The purpose of the work is to check the validity of generalizing the an-gular momentum ћ of a Cooper pair to any many-particle quantum system, including positronium. Methodology. For the purposes of this work, a semiclassical consideration is quite sufficient. To increase the reliability for helium, a quantum solution is additionally given. It is believed that the total minimum angular momentum of a two-particle quan-tum system – positronium is equal to ћ. When generalizing the angular momentum ћ of a Cooper pair and an electron-positron pair to a pair of helium electrons, its radius turns out to be almost four times less than the established value. And this means only one thing – the generalization of the angular momentum ћ of a Cooper pair to a pair of helium electrons is illegal. Therefore, each helium electron has a minimum angular mo-mentum equal to ћ. And the angular momentum of a pair of helium electrons is equal to the sum of the angular momentum of each electron. The double standards lie in the fact that, on the one hand, a quantum system – an electron-positron pair – is endowed with a quantum of angular momentum ћ, and on the other hand, a quantum system – a pair of helium electrons – is not endowed with this quantum (it is endowed with two quanta). And since double standards are unacceptable, an elementary dilemma arises – either the radius of helium should be quadrupled, or the radius of positronium should be quadrupled. The task is simplified by the fact that recently the value of the radius of positronium was completely unambiguously corrected, which turned out to be exactly four times larger than was commonly believed. Thus, the helium atom retains its size. It has been established that endowing a Cooper pair of electrons with a quantum of angular momentum ћ is just as illegitimate as in the case of positronium. The angular momentum of a many-particle quantum system, due to its additivity, must be divided between the particles of the system. It has been established that the quantum of angular momentum ћ can be attributed to only one quantum particle, and cannot be attributed to a quantum collection of particles.
The purpose of the work is to check the validity of generalizing the an-gular momentum ћ of a Cooper pair to any many-particle quantum system, including positronium. Methodology. For the purposes of this work, a semiclassical consideration is quite sufficient. To increase the reliability for helium, a quantum solution is additionally given. It is believed that the total minimum angular momentum of a two-particle quan-tum system – positronium is equal to ћ. When generalizing the angular momentum ћ of a Cooper pair and an electron-positron pair to a pair of helium electrons, its radius turns out to be almost four times less than the established value. And this means only one thing – the generalization of the angular momentum ћ of a Cooper pair to a pair of helium electrons is illegal. Therefore, each helium electron has a minimum angular mo-mentum equal to ћ. And the angular momentum of a pair of helium electrons is equal to the sum of the angular momentum of each electron. The double standards lie in the fact that, on the one hand, a quantum system – an electron-positron pair – is endowed with a quantum of angular momentum ћ, and on the other hand, a quantum system – a pair of helium electrons – is not endowed with this quantum (it is endowed with two quanta). And since double standards are unacceptable, an elementary dilemma arises – either the radius of helium should be quadrupled, or the radius of positronium should be quadrupled. The task is simplified by the fact that recently the value of the radius of positronium was completely unambiguously corrected, which turned out to be exactly four times larger than was commonly believed. Thus, the helium atom retains its size. It has been established that endowing a Cooper pair of electrons with a quantum of angular momentum ћ is just as illegitimate as in the case of positronium. The angular momentum of a many-particle quantum system, due to its additivity, must be divided between the particles of the system. It has been established that the quantum of angular momentum ћ can be attributed to only one quantum particle, and cannot be attributed to a quantum collection of particles.
Aim is to correct the value of the radius of the helium atom.Methodology. A semi-classical approach is used, which is quite sufficient for the purpose of the work.Results. The need to eliminate double standards when determining the angular momentum of quantum systems leads to the fact that the quantum of angular momentum ћ is assigned not to each electron of the helium atom, but to their quantum set. The consequence of this is the correction of the value of the radius of the helium atom, which turned out to be almost four times smaller than the tabulated value. A corrected formula for the radius of the helium atom has been found: r1,2 = [1 + (1 + 7α2)]a0/14 = 7,56∙10–12 (м).Research implications lie in the fact that the size of the helium atom is of significant importance, in particular for membrane technologies.
Models for the Helium Atom.---(1) Bohr's model is unsatisfactory because it gives too great a value for the ionizing potential and is not in accord with the optical and magnetic properties of helium. Now the chemical evidence suggests that each electron in an atom has its own separate orbit, and that these orbits are closely interrelated. Accordingly two new models are considered. (2) In the double circle model the two electrons are assumed to move in two separate, parallel, circular orbits. The model, however, is unstable, for the ionizing potential, computed by applying the quantum theory, comes out negative. Another objection is that the magnetic moment is not zero. (3) In the semi-circular model each electron is assumed to oscillate back and forth along an approximately semi-circular path in accord with classical mechanics, each being brought to rest at the end of its path by the repulsion of the other. Assuming the maximum angular momentum of each equal to $\frac{h}{2\ensuremath{\pi}}$, the absolute dimensions of the model were computed. The total energy comes out less than for the Bohr model, and the ionizing potential, 25.62 volts, agrees closely with the experimental value. The magnetic moment is zero.Application of the Quantum Theory to Coupled Electrons.---The success of the semi-circular model in giving the correct value for the ionizing potential of helium suggests that in the case of coupled electrons the quantum theory should be applied not to the momentum of the individual electrons according to the relation $\ensuremath{\int}\mathrm{pdq}=\frac{h}{2\ensuremath{\pi}}$, but rather to the momentum which, by being relayed from one electron to another, passes in each direction around the nucleus.
