half of 1926 the editorial board of Annalen der Physik received four parts of Schrödinger”s famous paper “Quantization as a problem about eigenvalues”. In the first part (received by the editorial board on January 27, 1926), starting from Hamilton”s optical-mechanical analogy, the author derived the wave equation, now known as the time-independent (stationary) Schrödinger equation, and applied it to finding the discrete energy levels of the hydrogen atom. The scientist considered the main advantage of his approach to be that “quantum rules no longer contain the mysterious ”integrability requirement”: it is now traceable, so to speak, one step deeper and finds justification in the boundedness and uniqueness of a certain spatial function. This function, later called the wave function, was formally introduced as a quantity logarithmically related to the action of the system. In the second communication (received February 23, 1926) Schrödinger addressed the general ideas underlying his methodology. Developing the opto-mechanical analogy, he generalized the wave equation and came to the conclusion that the velocity of a particle is equal to the group velocity of the wave packet. According to the scientist, in the general case “it is necessary to depict the variety of possible processes, based on the wave equation, rather than on the basic equations of mechanics, which for explaining the essence of the microstructure of mechanical motion is as unsuitable as geometric optics for explaining diffraction. Finally Schrödinger used his theory to solve some specific problems, in particular the harmonic oscillator problem, obtaining a solution consistent with the results of Heisenberg”s matrix mechanics.

In the introduction to the third part of the paper (received May 10, 1926) the term “wave mechanics” (Wellenmechanik) first appeared to denote the approach developed by Schrödinger. Generalizing the method developed by Lord Rayleigh in the theory of acoustic oscillations, the Austrian scientist developed a method of obtaining approximate solutions of complex problems within the framework of his theory, known as the theory of time-independent perturbations. He applied this method to the description of the Stark effect for the hydrogen atom and gave a good agreement with experimental data. In the fourth communication (received June 21, 1926), the scientist formulated the equation later called the non-stationary (time) Schrödinger equation and used it to develop the theory of time-dependent perturbations. As an example, he considered the problem of dispersion and discussed related questions, in particular in the case of a time periodic perturbation potential, he concluded that there were Raman frequencies in the secondary radiation. In the same paper a relativistic generalization of the basic equation of the theory, which had been obtained by Schrödinger at the initial stage of the work (the Klein-Gordon equation), was presented.

Schrödinger”s work immediately after its appearance attracted the attention of the world”s leading physicists and was greeted with enthusiasm by such scientists as Einstein, Planck and Sommerfeld. It seemed surprising that the description by means of continuous differential equations gave the same results as the matrix mechanics with its unusual and complicated algebraic formalism and reliance on discreteness of spectral lines known from experience. Wave mechanics, close in spirit to classical continuum mechanics, seemed preferable to many scientists. In particular, Schrödinger himself was critical of Heisenberg”s matrix theory: “Of course I knew about his theory, but I was discouraged, if not repulsed, by the very difficult methods of transcendental algebra and the lack of any clarity. Nevertheless, Schrödinger was convinced of the formal equivalence of the formalisms of wave and matrix mechanics. The proof of this equivalence was given by him in an article “On the relation of Heisenberg-Borne-Jordan quantum mechanics to mine,” received by the editors of Annalen der Physik on March 18, 1926. He showed that any equation of wave mechanics can be represented in matrix form and, conversely, from given matrices one can go to wave functions. Independently, the connection between the two forms of quantum mechanics was established by Carl Eckart and Wolfgang Pauli.

The importance of Schrödinger”s wave mechanics was immediately realized by the scientific community, and already in the first months after the appearance of the basic works in various universities in Europe and America activities were begun to study and apply the new theory to various private problems. Schrödinger”s speeches at meetings of the German Physical Society in Berlin and Munich in the summer of 1926, as well as an extensive tour of America undertaken by him in December 1926 – April 1927, helped to propagate the ideas of wave mechanics. During this trip he gave 57 lectures at various scientific institutions in the United States.

Soon after the appearance of Schrödinger”s fundamental articles, the convenient and consistent formalism outlined in them began to be widely used for solving a wide variety of problems of quantum theory. However, at that time the formalism itself was not yet clear enough. One of the main questions posed by Schrödinger”s seminal work was the question of what vibrates in the atom, that is, the problem of the meaning and properties of the wave function. In the first part of his paper he considered it as a real, unambiguous and everywhere twice differentiable function, but in the last part he allowed for the possibility of complex values. He treated the modulus square of this function as a measure of electric charge density distribution in configuration space. The scientist believed that now the particles could be clearly represented as wave packets, properly composed of a set of eigenfunctions, and thus he completely abandoned the corpuscular representations. The impossibility of such an explanation became clear very soon: in the general case wave packets are inevitably blurred, which is in contradiction with the clearly corpuscular behavior of particles in experiments on electron scattering. The solution to the problem was given by Max Born, who proposed a probabilistic interpretation of the wave function.

For Schrödinger this statistical interpretation, which contradicted his ideas about real quantum-mechanical waves, was totally unacceptable, because it left in force quantum jumps and other elements of discontinuity, from which he wanted to get rid. The scientist”s rejection of the new interpretation of his results was most evident in a discussion with Niels Bohr that took place in October 1926 during Schroedinger”s visit to Copenhagen. Werner Heisenberg, a witness to these events, later wrote:

The discussion between Bohr and Schrödinger began already at the train station in Copenhagen and continued daily from early morning until late at night. Schrödinger stayed at Bohr”s house, so that by purely external circumstances there could be no interruption of the argument… After a few days Schrödinger fell ill, probably due to extreme exertion; fever and a cold forced him to lie down in bed. Frau Bohr nursed him, brought him tea and sweets, but Niels Bohr sat on the edge of the bed and implored Schrödinger: “You still have to understand that…”… It was impossible to come to a true understanding then, because neither side could offer a complete and coherent interpretation of quantum mechanics.

Such an interpretation, which was based on the Born probabilistic treatment of the wave function, the Heisenberg uncertainty principle and the Bohr additionality principle, was formulated in 1927 and became known as the Copenhagen interpretation. However, Schrödinger was never able to accept it, and to the end of his life he defended the need for a visual representation of wave mechanics. However, after his visit to Copenhagen he noted that, despite all scientific disagreements, “the relationship with Bohr and especially with Heisenberg … was absolutely, uncloudedly friendly and cordial.

After completing the formalism of wave mechanics, Schrödinger was able to use it to obtain a number of important private results. By the end of 1926 he had already used his technique to describe the Compton effect, and he also attempted to combine quantum mechanics and electrodynamics. Starting from the Klein-Gordon equation, Schrödinger obtained an expression for the energy-momentum tensor and the corresponding conservation law for combined matter and electromagnetic waves. However, these results, as well as the original equation, were inapplicable to the electron, because they did not allow to take into account its spin (this was later done by Paul Dirac, who obtained his famous equation). Only many years later it became clear that the results obtained by Schrödinger were valid for particles with zero spin, such as mesons. In 1930, he obtained a generalized expression of the Heisenberg uncertainty relation for any pair of physical quantities (observables). In the same year he first integrated the Dirac equation for the free electron, concluding that its motion is described by the sum of a rectilinear uniform motion and a high-frequency trembling motion (Zitterbewegung) of small amplitude. This phenomenon is explained by the interference of parts of the wave packet corresponding to the electron, relating to positive and negative energies. In 1940-1941 Schrödinger developed in detail within the framework of wave mechanics (that is, the Schrödinger representation) the factorization method for solving problems on eigenvalues. The essence of this approach consists in the representation of the Hamiltonian of the system as a product of two operators.

Schrödinger returned to the criticism of various aspects of the Copenhagen interpretation many times since the late 1920s, discussing these problems with Einstein, with whom they were colleagues at the University of Berlin at the time. Their communication on the subject continued in later years through correspondence, which intensified in 1935 after the famous Einstein-Podolsky-Rosen (EPR) paper on the incompleteness of quantum mechanics was published. In one of his letters to Einstein (dated August 19, 1935), as well as in an article sent on August 12 to the journal Naturwissenschaften, he presented the first mental experiment, which became known as the paradox of “Schrödinger”s cat. The essence of this paradox, according to Schrödinger, was that uncertainty at the atomic level can lead to uncertainty on the macroscopic scale (“mixture” of a living and a dead cat). This does not meet the requirement of definiteness of states of macroobjects independently of their observation and therefore “prevents us from accepting in this naive way the ”blur model” [i.e. the standard interpretation of quantum mechanics] as a picture of reality. Einstein saw this mental experiment as an indication that the wave function is relevant to describing a statistical ensemble of systems, not a single microsystem. Schrödinger disagreed, seeing the wave function as directly related to reality rather than to its statistical description. In the same article, he also analyzed other aspects of quantum theory (e.g., the problem of measurement) and concluded that quantum mechanics “is still just a convenient trick, which, however, has gained … an extremely great influence on our fundamental views of nature. Further reflection on the EPR paradox led Schrödinger to the complex problem of quantum entanglement (Verschränkung, Entanglement). He managed to prove the general mathematical theorem that after splitting a system into parts, their total wave function is not a simple product of the functions of individual subsystems. According to Schrödinger, this behavior of quantum systems is a significant drawback of the theory and a reason to improve it. Although the arguments of Einstein and Schrödinger could not shake the position of supporters of the standard interpretation of quantum mechanics, represented primarily by Bohr and Heisenberg, they stimulated the clarification of some fundamentally important aspects of it and even led to a discussion of the philosophical problem of physical reality.

In 1927 Schrödinger proposed the so-called resonance concept of quantum interactions, based on the hypothesis of a continuous exchange of energy between quantum systems with close natural frequencies. However, this idea, despite all the hopes of the author, could not replace the concept of stationary states and quantum transitions. In 1952, in the article “Do quantum jumps exist?” he returned to the resonance concept, criticizing the probabilistic interpretation. In a detailed response to the comments contained in this paper, Max Born came to the following conclusion:

…I would like to say that I consider Schrödinger”s wave mechanics one of the most remarkable achievements in the history of theoretical physics… I am far from saying that the interpretation known today is perfect and definitive. I applaud Schrödinger”s attack on the satisfied indifference of many physicists who accept the modern interpretation simply because it works, without worrying about the accuracy of the reasoning. However, I do not think that Schrödinger”s article has made a positive contribution to solving philosophical difficulties.

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## Electromagnetism and the General Theory of Relativity

Schrödinger became acquainted with Einstein”s work on general relativity (GR) in Italy, on the shore of the Gulf of Trieste, where his military unit was stationed during World War I. He understood in detail the mathematical formalism (tensor calculus) and the physical meaning of the new theory, and in 1918 he published two small papers with his own results, in particular taking part in the discussion of the energy of the gravitational field in the framework of GR. The scientist returned to the general relativistic themes only in the early 1930s, when he made an attempt to consider the behavior of matter waves in curved space-time. The most fruitful period for Schrödinger was when he was working in Dublin. In particular, he obtained a number of specific results in the de Sitter cosmological model, including a reference to the processes of matter production in such a model of the expanding universe. In the 1950s, he wrote two books on GR and cosmology – “Spacetime Structure” (1950) and “The Expanding Universe” (1956).

Another direction of Schrödinger”s work was the attempt to create a unified field theory by combining the theory of gravitation and electrodynamics. This activity was immediately preceded, beginning in 1935, by the Austrian scientist”s study of the possibility of a nonlinear generalization of Maxwell”s equations. The purpose of this generalization, first undertaken by Gustav Mie (1912) and later by Max Born and Leopold Infeld (1934), was to limit the magnitude of the electromagnetic field at small distances, which was to provide a finite value of the intrinsic energy of charged particles. The electric charge within this approach is treated as an intrinsic property of the electromagnetic field. Since 1943 Schrödinger continued the attempts of Weyl, Einstein and Arthur Eddington to derive the unified field equation from the principle of least action by correctly choosing the type of the Lagrangian within affine geometry. Limiting himself, like his predecessors, to a purely classical consideration, Schrödinger proposed the introduction of a third field which was to compensate for the difficulty of combining gravitation and electromagnetism, represented in the Born-Infeld form. He associated this third field with nuclear forces, the carrier of which at the time were considered to be hypothetical mesons. In particular, the introduction of the third field into the theory allowed one to preserve its gauge invariance. In 1947 Schrödinger made another attempt to unite the electromagnetic and gravitational fields by selecting a new form of the Lagrangian and deriving new field equations. These equations contained a connection between electromagnetism and gravitation, which, according to the scientist, could be responsible for the generation of magnetic fields by rotating masses, such as the Sun or the Earth. The problem, however, was that the equations did not allow a return to a pure electromagnetic field when gravitation was “off”. Despite much effort, the many problems facing the theory could not be solved. Schrödinger, like Einstein, did not succeed in creating a unified field theory by geometrizing classical fields, and by the mid-1950s he withdrew from this activity. According to Otto Hittmair, one of Schrödinger”s Dublin collaborators, “high hopes were replaced by clear disappointment during this period of the great scientist”s life.

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## “What is life?”

The creation of quantum mechanics allowed to lay a reliable theoretical foundation for chemistry, with the help of which the modern explanation of the nature of chemical bonding was obtained. The development of chemistry, in turn, had a profound influence on the formation of molecular biology. The famous scientist Linus Pauling wrote in this connection:

In my opinion, it is fair to say that Schrödinger, having formulated his wave equation, is mainly responsible for modern biology.

Schrödinger”s immediate contribution to biology is his book “What is Life?” (1944), based on lectures given at Trinity College Dublin in February 1943. These lectures and the book were inspired by an article by Nikolai Timofeev-Ressovsky, Karl Zimmer and Max Delbrück, published in 1935 and given to Schrödinger by Paul Ewald in the early 1940s. This paper is devoted to the study of genetic mutations that arise under the influence of X-rays and gamma rays and for the explanation of which the authors developed the theory of targets. Although at that time the nature of heredity genes was not yet known, a view of the problem of mutagenesis from the standpoint of atomic physics made it possible to reveal some general laws of the process. Timofeev-Zimmer-Delbrück”s work was the basis of Schrödinger”s book, which attracted the attention of young physicists. Some of them (for example, Maurice Wilkins) under its influence decided to take up molecular biology.

The first few chapters of “What is Life?” are devoted to a review of information on the mechanisms of heredity and mutations, including the ideas of Timofeyev, Zimmer, and Delbrück. The last two chapters contain Schrödinger”s own thoughts on the nature of life. In one of them the author introduced the concept of negative entropy (probably going back to Boltzmann), which living organisms must receive from the outside world in order to compensate for the growth of entropy leading them to thermodynamic equilibrium and hence to death. This, according to Schrödinger, is one of the main differences between life and inanimate nature. According to Pauling, the notion of negative entropy, as formulated in Schrödinger”s work without the necessary rigor and clarity, adds little to our understanding of the phenomenon of life. Francis Simon pointed out shortly after the book was published that free energy must play a much larger role for organisms than entropy. In later editions, Schrödinger addressed this observation, noting the importance of free energy, but still left the discussion of entropy in this, in the words of Nobel laureate Max Perutz, “misleading chapter” unchanged.

In the last chapter, Schrödinger returned to his idea, which runs through the entire book, that the mechanism of functioning of living organisms (their exact reproducibility) is inconsistent with the laws of statistical thermodynamics (randomness at the molecular level). According to Schrödinger, the findings of genetics suggest that there is no place for probabilistic laws to which the behavior of individual molecules must obey; the study of living matter can thus lead to some new non-classical (but deterministic) laws of nature. To solve this problem, Schrödinger turned to his famous hypothesis of the gene as an aperiodic one-dimensional crystal, going back to the work of Delbrück (the latter wrote about a polymer). Perhaps it is the molecular aperiodic crystal in which the “program of life” is written that avoids the difficulties associated with thermal motion and statistical disorder. However, as the further development of molecular biology has shown, the existing laws of physics and chemistry were sufficient for the development of this field of knowledge: the difficulties Schrödinger argued about are solved by the principle of complementarity and enzymatic catalysis, which allows the production of large quantities of one or another substance. Recognizing the role of “What is Life?” in popularizing the ideas of genetics, Max Perutz, however, came to the following conclusion:

…A close examination of his book and related literature has shown me that what was correct in his book was not original, and much of what was original as was known even by the time the book was written was not correct. Moreover, the book ignores some crucial discoveries that were published before it went to press.

In 1960, Schrödinger recalled the time after the end of World War I:

I intended to teach theoretical physics, taking as my model the excellent lectures of my favorite teacher, Fritz Hasenörl, who had been killed in the war. For the rest, I intended to study philosophy. At that time I delved deeper into the works of Spinoza, Schopenhauer, Richard Zemon, and Richard Avenarius… Nothing came of this endeavor. I was forced to stay with theoretical physics and, to my surprise, sometimes something came out of this.

It was only after his arrival in Dublin that he was able to devote sufficient attention to philosophical questions. From his pen came a number of works not only on philosophical problems of science, but also of a general philosophical nature – Science and Humanism (1952), Nature and the Greeks (1954), Reason and Matter (1958) and My World View, an essay he completed shortly before his death. Schrödinger paid particular attention to ancient philosophy, which appealed to him for its unity and the importance it could play in solving the problems of our time. In this connection he wrote:

With a serious attempt to return to the intellectual milieu of the ancient thinkers, who knew far less about the actual behavior of nature, but who were also often far less prejudiced, we can regain freedom of thought from them, if only perhaps to use it, with our better knowledge of the facts, to correct their early mistakes, which may still put us on the spot.

In his writings, drawing also on the legacy of Indian and Chinese philosophy, Schrödinger tried to take a unified view of science and religion, human society and ethical problems; the problem of unity represented one of the main motives of his philosophical work. In his works, which can be classified as philosophy of science, he pointed to the close connection between science and the development of society and culture in general, he discussed problems of the theory of cognition, participated in discussions on the problem of causality and modification of this concept in the light of new physics. A number of books and collections of articles have been devoted to discussion and analysis of specific aspects of Schrödinger”s philosophical views on various issues. Although Karl Popper called him an idealist, in his works Schrödinger consistently defended the possibility of studying nature objectively:

There is a widespread scholarly opinion that an objective picture of the world, as it was previously understood, is impossible to obtain at all. Only the optimists among us (of which I include myself) believe that this is philosophical exaltation, a sign of cowardice in the face of crisis.

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## Some works in Russian translation

**Sources**