Erwin Schrödinger, Anschaulichkeit and Quantum Theory

Pergamon Stud. Hist. Phil. Mod. Phys., Vol. 28, No. 4, pp. 461481, 1997 0 1997 Elsevier Science Ltd

All rights reserved. Printed in Great Britain 1335-2198/97 $17.00+0.00

Erwin Schrtidinger, Anschaulichkeit, and Quantum Theory

Henk W. de Regt*

1. Introduction

Early in 1926 Erwin Schrodinger presented his famous theory of wave mechanics to account for atomic phenomena. It is often assumed that Schrodinger’s work reflected a realist philosophy.* In this article, I will argue that this assumption is incorrect. To substantiate this claim, I will offer a detailed analysis of Schriidinger’s philosophical views, which shows that he was heavily influenced by Ernst Mach’s anti-realist views concerning epistemology and ontology. His views on scientific method, however, differed considerably from Mach’s, and it is this aspect of his philosophy that was a driving force behind the construction of wave mechanics. Contrary to the Machian empiricist, who holds that the aim of science is to describe the observable phenomena in the most economical way, Schrodinger considered the aim of science to be the understanding of these phenomena. Such understanding could, according to Schrodinger, only be provided by theories that are visualisable in space and time. Because Schriidinger placed such strong emphasis on the demand that theories provide visualisable pictures, it is tempting to infer that he also believed that these pictures are direct representations of reality. However, this is not a necessary consequence. It will be shown that Schrodinger did not adhere to such a realist interpretation of theories.

The association of visualisability with understanding rather than with realism may be elucidated by considering the German word Anschaulichkeit, which is the term Schrbdinger used in his writings. This word does not only mean ‘visualisability’ but also ‘intelligibility’. In the 1920s the difficult question of whether both these notions could be applied to quantum theory led to serious

(Received 17 September 1996.) *Department of History and Foundations of Mathematics and Science, University of Utrecht,

P.O. Box 80000, 3508 TA Utrecht, The Netherlands (e-mail: [email protected]). ‘See e.g. Cushing (1994, pp. 106, 124) and Harm (1992, pp. 1477148). An alternative view on

Schrlidinger’s philosophy, which anticipates my analysis to some extent, is provided by Wessels (1983, pp. 269-271). Furthermore, Michel Bitbol’s book Schriidinger’s Philosophy of Quantum Mechanics. (1996), which appeared after the present article was completed, contains an analysis of Schrodinger’s position with respect to the realism issue that is similar to mine.

PII: S1355-2198(97)00017-S

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discussions by Schrodinger and other leading physicists. Indeed, the notion of Anschaulichkeit played a crucial role in the genesis of quantum mechanics, a role which is somewhat neglected by historians and philosophers of science. The present article attempts to contribute to filling this lacuna.

In Section 2 the notion of Anschaulichkeit, and its application to quantum theory, is discussed in more detail. Section 3 presents an analysis of Schrodinger’s philosophical position, with special attention to Anschaulichkeit.

Section 4 describes Schrodinger’s discovery of wave mechanics and analyses the role of Anschaulichkeit in this development.

2. Quantum Theory, Anschaul~cMwit, and Philosophy

From a conceptual point of view, quantum theory is a very strange theory. The main reason for this is that the theory appears to break with some fundamental conceptions which are characteristic of classical (i.e. pre-quantum) physics and which are in accordance with everyday experience. Examples are continuity, causality, and visualisability. Compared to the first-mentioned notions, visualisability has received scant attention in the literature. Still, discussions about visualisability were equally important to the historical development of quantum theory.*

The possibility and desirability of visualisation in quantum theory was debated chiefly by physicists who wrote in German. The German word for visualisability is Anschaulichkeit. This is of interest because the latter term also has a non-literal, metaphorical meaning, namely ‘intelligibility’. Actually, also many English expressions employ visual metaphors for concepts related to understanding: ‘insight’, ‘clarity’, ‘I see’, etc. It seems clear that there is some

connection between visualisation and understanding. But precisely what does it consist of?

In order to answer this question, note that there are two ways in which one can ‘understand’ a phenomenon. Firstly, if one observes a phenomenon, e.g. visually, then one has ‘direct’ understanding of it, in the sense of immediately grasping and knowing it. Obviously, this does not imply that one knows exactly how and why the observed phenomenon occurred. This latter form of knowledge constitutes the second sense of understanding: an ‘indirect’ but

deeper form, which science typically claims to be able to provide. Understand- ing in the first senseairect grasping by perception-always takes place in space and time: all our sensory experience can be described in a spatio-temporal frame. Thus, if ‘visualisability’ is conceived as the possibility of forming a

space-time picture, then direct understanding is necessarily connected with visualisability.

‘The importance of the notion of ~~sc~a~t~c~~eif in the development of quantum theory has been acknowledged by a few authors, notably Miller (1978, 1984), Forman (1979, 1984), and Belier (1983a). These authors, however, focus mainly upon Heisenberg’s views.

Schriidinger and Anschaulichkeit 463

Consider now understanding in the second sense. If one holds that science aspires to furnish such a deeper understanding of phenomena, then one must

specify how this is reached. The standard view is that science makes our perceptions more intelligible by presenting theories to account for them. Vari- ous criteria may be proposed for theories with this explanatory aim: unification, causality, locality, and so on. Since visualisability in space and time is a necessary condition for the direct understanding of phenomena, one may argue that it is also a condition for the indirect, deeper understanding provided by scientific theories. This is the central question in the debates on quantum theory and Anschaulichkeit: Is visualisability a necessary condition for physical theories aiming to provide understanding of nature? It appears that scientists disagree on this issue. Schrodinger answered the question in the affirmative, whereas other physicists, most notably Wolfgang Pauli, answered it in the negative.

In classical physics Anschaulichkeit is unproblematic because all classical theories are visualisable, in the sense that they are formulated in the same spatio-temporal framework as our visual experiences. When classical physics deals with unobservable entities, it does so in a manner that is, at least in this respect, completely analogous to the way it describes observable entities. A clear example is Rutherford’s atomic model, which depicts the atom as a nucleus with electrons orbiting around it, that is, as a miniature solar system. The visualisability which is so characteristic of classical physics was challenged by quantum theory in at least two fundamentally different ways. Firstly, certain processes inside the atom, particularly the so-called ‘quantum jumps’ of electrons between orbits, seemed to completely defy spatio-temporal description, and thus visualisation. Secondly, the wave-particle duality prohibited unambiguous visualisation of radiation and later also of matter. These were important reasons that Anschaulichkeit, which was hitherto a relatively unproblematic notion, came to be a topic of debate among physicists in the quantum era.