Показано, что квантование момента импульса двухчастичных квантовых систем в атоме гелия и позитрония выполнено существенно по-разному. Установлено, что для гелия оно выполнено корректно, а для позитрония - нет. It is shown that the quantization of the angular momentum of two-particle quantum systems in a helium andpositronium atom is carried out significantly differently. It has been established that for helium it is carried out correctly, but for positronium it is not.
Models for the Helium Atom.---(1) Bohr's model is unsatisfactory because it gives too great a value for the ionizing potential and is not in accord with the optical and magnetic properties of helium. Now the chemical evidence suggests that each electron in an atom has its own separate orbit, and that these orbits are closely interrelated. Accordingly two new models are considered. (2) In the double circle model the two electrons are assumed to move in two separate, parallel, circular orbits. The model, however, is unstable, for the ionizing potential, computed by applying the quantum theory, comes out negative. Another objection is that the magnetic moment is not zero. (3) In the semi-circular model each electron is assumed to oscillate back and forth along an approximately semi-circular path in accord with classical mechanics, each being brought to rest at the end of its path by the repulsion of the other. Assuming the maximum angular momentum of each equal to $\frac{h}{2\ensuremath{\pi}}$, the absolute dimensions of the model were computed. The total energy comes out less than for the Bohr model, and the ionizing potential, 25.62 volts, agrees closely with the experimental value. The magnetic moment is zero.Application of the Quantum Theory to Coupled Electrons.---The success of the semi-circular model in giving the correct value for the ionizing potential of helium suggests that in the case of coupled electrons the quantum theory should be applied not to the momentum of the individual electrons according to the relation $\ensuremath{\int}\mathrm{pdq}=\frac{h}{2\ensuremath{\pi}}$, but rather to the momentum which, by being relayed from one electron to another, passes in each direction around the nucleus.
The present approach is an advancement of the author’s former attempts to develop an atom model of Helium with well-defined electron trajectories. Thus it calls in question the traditional quantum mechanics which assume—in contrast and as a consequence of Heisenberg’s uncertainty principle—electronic probabilities of presence. Its basic idea consists of the assumption that the motions of the two electrons are influenced by their spins exhibiting the value h/2π, but in two different ways: on the one hand, one spin induces a rotation; and on the other hand, the other spin induces a harmonic oscillation. A second important relation is given by the fact that the retroactive force of the oscillation process is due to the centrifugal force when the process runs along the surface of a sphere, whereas in usual oscillation processes—such as the one of a spring pendulum—it is due to a permanent shift between potential and kinetic energy. Therefore, in the present case, the potential energy remains constant since the distance between the nucleus and the—diametrically positioned—electrons remains constant. Considering these two conditions and the usual physical relations such as Coulomb attraction, centrifugal force, and the conservation laws of the angular momentum and of the energy, it was possible to compute the respective key values. Thereby, the deflection of the oscillation angle ψ = 45˚ is remarkable. Finally, the process is described using a Cartesian coordinate system with z as the rotation axis, a variable oscillation distance d and variable rotation velocities rrot. Thereby, the projections onto the x-axis and on the y-axis are not identically equal, leading to an elliptic projection shape. Thus this system is anisotropic, in contrast to the isotropic array of the conventional quantum mechanics according to Schrödinger, where the 1s-orbital is spherically symmetrical. This anisotropy explains the existence of interatomic Van der Waals forces, which enable the condensation of Helium, even though the condensation temperature is very low. But in particular, it exhibits well-defined electron waves, thus finally delivering the explanation of the hypothesis of Louis de Broglie, which has been established 100 years ago.
The original intention of the author’s preoccupation with the quantum-mechanical behaviour of simple atoms and molecules such as Hydrogen and Helium was, on the one hand, the elegant simplicity of Niels Bohr’s atom model for Hydrogen, describing the metastable states of the excited electrons by planar concentric electron orbits, and, on the other hand, the hardly intelligible wave mechanical approach of Heisenberg, Schrödinger and others, describing mainly atoms with multiple electrons by three-dimensional orbitals which were characterized by probabilities of presence. Thereby the question arose whether it would be possible to find alternative atom models with well-defined electron trajectories. Therein, Louis de Broglie’s thesis of the wavy nature of electron motion implicating standing waves would have to be implemented. Nevertheless, as reviewed in the introduction, the orthodox three-dimensional concept influenced the own thinking in such a way that three-dimensional constellations for the electronic excited states were conceived. The break-through was achieved for the electronic ground state in the form of the spin-orbit coupling where the spin acts as a perpetuum mobile, inducing the orbital angular momentum h/2π. Furthermore, the insight was gained that a circularly rotating electron intrinsically corresponds to a harmonic oscillator, thus fulfilling the condition of a standing wave. Based on this concept, a double planar model was established for the H2-molecule which could be empirically verified by X-ray data from literature. However, for the two electrons containing Helium a 2D-array seemed impossible since the Pauli-principle seemed to be violated. After a long stepwise succession of 3D-attempts which turned out to be impossible—not least since eccentric forces are not possible in such a system—the here presented 2D-version for Helium was found, composed by two imaginary orthogonal electron orbits. It will enable in a subsequent publication the quantum mechanical interpretation of the thermal-radiative behaviour of Helium which was reported in the author’s publication nine years ago.