From the very beginning of his development, Einstein was very critical of the main results of quantum mechanics. That is why in 1925, when the main papers on matrix mechanics were published, he wrote to his friend Besso [1] as follows: "The most interesting theoretical achievement of recent times is the Heisenberg-Born-Jordan theory of quantum states. A real witchcraft calculus in which infinite determinants (matrices) appear instead of Cartesian coordinates. It is highly witty and, due to its complexity, is insured against proving erroneous." In this connection I wish to say the following. A lot of time has passed since this thought was made. Therefore it makes sense to try to prove that the results of matrix mechanics actually contain defects. In the paper an attempt is made to prove that it is really so.
From the very beginning of his development, Einstein was very critical of the main results of quantum mechanics. That is why in 1925, when the main papers on matrix mechanics were published, he wrote to his friend Besso [1] as follows: "The most interesting theoretical achievement of recent times is the Heisenberg-Born-Jordan theory of quantum states. A real witchcraft calculus in which infinite determinants (matrices) appear instead of Cartesian coordinates. It is highly witty and, due to its complexity, is insured against proving erroneous." In this connection I wish to say the following. A lot of time has passed since this thought was made. Therefore it makes sense to try to prove that the results of matrix mechanics actually contain defects. In the paper an attempt is made to prove that it is really so.
It is known that there was a disagreement between Bohr and Einstein about the nature of the fundamental equations of quantum mechanics. In 1949, in the work "Remarks on Articles," Einstein clearly expressed his thoughts on this issue. He wrote: "All this discussion was needed only to clarify the following. By attempting to defend the thesis that statistical quantum theory can in principle give a complete description of individual systems, we come to highly improbable theoretical concepts. On the other hand, the aforementioned interpretive difficulties of the theory disappear if quantum-theoretical descriptions are considered as descriptions of ensembles of systems".
The report by Born and Heisenberg on ‘quantum mechanics’ may seem surprisingly difficult to the modern reader. This is partly because Born and Heisenberg are describing various stages of development of the theory that are quite different from today's quantum mechanics. Among these, it should be noted in particular that the theory developed by Heisenberg, Born and Jordan in the years 1925–6 and known today as matrix mechanics (Heisenberg 1925b [1], Born and Jordan 1925 [2], Born, Heisenberg and Jordan 1926 [4]) a differs from standard quantum mechanics in several important respects. At the same time, the interpretation of the theory (the topic of Section II of the report) also appears to have undergone important modifications, in particular regarding the notion of the state of a system. Initially, Born and Heisenberg insist on the notion that a system is always in a stationary state (performing quantum jumps between different stationary states). Then the notion of the wave function is introduced and related to probabilities for the stationary states. At a later stage, probabilistic notions (in particular, what one now calls transition probabilities) are extended to arbitrary observables, but it remains somewhat unclear whether the wave function itself should be regarded as a fundamental entity or merely as an effective one. This may reflect the different routes followed by Born and by Heisenberg in the development of their ideas. The common position presented by Born and Heisenberg emphasises the probabilistic aspect of the theory as fundamental, and the conclusion of the report expresses strong confidence in the resulting picture.
Born's reaction In his reminiscences, Born recounted his memories of these exciting days (Born, 1978): ‘Meanwhile Heisenberg pursued some work of his own, keeping its idea and purpose somewhat dark and mysterious. Towards the end of the summer semester, in the first days of July 1925, he came to me with a manuscript and asked me to read it and decide whether it was worth publishing … He added that though he had tried hard, he could not make any progress beyond the simple considerations contained in his paper, and he asked me to try myself, which I promised … His most audacious step consists in the suggestion of introducing the transition amplitudes of the coordinates q and momenta p in the formulae of mechanics … I was most impressed by Heisenberg's considerations, which were a great step forward in the programme which we had pursued … After having sent Heisenberg's paper to Zeitschrift für Physik for publication, I began to ponder about his symbolic multiplication, and was soon so involved in it that I thought the whole day and could hardly sleep at night. For there was something fundamental behind it … And one morning … I suddenly saw the light: Heisenberg's symbolic multiplication was nothing but matrix calculus, well known to me since my student days from the lectures of Rosanes at Breslau.’
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
As mentioned in Chapter 1 , the Heisenberg matrix mechanics provides one of the three main avenues to quantum mechanics. This approach to quantum mechanics was disclosed in three papers authored by Heisenberg (1925) , Born and Jordan (1925) , and Born et al. (1926) . An iconic result from the Heisenberg–Born–Jordan contribution was the commutation rule ( 14.1 ) p q − q p = h 2 π i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315216720/70121b16-f2c6-4331-b1ba-eecb806b93da/content/eq595.tif"/>
It is well known that in 1900, after Planck obtained the basic equation of quantum theory, there came a moment when it became necessary to obtain a theoretical justification of this equation. In 1911, based on the possibility of Gibbs' canonical distribution function (CDF), he solved this problem partially: he obtained a justification for the second multiplier of his equation. Then on the basis of the analysis of these results he stated that henceforth in the role of the basic concept of quantum theory there is a necessity to accept the possibility of the concept of the quantum of action. However, Poincaré at the same congress criticized these new ideas. For he recognized the value of the idea of the quantum of action in solving only frequent problems. Therefore, he further, in his article entitled "On the Theory of Quanta", made the following conclusion: in his opinion, the main essence of quantum theory is connected with the concept of the quantum of energy [1]. Thus, he wanted to say that in the future the main results of Planck's quantum theory have the possibility to substantiate only if the possibility of the concept of energy quantum is taken as more general. In this paper an attempt is made to prove that it is really so.