This article attempts to relate Schrodinger’s views on Anschaulichkeit to his more general philosophical position, The analysis employs a distinction between three domains of philosophy: ontology, epistemology and methodology. Ontology is concerned with the general nature of what exists; epistemology deals with the nature, the source, and the scope of knowledge; methodology concerns the specific aims and procedures of science. While boundaries between these domains may be blurry, it is important to distinguish between them. For example, in the three domains realism has a different meaning. Ontological realism asserts that the world (whatever its specific constitution) exists independently of our knowledge of it. Epistemological realism asserts that scientific theories aim to provide knowledge about an unobservable reality behind the phenomena. Finally, methodological realism merely declares that scientists, when working with a theory, should act as if it truly represents reality


in all respects.3 Schrodinger’s commitment to Anschaulichkeit is classified as a methodological view, because it concerns a specific aim of science. Obviously, scientists’ views on Anschaulichkeit will usually be connected with their epistemological and ontological views. However, contrary to what might be believed, there does not exist a universal and necessary relationship. The following analysis of Schrodinger’s philosophy purports to show that it is possible to value Anschaulichkeit without a commitment to ontological and epistemological realism.

3. The Philosophy of Erwin Schriidinger

3.1. Biographical introduction In 1906 Erwin Schrodinger (1887-1961) entered the University of Vienna to

study physics. The memories of Boltzmann, who had just taken his own life, and Mach, who had retired in 1901, still lingered in the air, and their heated debates over the existence of atoms and the philosophical foundations of physical science were far from forgotten. Schrbdinger learned theoretical physics from Fritz Hasenohrl, the former pupil and successor of Boltzmann, and experimental physics from Franz Exner. Both men exerted a deep influence on him. Hasenohrl’s inaugural lecture of 1907, being a brilliant exposition of Boltzmann’s ideas and achievements, impressed him indelibly and directed his own thoughts towards a path from which he would never again depart. Schrodinger (1992, p. 168) later testified that ‘no other human being had a greater influence on me than Fritz Haseniihrl, except perhaps my father Rudolph’. This influence, however, consisted above all in the ideas of Boltzmann. Thus, in his 1929 inaugural speech, he remarked: ‘[Boltzmann’s] line of thought may be called my first love in science. No one has ever thus enraptured me or will ever do so again’ (Schrodinger, 1957, p. xiv).

The influence of Exner can be found in Schrbdinger’s early ideas on causality and indeterminism (see Hanle, 1979a). Furthermore, being involved in experimental physics is sure to have affected Schrodinger’s philosophy of science. Although he was chiefly interested in theoretical physics, Schrodinger worked in both fields until 1920, and his doctoral dissertation of 1910 was a purely experimental study. After obtaining his degree he was offered a one-year university assistantship under Exner in experimental physics. During this time Schrodinger discovered that he was not suited to be an experimentalist, but he later expressed his gratitude for having been able to experience directly the experimental side of physical research (see Schrodinger, 1992, p. 169). In his biography of Schrbdinger, Moore (1989, p. 59) states that ‘his work as a laboratory assistant helped to determine the philosophical framework that he was willing to accept for a physical theory’, namely that ‘physics is not

‘See De Regt (1993, pp. 29-38) for a more detailed survey of these different domains of philosophy.


based upon mathematical fantasies but on a solid ground of experimental observations’.

Just before the war, in 1914, Schrijdinger completed his Hub&&ion

in theoretical physics and obtained a position as Privatdozent at Vienna University. During the war he served in the army, but he still managed to carry out some theoretical research, especially on statistical dynamics, and he also carefully studied Einstein’s general theory of relativity. In September 1921 Schrbdinger settled in Zurich, having been offered a professorship in theoretical physics. There he worked mainly on statistical physics. Quantum theory did not have his special attention, although he made minor contributions to it (of which his 1922 paper is important for his later discovery of wave mechanics; see Section 4). The main reason for this lack of interest was that he regarded the old quantum theory of Bohr and Sommerfeld as unsatisfactory, because of ‘its inherent contradictions’ (Schriidinger, 1957, p. xiv). In 1925, however, his work on gas statistics led him to study Einstein’s paper on the quantum theory of monatomic gases and Bose statistics. Through this paper he became acquainted with Louis de Broglie’s hypothesis of matter waves. In late December 1925, in an outburst of creativity, Schrbdinger developed the essential foundations of wave mechanics, a theory of atomic structure in which only wave concepts were used. During the first half of 1926 he further elaborated his theory and presented a proof of its mathematical equivalence with matrix mechanics, a rival theory advanced by Heisenberg, Born, and Jordan. 4 Matrix mechanics had repelled Schrodinger because of its lack of Anschaulichkeit. However, the various physical, visualisable (anschauliche) interpretations that he proposed for his wave mechanics all ran into difficulties, and he did not succeed in defending them against the attacks from the Copenhagen-Giittingen school of quantum mechanics. The way in which this school then incorporated his theory was wholly unacceptable to Schriidinger: he considered Born’s statistical interpretation of the wave function a complete misinterpretation of his theory, and he was ‘concerned and disappointed that this transcendental, almost psychical interpretation of the wave phenomena [i.e. the Copenhagen interpretation] had become the almost universally accepted dogma’ (quoted in Hermann, 1975, p. 221). For the rest of his life he was to remain a critic of the Copenhagen interpretation, and an outsider in the field to which he himself had contributed so decisively.

Since his early days Schrodinger was seriously interested in philosophy in the broadest sense of the word. In the war he found the time to study philosophical works. Thus, as he wrote in the preface to Meine Weltansicht, around 1918 he was ‘deeply imbued with the writings of Spinoza, Schopenhauer, Mach, Richard Semon and Richard Avenarius’ (Schrbdinger, 1964, p. viii). Through

4The validity of Schriidinger’s proof has been recently contested by Muller (1997).


Schopenhauer’s work, which he read with fascination, he encountered the

ancient Indian philosophy of Vedanta and the Upanishads, of which the main

characteristic is the belief in an undivided reality of thought, a transcendent

unity of consciousness. He incorporated these philosophical doctrines into his

own philosophy, and in 1925 wrote a philosophical treatise, Suche nach dem

Weg, in which he analysed the desperate state of metaphysics in Western

thought and developed an alternative based on the Vedanta.s Although

Schrodinger presented his views as having no connection to the particular

philosophical problems of physics, his analyses (especially of Western philoso-

phy) shed some light upon his views concerning the philosophy of science, as

will be shown further on. In his later years Schrijdinger lectured and wrote on

a wide variety of subjects, but always with a philosophical flavour. Among his

books are Science and Humanism (1951), Nature and the Greeks (1954), and

Mind and Matter (1958).

3.2. Between Boltzmann and Mach

As mentioned above, the debate between Boltzmann and Mach was still a

lively subject of discussion when Schriidinger began his studies in Vienna. It is

therefore not surprising that both Mach’s and Boltzmann’s ideas affected

Schrodinger’s philosophy of science. Like many Viennese physicists of the

younger generation, Schrodinger did not consider the views of Boltzmann and

Mach irreconcilable:

Boltzmann’s idea consisted in forming absolutely clear, almost naively clear and detailed ‘pictures’-mainly in order to be quite sure of avoiding contradictory assumptions. Mach’s ideal was the cautious synthesis of observational facts that can, if desired, be traced back till to the plain, crude sensual perception (pointer reading). He was most anxious not to contaminate this absolutely reliable timber with any other one of a more doubtful origin.

However, we decided for ourselves that these were just different methods of attack, and that one was quite permitted to follow one or the other provided one did not lose sight of the important principles that were more strongly emphasized by the followers of the other one, respectively (Schriidinger, 1995, pp. 121-122).

Schrijdinger thus took from Boltzmann the Bild conception of physical

theory: the aim of physical theory is the construction of a ‘picture’ of the outer

world, that serves as a guiding star in thought and experiment (Boltzmann,

1974, p. 33). As I have pointed out elsewhere (De Regt, 1996, pp. 4143),

Boltzmann’s epistemological position in the 1890s was rather subtle: he was

cautious not to assert a simple identity relation between the theoretical picture

and the real world. At the methodological level, however, he strongly advised

a realist attitude and he forcefully rejected the ideas of Mach and his followers.

His main strategies in attacking Mach were to argue, firstly, that Mach’s

‘Suche nach dem Weg was published only in 1961 as part of Meine Weltansicht. An English translation of this book appeared as Schriidinger (1964).


phenomenological position was no less metaphysical than his own, and secondly, that Mach’s ideas were sterile and would hinder the progress of

science. We will see below that Schrodinger was to make use of both these Boltzmannian strategies in his work.

At this point it is interesting to note that Schrodinger possessed a copy of Boltzmann’s Pop&ire Schrzjien, in which he made notes on a number of articles (see Flamm, 1987, pp. 11-12). This fact is important for two reasons. Firstly, it shows that Schrodinger studied Boltzmann’s philosophical writings, and thus had a more than superficial understanding of his philosophy and, consequently, of the controversy between Boltzmann and Mach. That is to say, he was probably aware of the fact that at the epistemological level Boltzmann was not a naive correspondence realist, and that it was thus rather at the ontological and the methodological level that the struggle between Boltzmann and Mach was situated. Secondly, certain notes Schrodinger made in his copy show that he had doubts about some of Boltzmann’s stronger realist claims.

Apart from the fact that a considerable part of Schriidinger’s early work was in the field of statistical mechanics and gas theory, and thus was a direct continuation of Boltzmann’s line of research, an especially interesting influence of Boltzmann’s way of thought can be discerned in Schrodinger’s 1914 article ‘Zur Dynamik elastisch gekoppelter Punktsysteme’.6 In this paper he dealt with the controversy between atomistic and continuum theories of matter. Building upon Boltzmann’s above-mentioned strategies, he argued that the ‘idealised’ differential equations must be replaced by difference equations, and then he set out to state conditions under which differential equations actually lead to incorrect results, thus attempting to prove the superiority of the atomistic theory of matter.

Schrodinger’s admiration for Boltzmann notwithstanding, he was at the same time influenced by Mach. Mach’s philosophy affected first and foremost Schrodinger’s ontological views, and to a lesser extent his epistemological views. The ontological doctrine which Schrodinger adopted from Mach is known as neutral monism. It asserts that minds and bodies do not differ in their intrinsic nature, but that the difference between them lies in the way that a common (‘neutral’) material is arranged (Flew, 1983, p. 237). This common material may consist of many entities of the same fundamental kind, e.g. experiences. Neutral monism is opposed to subject-object (or mind-body) dualism. Schrodinger attributed the latter view to Kant and his opposition to dualism might explain his rejection of Bohr’s philosophy of complementarity (see MacKinnon, 1982, p. 221n). He upheld this doctrine all his life, as his 1954 book Nature and the Greeks shows (see Schriidinger, 1996, pp. 93394). Karl Popper (1976, p. 135) writes in his autobiography that, when discussing

6Reprinted in Schriidinger (1984, Vol. 1, pp. 124-142).


philosophical questions around 1954-1955, he was surprised to find that Schriidinger defended neutral monism, admitting that it was a form of idealism. Popper adds that ‘Schriidinger had absorbed his idealism from Schopenhauer’.

In the fourth chapter of his 1925 essay Suche nach dem Weg, Schrodinger (1964, pp. 12-18) presents a discussion of the mind-body problem that contains an affirmation of Mach’s ontological position. He begins by rejecting out of hand the assumption that a soul can exist independently of a body as naive and childish, and states that both the ‘I’ and the ‘World’ consist of the same ‘empirical elements’. Then he addresses the question of what happens when the ‘I’ observes something, for example a tree. Schrodinger criticises the well- known solution of distinguishing between the observation of the tree and the tree itself (the tree an sich), because it does not clarify in what the observation of the tree consists. Subsequently, he sketches the radical and simple answer to this problem, proposed by Mach and Avenarius: as the tree an sich is inaccessible for our experience, we might as well declare once and for all that it is uninteresting and that we will abandon it. Hence only one given aspect of the tree remains, and there is no reason not to call it ‘tree’. Schrodinger (1964, p. 15): ‘This one tree, then, is the one datum we have: it is at one and the same time the tree of physics and the tree of psychology’. He remarks approvingly that this implies that the ‘I’ and the ‘World’ consist of the same elements, leading to the elimination of many pseudo-problems and being a vindication of ‘naive realism’7 However, one problem remains, namely the question of what happens when different people observe the same tree. The only consistent answer, according to Schrodinger, and indeed the answer that Mach and Avenarius gave, is to say that the tree, being one complex of elements, is a common ingredient of different ‘1’s. Schrodinger claims that this conclusion, which may at first sight seem quite bizarre, is in fact closer to daily experience and to naive realism than the idea, common to Western thought, that the experiences and thoughts of different individuals belong to mutually exclusive spheres. In Chapter 5 of Suche nach dem Weg he proceeds to explain his alternative metaphysics, based on the doctrine of Vedanta, in which the manifold of appearances is reduced to a universal, undivided consciousness.

Thus, Schrodinger’s ontological views were based upon neutral monism, and, like Mach, he identified the neutral elements with ‘empirical elements’ (Mach’s ‘sensations’). Consequently, he implicitly confirmed Mach’s empiricist epistemology: all that we know is given by experience. However, Schriidinger’s views transcended Mach’s in two ways. On the one hand, as mentioned above, he transformed the Machian ontology into substantive monism, that is, the doctrine that ultimately there exists only a single entity, comprising all

‘Note that SchrBdinger’s term ‘naive realism’ does not refer to naive correspondence realism with respect to scientific theories, but to the common sense conception of the world as it presents itself to us.


empirical elements. On the other hand, he strongly disagreed with Mach’s narrow view of scientific method. Mach claimed that the only aim of science is to describe the complexes of sensations in the most economical manner. Contrary to this, SchrGdinger believed that the final aim of science is not merely to describe the phenomena, but also to explain them, to gain understanding of them. Arguing against Mach’s methodological views, Schrddinger stated, in the first chapter of Suche nach dem Weg, that ‘elimination of metaphysics means taking the soul out of both art and science, turning them into skeletons incapable of any further development’, and urged the reader to

call to mind that sense of misgiving, that cold clutch of dreary emptiness which comes over everybody, I expect, when they first encounter the description given by Kirchhoff and Mach of the task of physics (or of science generally): ‘a description of the facts, with the maximum of completeness and the maximum economy of thought’; a feeling of emptiness which one cannot master, despite the emphatic and even enthusiastic agreement with which one’s theoretical reason can hardly fail to accept this prescription. In actual fact (let us examine ourselves honestly and faithfully), to have only this goal before one’s eyes would not suffice to keep the work of research going forward in any field whatsoever (Schriidinger, 1964, pp. 34).

In these passionate lines one clearly hears the echo of Boltzmann’s opinion.

As mentioned above, Boltzmann pleaded for methodological realism, arguing that Mach’s phenomenalism was bound to remain sterile. Accepting this thesis, SchrGdinger immersed himself in the world-picture of science (which explains why he persistently used the word ‘real’ in his scientific writings despite his Machian epistemology). He also employed Boltzmann’s thesis when in 1926 his theory of wave mechanics had to compete with matrix mechanics. Although these two theories are very different in character (the former: ‘physical’, visualisable, and ‘realistic’; the latter: ‘mathematical’, non-visualisable, and ‘positivistic’), Schriidinger himself proved their mathematical equivalence (Schriidinger, 1928, pp. 45-61). Having proved this, he went on to compare the physical content of the two theories, and observed that:

Today there are not a few physicists who, like Kirchhoff and Mach, regard the task of physical theory as being merely a mathematical description (as economical as possible) of the empirical connections between observable quantities, i.e. a description which reproduces the connection, as far as possible, without the intervention of unobservable elements. On this view, mathematical equivalence has almost the same meaning as physical equivalence. In the present case there might perhaps appear to be a certain superiority in the matrix representation because, through its Unanschauli- chkeit, it does not tempt us to form space-time pictures of atomic processes, which must perhaps remain uncontrollable (SchrBdinger, 1928, p. 58).8

Against this conclusion, however, Schriidinger presented two arguments.

Firstly, because the mathematical equivalence is completely symmetrical, the

*Note concerning translations: in quotations I have replaced the original German terms unschaulich, Anschaulichkeit, etc., because they are often inadequately translated (e.g. as ‘intuitive’).


formalism of wave mechanics does not contain a superfluous, ‘metaphysical’ part that transcends the physical (empirical) content of matrix mechanics. In Schrodinger’s own words (1928, p. 58): the wave functions ‘do not form, as it were, an arbitrary and special “fleshy clothing” for the bare matrix skeleton, provided to pander to the need of Anschaulichkeit, [which] would establish the superiority of the matrices, from the epistemological point of view’. This argument is interesting also for its Machian flavour: it assumes that those parts of a theory which transcend empirical evidence are epistemologically irrelevant. As wave mechanics equals matrix mechanics in this respect, Schrijdinger continued by stating his second argument, which was meant to prove the superiority of wave mechanics over matrix mechanics. He argues that mathematically equivalent theories may differ in their possible extensions, and that they can therefore differ in their fruitfulness. According to Schriidinger (1928, p. 59) atomic theory is confronted with problems that seem ‘extra- ordinarily difficult to tackle [...I as long as we feel obliged on epistemological grounds to repress Anschauung in atomic dynamics, and to operate only with such abstract ideas as transition probabilities, energy levels, etc.‘. Thus, he contends that anschauliche theories are preferable because they are heuristically more powerful. In the case of wave mechanics versus matrix mechanics this indeed proved to be true: as Beller (1983b, pp. 489490) reports, wave mechanics ‘was successfully applied to a great variety of problems unamenable to matrix treatment’.

3.3. The Import of Anschaulichkeit The fact that Schrodinger agreed with the ontological and epistemological

views of Mach, while he based his scientific methodology on Boltzmann’s ideas, leads to a somewhat paradoxical conclusion, noted by Wessels (1983, p. 271) and Moore (1989, p. 217). Schrodinger held that good physical theories should provide visualisable pictures in space and time, but apparently he did not think that such theories represent reality. Since he rejected any form of epistemological realism, one is tempted to ask why Schrddinger valued Anschaulichkeit so highly. Why did he so desperately try to find a (realistically interpretable) visualisation of his wave mechanics, as we will see in the next section?

MacKinnon (1980, pp. 3, 18); (1982, pp. 246247) has argued that it was not a prior philosophical commitment to Anschaulichkeit, but the direct competition with Heisenberg that was the driving force behind Schrodinger’s attempts to find an interpretation establishing the scientific superiority of his wave mechanics over matrix mechanics. Although this hypothesis does indeed permit a simple resolution of the apparent tension between Schriidinger’s epistemological and methodological views, it seems that there is no independent support for it. Moreover, there are several arguments against it. Firstly, long before the competition with Heisenberg emerged, Schriidinger had already committed


himself to Anschaulichkeit (cf. Wessels, 1980). Secondly, wave mechanics was enthusiastically received by the physics community, both on philosophical and scientific grounds, and SchrGdinger was therefore, at least in the first months of 1926, not in need of extra support. Indeed, it was rather matrix mechanics that was threatened by the success of wave mechanics (see Belier, 1983b).

It appears more plausible that it was Schriidinger’s prior commitment to Anschaulichkeit, which had its source in Boltzmann’s Bild conception of physical theory, that played a role here. In that case, however, we still need to clarify the relation between this methodological commitment and Schrddinger’s Machian anti-realism at the epistemological level. Some clues can be found in a letter from SchrGdinger to Wilhelm Wien (25 August 1926), in which he criticised Born’s and Bohr’s interpretations of quantum mechanics. Concerning Born’s statistical interpretation of the wave function, he remarked that

Born [...I overlooks that [...I it would depend on the taste of the observer which he now wants to regard as real, the particle or the guiding field [Born favours the former. There is certainly no criterion for reality if one does not want to say: the real is only the complex of sense impressions, all the rest are only pictures (quoted in Moore. 1989, p. 225).

Note that SchrGdinger did not argue for the reality of the guiding field (i.e. the waves). On the contrary, he asserted that no criterion for reality can be given, except for the observable consequences of the theory. This is clearly an endorsement of Machian empiricism. Immediately after this statement Schriidinger went on to criticise Bohr:

Bohr’s standpoint, that a space-time description is impossible, I reject a limine. Physics does not consist only of atomic research, science does not consist only of physics, and life does not consist only of science. The aim of atomic research is to fit our empirical knowledge concerning it into our other thinking. All of this other thinking, so far as it concerns the outer world, is active in space and time. If it cannot be fitted into space and time, then it fails in its whole aim and one does not know what purpose it really serves (quoted in Moore, 1989, p. 226).

These statements reveal why for SchrGdinger there was no tension between

his insistence on Anschaulichkeit and his anti-realism. He intended the former criterion not as a restriction on the possible character of reality, but as a restriction on the possible character of understanding. Scientific theories should be designed to understand nature, and only a theory which is anschaulich can provide understanding. Although anschaulich may be translated as ‘intelligible’, it cannot be doubted that SchrGdinger equated intelligibility with visualisability in space and time, because:

we cannot really alter our manner of thinking in space and time, and what we cannot comprehend within it we cannot understand at all. There are such things-but I do not believe that atomic structure is one of them (Schriidinger, 1928, p. 27).


Thus, Schrodinger believed in the ultimate intelligibility of nature, and he held that Anschaulichkeit is a necessary criterion for understanding.9 Besides its heuristic power, this was his main reason for valuing Anschaulichkeit so highly. It follows that Schrodinger considered it possible to understand phenomena by means of a theory of which one, at the same time, admits that its truth (i.e. correspondence with reality) cannot be justified epistemologically. While this position may at first sight appear outlandish and even inconsistent, it turns out to be tenable in the light of the distinction between epistemology and methodology, introduced in Section 2. The question of whether a theory provides understanding-and thus also the criterion of Anschaulichkeit-

belongs to the latter domain, and should not be confused with epistemological questions. A modern defence of this position is presented by Van Fraassen (1980, esp. pp. 87-96). In his terminology, the explanatory power of a theory is labelled a pragmatic virtue (as opposed to epistemic virtues). Pragmatic virtues, Van Fraassen (1980, p. 88) states, ‘do not concern the relation between the theory and the world, but rather the use and usefulness of the theory; they provide reasons to prefer the theory independently of questions of truth’. In Section 5, I will expand on the similarities between Schriidinger’s and Van Fraassen’s views.

3.4. Anschaulichkeit and the BKS Theory

In the early 1920s Schrijdinger came out in support of acausality. He endorsed the views of his teacher Exner, who in 1919 had suggested that macroscopic laws of nature have merely statistical validity, the underlying microscopic processes being probabilistic (see Hanle, 1979a). Accordingly, Schrodinger appreciated the new radiation theory of Bohr, Kramers and Slater (BKS), which implied a renunciation of conservation laws at the microscopic level. On 24 May 1924, he wrote to Bohr:

I have just read with the greatest interest the interesting change in your ideas in the May issue of the Phil. Msg. I am extremely sympathetic to this change. As a pupil of old Franz Exner, I have long been fond of the idea that the basis of our statistics is probably not microscopic ‘regularity’ but perhaps ‘pure chance’ and that perhaps even the laws of energy and momentum have merely statistical validity (Bohr, 1976,

Vol. 5, p. 490; translation in Moore, 1989, p. 162).‘0

SchrGdinger’s opinion of the BKS theory is interesting not only for its

aflirmation of acausality, but also because it illustrates his criteria for theory

acceptance. The BKS theory contained ‘virtual’ elements, which were not

‘Schrodinger’s belief in the intelligibility of nature is even more apparent in his later philosophical writings, for example in Nature and the Greeks. There Schrodinger (1996, p. 90) argues that all science is based on ‘the hypothesis that the display of Nature can be understood, a fundamental principle which originated from ancient Greek thought’.

“Cf. Schrodinger’s 1924 paper ‘Bohrs neue Strahlungshypothese und der Energiesatz’, reprinted in Schrijdinger (1984, Vol. 3, pp. 2630).


interpreted realistically. I* I hold that Schrodinger’s sympathy for the theory

confirms my account of the relation between his epistemological and methodo-

logical ideas. Although Anschaulichkeit is a necessary criterion for a scientific

theory, acceptance of an anschaulich theory does not imply an epistemological

commitment to the idea that there is a correspondence between theory and

reality. Accordingly, Schrodinger valued the Anschaulichkeit of the BKS

theory, since it provided a space-time description of atomic structure, but he

did not interpret it realistically. This can be gleaned from another passage from

his letter to Bohr:

I cannot completely go along with you when you keep calling [this radiation] ‘virtual’ [...I For what is the ‘real’ radiation if it is not that which ‘causes’ transitions, i.e., which creates the transition probabilities? Another sort of radiation will surely not be assumed in addition. Indeed, from a purely philosophical standpoint, one might even venture to wonder which of the two electronic systems has a greater reality-the ‘real one’ which describes the stationary states or the ‘virtual one’ that supplies virtual radiation and scatters impinging virtual radiation (Bohr, 1976, Vol. 5, p. 490; translation in Wessels, 1979, p. 313).

Quoting this passage, Wessels (1979, p. 3 13) claims that Schrodinger

‘objected to the authors’ reluctance to give a coherent physical picture for the

theory’. I suggest that Schrodinger indeed criticised the incoherence of the BKS

picture, but that this incoherence is due to the distinction between ‘real’ and

‘virtual’ radiation. He argues for abandoning this distinction, not because he

believes that the BKS description faithfully portrays atomic reality, but, on the

contrary, because he believes that no theory can do so. Thus, there is no point

in distinguishing between a ‘real’ and a ‘virtual’ part of theoretical description,

and we might as well call all parts ‘real’. From the philosophical (read:

epistemological) point of view, according to Schrodinger, no criterion for

reality is possible, except the Machian criterion of attributing reality only to

sense impressions. At the methodological level, however, he sees no objection

to calling theoretical terms ‘real’. Precisely because his epistemological

position amounts to Machian anti-realism, Schrodinger is in a position to

object to calling some terms in the theory ‘virtual’. If he had adhered to a

hard-headed correspondence realism, he would have dismissed the BKS theory

out of hand.

4. The Quest for a Visual&able Quantum Mechanics

In the first half of 1926 Schriidinger published four papers ‘Quantisierung als

Eigenwertproblem’ (Part I-IV) and two additional papers. In these papers he

developed the formalism of wave mechanics, which he presented as an

alternative theory of atomic structure, a theory in which there was no need

“For an illuminating account of the BKS theory, see Dresden (1987, pp. 159-178).


for ‘quantum jumps’ and for a renunciation of space-time description. In other words, Schrbdinger’s theory was an attempt to restore the place of Anschaulichkeit in physical science. In this section I will discuss the development of wave mechanics and the role of Anschaulichkeit in it. Before reviewing the crucial years 1925-1926, I will first briefly sketch Schrodinger’s early work on quantum theory. l2

4.1. Towards Wave Mechanics

Before 1925 Schrodinger was not deeply involved in quantum theory, but he did make some contributions to it. From the viewpoint of his later work, the most important of these is his 1922 article ‘ober eine bemerkenswerte Eigenschaft der Quantenbahnen eines einzelnen Elektrons’.13 In this paper Schrodinger suggested that a ‘phase wave’ could be assigned to the orbiting electron, an idea which was conceived independently by Louis de Broglie one year later. However, Schrbdinger did not pursue his discovery. In 1925 his interest in the statistical foundations of gas theory led him to study Einstein’s paper ‘Quantentheorie des einatomigen idealen Gases’, in which a new statistical counting procedure (‘Bose-Einstein statistics’) was employed to derive the entropy of a monatomic gas. Searching for an explanation of Bose-Einstein statistics, Schrodinger read de Broglie’s work, which was cited in Einstein’s article. He was excited about it for two reasons: firstly, because he recognised that his discovery of 1922 fitted into de Broglie’s theory of matter waves; and secondly, because de Broglie’s ideas provided new insight into Einstein’s gas theory. l4

Einstein had referred to de Broglie in connection with the analogy between radiation and gases. In his paper ‘Zur Einsteinschen Gastheorie’,IS Schrodinger pursued this analogy and treated the whole gas as a quantised wave phenomenon instead of as a collection of particles. In this way he was able to derive Einstein’s theory using only Boltzmann-Gibbs statistics, a success that led him to conclude: ‘This means nothing other than taking seriously the de Broglie-Einstein undulatory theory of moving particles, according to which a particle is nothing but a kind of “wave crest” on a wave radiation that forms the basic substance of the world’ (Schrodinger, 1984, Vol. 1, p. 358; my translation). This statement goes beyond de Broglie’s ideas, for it does not treat the concepts of wave and particle on equal footing: whereas de Broglie’s theory was truly dualistic, ascribing both wave and particle aspects to matter (and light), Schrodinger wanted to reduce the particle aspect to a wave characteristic.

‘*This sketch draws chiefly on the detailed analysis of Schrodinger’s route to wave mechanics provided by Wessels (1979).

13Reprinted in Schrodinger (1984, Vol. 3, pp. 14-24). Raman and For-man (1969) analyse the importance of this article for the development of wave mechanics.

‘%ee Schrodinger’s letter to Einstein of 3 November 1925, quoted in Hanle (1979b, p, 645). IsReprinted in Schriidinger (1984, Vol. 1, pp. 358-364).


Here one may notice Schriidinger’s predilection for substantive monism. In the

final section of his paper he elaborated his idea and considered the question of

whether particles could be reduced to packets of phase waves. The dispersion of

such wave packets turned out to be an obstacle. However, Schriidinger hoped

that this problem could be solved in the future.

4.2. The Discovery of Wave Mechanics

Schrodinger then tried to apply his new idea to atomic theory. Already before

his final paper on gas theory, he had, motivated by the parallel between de

Broglie’s theory and his own 1922 paper, attempted in vain to conceive of the

electron as a wave travelling in the orbit. In December 1925 he took a different

approach: he treated the bound electron as a wave theoretic eigenvalue

problem, and tried to find a wave equation; this implied a change from

travelling to standing waves. Schrodinger first found a relativistic wave

equation, and solved it for the hydrogen atom. Unfortunately, the obtained

solutions gave incorrect results for the fine structure of the spectrum.‘6 Soon,

however, Schrodinger discovered that the non-relativistic version of his

equation did yield correct solutions. From January 1926 on he developed a

complete non-relativistic wave mechanics, which he rapidly published in the

papers ‘Quantisierung als Eigenwertproblem’ (Part I-IV).17 In the first paper

Schrodinger derived the time-independent equation

AY+@(E- vy=o, K2

with ‘I’ a real, single-valued, continuous function, and showed that for K= hl2n

the solutions of the equation correspond to the energy levels of the hydrogen

spectrum. The most remarkable feature of the derivation is that it is purely

mathematical and does not refer to the wave picture. It was only in the last

section of the paper that Schrodinger discussed a possible interpretation:

It is, of course, strongly suggested that we should try to connect the function Y with some vibrationalprocess in the atom, which would more nearly approach reality than the electronic orbits, the real existence of which is very much questioned today. I originally intended to found the new quantum conditions in this more anschaulich manner, but finally gave them the above neutral mathematical form, because it brings more clearly to light what is really essential (Schriidinger, 1928, p. 9).

What is ‘essential’, according to Schrodinger, is that his analysis shows that

the ad hoc quantum conditions can be explained by the conditions for the

function Y. Wessels (1979, pp. 332-333) suggests two possible reasons that

Schrbdinger might have had for presenting his theory in this way. First, a

16The reason for this failure is that the relativistic equation (the Klein-Gordon equation) applies only to particles without spin.

17Reprinted in Schriidinger (1984, Vol. 3, pp. 82-136, 166250). English translations in Schradinger (1928).


strategic one: in 1926 the leading quantum physicists believed that quantum theory required the renunciation of any visualisation, and Schrodinger’s theory, if presented in the ‘more anschaulich manner’, might have seemed a reactionary attempt to return to classical physics. Second, Schrodinger realised that his interpretation confronted problems, namely, the dispersion of wave packets and the fact that Y was a function in configuration space and not in real space. Nevertheless, Schrodinger did not completely suppress his enjoyment over the possibilities for visualisable (anschaulich) interpretations. He envisaged an understanding of the Bohr frequency condition by conceiving of the emission of light during transitions as a kind of ‘beat’ caused by the changing of vibrations, and added:

It is hardly necessary to emphasize how much more congenial it would be to imagine that at a quantum transition the energy changes over from one form of vibration to another, than to think of a jumping electron. The changing of the vibration form can take place continuously in space and time (Schrodinger, 1928, pp. 10-11).

In the second paper Schrodinger offered a new derivation of his wave equation, based on more physical, anschaulich considerations. The starting point was Hamilton’s analogy between particle mechanics and geometrical optics. Schrodinger suggested that classical mechanics is, in complete analogy to geometrical optics, an idealisation which holds only for a certain domain. Finding the correct theory is therefore ‘a question of searching for an undulatory mechanics, and the most obvious way is the working out of the Hamiltonian analogy on the lines of undulatory optics’ (Schriidinger, 1928, p. 18). Traditionally it is assumed, e.g. by Jammer (1966, pp. 236ff.), that this presentation mirrors Schrodinger’s original path of discovering wave mechanics. Wessels (1979, pp. 336338) argues against this account, showing that certain features of the first paper indicate that Schrodinger had not yet elaborated Hamilton’s analogy. Still, she deems it possible he had a superficial knowledge of the analogy, which might have been a motivating force in the background.

Schrodinger also compared his wave interpretation with the views of the Copenhagen-Giittingen school. He argued that his interpretation allows for the apparently necessary renunciation of the concepts of ‘position’ and ‘path’ of the electron, while it avoids the conclusion that a space-time description of atomic structure must be given up altogether, a conclusion that Schrodinger (1928, pp. 2627) regarded, from the philosophical standpoint, ‘as equivalent to a complete surrender’. A surrender because, as already quoted above: ‘we cannot really alter our manner of thinking in space and time, and what we cannot comprehend within it we cannot understand at all’. Also, when discussing matrix mechanics, which he regarded as ‘in its tendency’ very close to his

own theory, Schrodinger (1928, p. 30) stressed that the strength of wave

SchrGdinger and Anschaulichkeit 411

mechanics lies in its ‘guiding, physical point of view’, promising ‘anschaulich

understanding’. A few weeks later SchCdinger elaborated on the relation between

matrix mechanics and his own theory. In his article ‘ober das Verhgltnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen’, he claimed that they were formally, i.e. mathematically, equivalent. In an oft- quoted footnote, Schriidinger (1928, p. 46n) remarked: ‘I naturally knew about his [Heisenberg’s] theory, but was discouraged, if not repelled, by what appeared to me as very difficult methods of transcendental algebra, and by the want of Anschaulichkeit’. Furthermore, after his proof of mathematical equivalence, he discussed the physical relation, and defended the superiority of his anschaulich theory (his arguments were described in more detail in Section 3).

In the third and fourth papers Schriidinger elaborated the theory of wave mechanics (e.g. perturbation theory, the time-dependent equation and its solution, the relativistic equation). He also proposed a new interpretation of the wave function, because Lorentz had in correspondence convinced him that his original interpretation was untenable. He now suggested that ‘Y.Y* is a kind of weight-function in the system’s configuration space’. Schr6dinger admitted that this new interpretation seems to be something of a defeat, ‘since we have previously often spoken in such an anschaulich concrete way of the “Y- vibrations” as though of something quite real’. However, he maintained that

there is something tangibly real behind the present conception also, namely, the very real effective fluctuations of the electric space-density. The Y-function is to do no more and no less than permit of the totality of these fluctuations being mastered and surveyed mathematically by a single partial differential equation (Schrsdinger. 1928, p. 120).

Thus, although the wave function itself does not describe real vibrations in space and time, Schrcdinger still believed it is in some way related to real spatio-temporal processes. In other words, he believed that a visualisable, anschaulich description of atomic structure remains possible. In spite of his persistent use of the word ‘real’ it is, in my opinion, SchrGdinger’s philosophical conviction that Anschaulichkeit is a necessary criterion for understanding, rather than a realist epistemology, that determines this choice. This can be inferred from his negative response to Born’s statistical interpretation of the wave function, expressed in the letter to Wien analysed in Section 3. Born interpreted the product Y.Y* as a ‘weight-function’ also, but as one determining probabilities for the behaviour of particles. Schrijdinger’s criticism was not directed against the claim of reality of particles as such (for there is no criterion for reality at all>, but against the fact that Born’s interpretation implies that a visualisable space-time description is fundamentally impossible.


4.3. Aftermath

The second half of 1926 saw intense discussions over the interpretation

of quantum mechanics, especially in connection with Anschaulichkeit. The

Copenhagen-Gottingen school accepted Schrodinger’s theory but not his

interpretation. Heisenberg was the most fervent opponent in this respect. On

8 June 1926, he wrote to Pauli: ‘The more I think of the physical part of

Schrodinger’s theory, the more abominable I find it. What Schrodinger writes

about Anschaulichkeit makes scarcely any sense, in other words I think it is

bullshit [Mist]. The greatest result of his theory is the calculation of the matrix

elements’.18

In October 1926 Schrddinger visited Copenhagen, at the invitation of Bohr.

The heated and often emotional discussions on interpretational questions

have been described by Heisenberg (1967). Bohr had a more favourable

attitude towards Schrodinger’s ideas than Heisenberg. However, he rejected

Schriidinger’s interpretation because it failed to explain the corpuscular aspects

of matter. Bohr considered wave-particle duality as the most fundamental

feature of quantum theory. Schrbdinger’s ideas were demolished by Bohr and

Heisenberg, and he returned rather discouraged. Reporting his defeat he wrote

to Wien, on 21 October:

Certainly the standpoint of anschaulich pictures, taken by de Broglie and me, is not yet far enough developed to account even for the most important facts. Moreover, it is quite probable that here and there we have taken a wrong turn, which must be left again (quoted in Pauli, 1979, p. 339; my translation).

However, two days later he wrote to Bohr:

even if a hundred attempts miscarry, one would not give up the hope of reaching the goal of a representation of the true properties of space-time events through-I do not say classical pictures-but through representations that are free of logical contradictions. It is extremely probable that this is possible (quoted in MacKinnon, 1982, p. 249).

Schrddinger never relinquished his ideal of visualisation. Until the end of his

life he was to remain a critic of the accepted Copenhagen interpretation (see e.g.

Schriidinger, 1952, 1995).

5. Conclusion

In this article, the philosophical views of Erwin

in his discovery of wave mechanics have been

Anschaulichkeit, which was a crucial factor in

Schrodinger and their role

examined. The notion of

Schrodinger’s attitude to

quantum theory, has received special attention. I have argued that Schrodinger

“Heisenberg to Pauli (8 June 1926), in Pauli (1979, p. 328); translation in Moore (1989, p. 221). Cf. Heisenberg’s letter to Pauli of 28 July 1926, in which he described his first personal encounter with Schriidinger in Munich on 23 July 1926 (Pauli, 1979, pp. 337-338).


was, at the epistemological and ontological level, less of a realist than often is

assumed. Not a return to some form of classical realism but a strong belief in the intelligibility of nature (expressed by postulating Anschaulichkeit as a criterion for physical theories) was the driving force behind his attempts to construct an alternative foundation for quantum theory.

Schrbdinger’s commitment to Anschaufichkeit had its source in his admiration for Boltzmann’s philosophy of science. Boltzmann’s influence shaped his views on scientific methodology, and thus affected his scientific activities most directly. Schrodinger’s epistemological views, which were mainly based upon Mach’s philosophy, were less important from a heuristic point of view. His Machian position repeatedly emerges from his writings but does not appear to have had a decisive influence on the direction of his researches. The reason for this is that Schrodinger rejected the ‘strong’ methodological tenets of Machian positivism, and was thus left with the ‘weak’ empiricist methodology shared by all scientists: the precept that theories must be empirically adequate.

When reading Schrodinger’s scientific papers, one encounters expressions which, at first sight, seem to contradict my claim that he was committed to Machian anti-realism. For example, as we have seen in Section 4, Schrodinger freely spoke about ‘reality’ and the ‘real existence’ of objects in the atomic domain. This paradox can be resolved by making a clear distinction between methodological realism and epistemological realism. The former is merely a methodological advice to adopt a realist attitude towards scientific theories when practising science. The latter is an epistemological position which asserts that scientific theories are true descriptions of a real world behind the appearances. Whereas Schrbdinger adhered to the former view, he rejected the latter.

In this respect, Schrodinger’s attitude appears to be in striking agreement with the central tenets of Van Fraassen’s constructive empiricism. Van Fraassen (1980, pp. 80-83) argues that an empiricist epistemology does not forbid a working scientist to be ‘immersed in the scientific world-picture’. Indeed, ‘nothing is more natural, or more to be recommended than this total immersion’ (p. 81). Such immersion is, of course, precisely the attitude I have labelled methodological realism. It should not be confused, Van Fraassen emphasises, with the epistemic commitment of the scientist: ‘it is possible even after total immersion in the world of science to distinguish possible epistemic attitudes to science, and to state them, and to limit one’s epistemic commitment while remaining a functioning member of the scientific community’ (p. 83).

When Schrddinger reflected on his scientific activities, stepping back to describe his epistemic commitment, he took an empiricist stance. Therefore, he might be described, at least in this respect, as constructive empiricist avant la

Zettre. I have elsewhere compared Boltzmann’s position to Giere’s constructive realism (De Regt, 1996, p. 43). It appears that the difference between the


philosophical views of Boltzmann and Schrodinger mirrors that between Giere’s and van Fraassen’s views. It amounts to a shift from a subtle version of realism to a subtle version of empiricism.

One notices a further parallel between Schrddinger and van Fraassen when considering the latter’s argumentation against ‘inference to the best explanation’. Van Fraassen (1980, pp. 19-23) claims that we can admit that some theory is the best explanation for the observed phenomena, without having to believe that this theory is a description of reality. In exactly the same vein, Schrodinger held that we may accept a theory because of its Anschaulichkeit,

while we do not have to believe the theory to be true.

Acknowledgements-I wish to thank Dennis Dieks, Peter Kirschenmann, Fred Muller, Hans Radder, Jos Uffink, and Michael van Kempen for helpful suggestions. The investigations were supported in part by the Foundation for Research in the Field of Philosophy and Theology, which is subsidised by the Netherlands Organisation for Scientific Research (NWO).

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Erwin Schrödinger, Anschaulichkeit and Quantum Theory

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