This chapter summarizes the main results of this book and draws some conclusions. Special attention is paid to the axiomatic basis, i.e., to Schrödinger’s requirements and Hertz’s program, which lead to the novel paradigm of quantization as selection problem (rather than eigenvalue problem). Both ones are realized by means of limiting (Schrödingerian weight) functions, which guarantee the validity of the energy conservation law in the classically forbidden regions of configuration and momentum configuration spaces (e.g, beyond the turning points of classical oscillators). Further topics are the separation of external and internal parameters as initiated by Euler, the differences between classical and quantum wave equations, causality and determinism, hidden variables, symmetry and field quantization.
This chapter summarizes the main results of this book and draws some conclusions. Special attention is paid to the axiomatic basis, i.e., to Schrödinger’s requirements and Hertz’s program, which lead to the novel paradigm of quantization as selection problem (rather than eigenvalue problem). Both ones are realized by means of limiting (Schrödingerian weight) functions, which guarantee the validity of the energy conservation law in the classically forbidden regions of configuration and momentum configuration spaces (e.g, beyond the turning points of classical oscillators). Further topics are the separation of external and internal parameters as initiated by Euler, the differences between classical and quantum wave equations, causality and determinism, hidden variables, symmetry and field quantization.
The "quantum equation" should "carry the quantum conditions in itself" (Second Commun., p. 511); 2. There should be a special mathematical method for solving the stationary Schrödinger equation, which accounts for the non-classical character of the quantization problem, ie, which is different from the classical eigenvalue methods for calculating the (eigen)modes of strings, resonators and so on (ibid., p. 513);3. The derivation should uniquely decide, that the energy rather than the frequency values are discretized (ibid., pp.511, 519), since only the former means true quantization, while the latter corresponds to the classical discretization mentioned in requirement 2;4. The use of the classical expressions for the kinetic and potential energies should be justified (Fourth Commun., p. 113).Schrödinger's requirements 1 and 2 mean, that -contrary to the very title of the papersquantization is actually not an eigenvalue problem.For in the latter the discretization is imposed not by the differential equation itself, but by the boundary conditions, and this is the classical discretization for standing waves in organ pipes etc.I will fulfill all four requirements by treating quantization as a selection problem.The number of stationary states of a quantum system is smaller than that of a classical system [6].I will describe, (i), the selection of quantum systems out of the set of all mechanical systems and, (ii), the selection of their stationary states out of the continuum of classical stationary states.Earlier arguing [19][10] is improved and extended in several essential points. Elements of an Eulerian representation of classical mechanics Euler's axiomsLeonhard Euler [20][21][22][23][24][25] was the first to apply the calculus to all areas of mathematics and mechanics of his time, and he developed novel methods, such as the calculus of variations and topology.Moreover, he worked out an axiomatic of mechanics, where only Newton's Law (axiom) 1 concerning the conservation of stationary states is retained as an axiom, while Newton's Laws (axioms) 2 and 3 concerning the change of stationary states are treated as problems to be solved (for a detailed account, see [19][10][45]).This allows for introducing alternative equations of motion without loosing the contact to CM.The existence of stationary states is postulated in the following axioms (as in Newton's axioms, rotatory motion is not considered). Axiom E0: Every body is either resting or moving.This means, that the subsequent axioms E1 and E2 are not independent of each other; they exclude each other and, at once, they are in harmony with each other [22].
In the past few chapters, the Schrödinger equation has been developed and applied to the solution of a number of quantum mechanical problems, mainly to develop experience with the results of simple and common systems. The results from these examples can be summarized relatively simply. In general, quantization enters the problem through the non-commuting nature of conjugate operators such as position and momentum as well as from classical boundary quantization. This has led to the Schrödinger equation itself being the primary equation of motion for the wave function solution to the problem. In essence, duality is incredibly important and is the basic method by which the electron is treated as a wave, rather than as a particle, and the wave equation of interest is the Schrödinger equation. When boundary conditions are applied, either through potential barriers or through the form of the potential (as in the harmonic oscillator), the time-independent Schrödinger equation yields solutions that are often special functions. Examples of this are the sines and cosines in the rectangular-barrier case, the Airy functions in the triangular-well case, and the Hermite polynomials in the harmonic oscillator case.
This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol, as defined in Eq. (3.6.6). The following steps are the local and global conservation laws associated to the Schrödinger equation, the probabilistic interpretation of the wave function, the spreading of wave packets and transformation properties of wave functions. The solution of the Schrödinger equation for a given initial condition is then studied. For this purpose, one first expands the initial condition in terms of eigenfunctions of the Hamiltonian operator. This is then ‘propagated’ to determine the evolution of the wave function. Thus, one is led to consider the Green kernel of the Schrödinger equation, and the boundary conditions on the wave function that lead to complete knowledge of stationary states. A dynamics involving an isometric non-unitary operator is also studied. The chapter ends with an elementary introduction to harmonic oscillators and to the JWKB method in wave mechanics, plus examples on the Bohr–Sommerfeld quantization. From classical to wave mechanics The correspondence between differential operators and polynomials in k relies on the notion of the symbol defined in Eq. (3.6.6). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is non-trivial because the map from symbols to operators contains some ambiguity. Such a transition is always possible if polynomials in p have constant coefficients. When this is not the case, i.e. if the coefficients are functions on the configuration space, there is an obvious ambiguity in the association of an operator with xp or px , for instance.
This chapter introduces the paradigm ‘quantization as selection problem’ by means of few historical remarks. They include Einstein’s 1907 reasoning, Schrödinger’s 1926 derivation of his wave equation, and Gödel’s theorem and Munchhausen’s trilemma. The latter ones concern questions, which can be posed, but not be answered within CM. Such questions transcendent CM. This book poses the question, how the mechanics of oscillators without turning points looks like. (“oscillators without turning points” means oscillators, which may assume configurations beyond the classical turning points.)
This chapter brings together Einstein’s 1907 vision of ‘quantization as selection problem’ (see Section 5.1) and our explorations of the possible and impossible (momentum) configurations in Chapter 4. The hierarchy is formulated in terms of the latter ones and consists of Newtonian systems, non-Newtonian classical systems, non-classical systems and non-mechanical systems. These explorations are assisted by an overview on selection problems in classical and quantum theory, by further reasonings for Einstein’s vision and by few explanations about alternative and harmony in Euler’s axioms of classical mechanics. Two fundamental conclusions for non-classical systems can be drawn already at this stage. (i), The notion of path as a point-wise relationship between the configurations and the momentum configurations looses its meaning. (ii), there are no longer coordinates describing the boundaries of a system in (momentum) like the classical turning points of an oscillator.
We discuss the axiomatic basis of quantum mechanics and show that it is neither general nor consistent, since its axioms are incompatible with each other and moreover it does not incorporate the magnetic quantization as in the cyclotron motion. A general and consistent system of axioms is conjectured which incorporates also the magnetic quantization. 1 The neccessity of quantum mechanics (QM) was a result of experimental data for the energy spectrum of atoms which could not be explained in accord with the laws of classical mechanics. Nevertheless in the axiomatic basis of QM there are axioms which are not supported by the experimental results, since the whole system of axioms (1) is introduced only as a ”plausible generalization ” of the first axiom [1]: