BENTHAM SCIENCE PUBLISHERS eBooks

Possible and Impossible (Momentum) Configurations

P. Enders

Published Feb 12, 2019• 0 citations

Abstract

This chapter prepares a smooth way from Eulerian classical mechanics to quantum mechanics. Starting from Helmholtz’s foundation of the energy conservation law using Leibniz’s theorem as described in the foregoing Chapter 2, the configurations a system can (cannot) assume in a given stationary state are explored. The following constraints are taken into account: Newton-Euler’s exclusion principle, d’Alembertian constraints and constraints imposed by conservation laws. The set of possible and impossible momentum configurations is also considered, using Nemorarius’ theorem of the foregoing chapter.

DOI: 10.2174/9781681084497119010006

Related papers

A Hierarchy of Selection Problems

P. Enders

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.

BENTHAM SCIENCE PUBLISHERS eBooksFeb 12, 20190 citations

Foundations of quantum mechanics revealed by the conservation laws

Richard Joseph Oldani

It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although statistical formulations of quantum mechanics correctly describe energy and momentum exchange with remarkable accuracy there are individual cases, paradoxically, that violate the conservation laws. A recent study finds that in order to account for observed violations frames of reference involved in preparation of the initial state must be explicitly taken into account [1]. We show here that the same conclusion was arrived at in a much simpler way by Einstein over 100 years ago. Deficiencies in the foundations of non-relativistic theory are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows quantum mechanical formalism in abstract space to be replaced by the superposition of particle field geometries in real space. Applying the calculus of variations to the Schrödinger wave equation reveals that the wave function is incomplete because it requires twice the allowable action minimum.

QeiosOct 10, 20231 citation

Conservation of Stationary Quantum States Stationary Schrödinger Equation

P. Enders

In this chapter, the stationary Schrödinger equations in configuration and momentum configuration spaces are derived from the non-classical expression of the total energy (7.15) found in the foregoing chapter. The new parameter in the Fourier transform is identified as ℏ ≡ ℎ/2. The roles of boundary conditions, probability concepts and independent dynamical variables are shortly discussed.

BENTHAM SCIENCE PUBLISHERS eBooksFeb 12, 20190 citations

Summary and Conclusions

Peter Enders

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.

BENTHAM SCIENCE PUBLISHERS eBooksFeb 12, 20190 citations

Conservation laws

Harry Paul

As in classical physics, also in quantum mechanics: physical processes are bound to obey conservation laws. However, there is an important difference. While in classical physics we can suppose that all variables have definite values (irrespective of whether we know them or not), quantum mechanics allows them to be intrinsically uncertain. In those cases conservation laws apply only to ensemble averages (expectation values), and hence are less informative. They show their full strength in measurement processes. Then the measured values satisfy the respective conservation laws in any individual experiment. Let me remind you of spin measurements on a system of two spin-½ particles, being in a state with zero total spin, which we used in Section 5.5 to illustrate the EPR paradox. The spin components, with respect to an arbitrarily chosen direction, of the two particles are indefinite; however, measuring them yields definite values (+½ or −½) that add up to zero, thus confirming the angular conservation law. In the following, we will consider some realistic experiments for which conservation laws provide a simple explanation. Energy and momentum conservation For an understanding of the interaction between matter and radiation, Einstein's photon concept (actually, he spoke of ‘light quanta’) proved to be very helpful. He assumed photons to be particle-like constituents of electromagnetic radiation, in particular light. They have the energy h ν, where ν is the frequency of the radiation field. (Since the electromagnetic spectrum extends over an extremely wide range, there is a huge number of different sorts of photons.) The crucial point is that photons are generated or annihilated only as a whole. This property gives rise to typical quantum effects.

Cambridge University Press eBooksJun 5, 20080 citations

The Conservation Laws in Quantum Mechanics

Richard Joseph Oldani

It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although quantum mechanics correctly predicts what it is possible to observe (the emissions) with remarkable accuracy it ignores the other half of natural phenomena, that which cannot be observed (the absorptions), thereby violating the conservation of energy. Similarly, the conservation of momentum is violated by disregarding unobservable molecular impulses. Despite irrefutable proof that molecular impulses transfer momentum asymmetrically as shown by Einstein’s derivation of the A and B coefficients, the wave function is assumed to apply symmetrically in time. The deficiencies are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows non-relativistic quantum mechanics to be described with a physical model and wave function behavior to be interpreted as the combined action of a particle and its associated quantized field. The calculus of variations is applied to the wave function to show that it is an incomplete equation of motion because it yields twice the allowable action minimum. The concept of reality is redefined by showing that reductionism applies to classical phenomena, but not quantum phenomena.

QeiosAug 16, 20231 citation