‘Aequat causa effectum’ (the cause is equal to the effect) is an old metaphysical rule, which seems, however, to be violated for the single trajectories in spherically symmetric force fields, as the Kepler orbits are not spherically symmetric. The reason for the latter is the lower symmetry of the initial conditions. The rule is reestablished, if appropriate sets of trajectories (i.e, sets of initial conditions) are considered. The corresponding hodographs are also explored. The fact, that the Kepler orbits are plane, is derived by means of logical and symmetry arguments, without using angular momentum conservation. Spherically symmetric sets of Bohr orbitals are constructed by means of a sum rule for the spherically harmonic functions (Unsöld’s theorem). This results in a less known similarity between classical and quantum mechanical systems. Metaphysical rules do not solve concrete physical problems, but encourage to explore them more profoundly.
‘Aequat causa effectum’ (the cause is equal to the effect) is an old metaphysical rule, which seems, however, to be violated for the single trajectories in spherically symmetric force fields, as the Kepler orbits are not spherically symmetric. The reason for the latter is the lower symmetry of the initial conditions. The rule is reestablished, if appropriate sets of trajectories (i.e, sets of initial conditions) are considered. The corresponding hodographs are also explored. The fact, that the Kepler orbits are plane, is derived by means of logical and symmetry arguments, without using angular momentum conservation. Spherically symmetric sets of Bohr orbitals are constructed by means of a sum rule for the spherically harmonic functions (Unsöld’s theorem). This results in a less known similarity between classical and quantum mechanical systems. Metaphysical rules do not solve concrete physical problems, but encourage to explore them more profoundly.
The problem of deriving macroscopic properties from the Hamiltonian of the hydrogen atom is resumed by extending previous results in the literature, which predicted elliptic orbits, into the region of hyperbolic orbits. As a main tool, coherent states of the harmonic oscillator are used which are continued to imaginary frequencies. The Kustaanheimo-Stiefel (KS) map is applied to transform the original configuration space into the product space of four harmonic oscillators with a constraint. The relation derived between real time and oscillator (pseudo) time includes quantum corrections. In the limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$, the time-dependent mean values of position and velocity describe the classical motion on a hyperbola and a circular hodograph, respectively. Moreover, the connection between pseudotime and real time comes out in analogy to Kepler's equation for elliptic orbits. The mean-square-root deviations of position and velocity components behave similarly in time to the corresponding ones of a spreading Gaussian wave packet in free space. To check the approximate treatment of the constraint, its contribution to the mean energy is determined with the result that it is negligible except for energy values close to the parabolic orbit with eccentricity equal to 1. It is inevitable to introduce a suitable scalar product in R4 which makes both the transformed Hamiltonian and the velocity operators Hermitian. An elementary necessary criterion is given for the energy interval where the constraint can be approximated by averaging.
When a non-relativistic quantum particle moving under a radial potential V(r) has either an extra conserved vectorial operator or an extra conserved tensorial operator then, in the first case, the corresponding operator must be proportional to the quantum Laplace-Runge-Lenz vector and the compatible potential has to be the Coulomb potential. While, in the second case, the compatible potential is the harmonic oscillator one and the conserved quantity has to be a tensor formed by a symmetric combination of the position and the momentum operators, which is the quantum version of the classically conserved tensor. Thus, the superintegrable condition on classical 3D radial problems producing only closed, periodic and plane orbits still manages, even in the quantum case, to impose strong restrictions on the interaction potentials.
In the context of two-dimensional spacetime within a helium atom, both 1s electrons are characterized by wave functions that observe duality equation. They are symmetric, orthogonal and interwoven, forming a dynamic rope structure at any moment. Instead of elliptical orbit of planets around the sun, electronic orbitals take the form of matter state transformation cycle. While the kinematic movement of planets is governed by Kepler's first law, electronic transformation obeys Pythagorean theorem, both being equivalent in physical principle. The atomic spacetime is a continuous medium of electron clouds in synchronized differential and integral processes that are implemented by smooth trigonometry. In order to integrate this new approach with conventional physics, the author translates the pattern of electronic motion in the atomic spacetime into spherical volume undulation in Euclidean geometry and calculates the probability density of an electron within the sphere from the classical perspective. From the primary wave function of a 1s electron, the author also tries to derive the mathematical expression of central force that guides the surrounding bodies along the orbits. The result is exciting and surprising that questions the exactness of the venerable Coulomb's law.
The relation between the solutions of the time-independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half-width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well-defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.
For undergraduate students newly introduced to quantum mechanics, solving simple Schrödinger equations is relatively straightforward. However, the more profound challenge lies in comprehending the underlying physical principles embedded in the solutions. During my academic experience, a recurring conceptual difficulty was understanding why only s orbitals, and not others like p orbitals, exhibit spherical symmetry. At first glance, this seems paradoxical, given that the potential energy function itself is spherically symmetric. Specifically, why do p orbitals adopt a dumbbell shape instead of a spherical one? For a hydrogen atom with an electron in the [Formula: see text] state, which specific [Formula: see text] orbital does the electron occupy, and how do the x, y, and z-axes in [Formula: see text], [Formula: see text], and [Formula: see text] connect to the real world? Additionally, is the atom still spherically symmetric in such a state? These questions relate to core concepts of quantum mechanics concerning symmetry, mixed state, and superposition. This paper delves into these questions by investigating this specific case, utilizing the advanced visualization capabilities offered by ChatGPT. This paper underscores the importance of emerging AI tools in enhancing students’ understanding of abstract principles.
Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.