Quantum mechanical Earth: where orbitals become orbits

Abstract

Macroscopic objects, although quantum mechanical by nature, conform to Newtonian mechanics under normal observation. According to the quantum mechanical correspondence principle, quantum behavior is indistinguishable from classical behavior in the limit of very large quantum numbers. The purpose of this paper is to provide an example of the correspondence principle appropriate for an undergraduate quantum mechanics course by showing that Earth conforms to quantum mechanics while it orbits the Sun. It is argued that the planarity of Earth's orbit implies that the quantum numbers l and ml are equal and very large. It is additionally shown that for Earth in its orbit, the quantum number n is relatively indistinguishable from l and ml. Furthermore, it is demonstrated that the very large common value for the three quantum numbers implies a spatial probability distribution for Earth that is consistent with its actual distribution. In other words, the wavefunction of Earth, whose orbit will be assumed for simplicity to be circular, is consistent with a planar orbit with all angles within the plane equally probable and with a precisely defined Earth–Sun distance.

Determination of classical behaviour of the Earth for large quantum numbers using quantum guiding equation

“…As we argued before the dynamics here are based on the energy relation (5), calculated by Keeports, that in high values of n are completely compatible with classical results [1]. Moreover, we previously computed the phase of the wave function S using the quantum Hamilton-Jacobi equation (3), regarding the energy relation (5) and the fact that the term − 2 2me ∇ 2 R R (quantum potential) is negligible because of the mass of the Earth.…”

“…Astonishingly, we see that the large quantum numbers directly ascertain the Earth dynamics. As Keeports has already shown [1], at large quantum numbers n and l, the classic energy of the Earth obtained. Also, we showed here that the Earth dynamics depends on the high values of the quantum number m which affects the Earth trajectory via the guiding equation (12).…”

“…The relation (3) is the well-known quantum Hamilton-Jacobi equation. Keeports pointed out that how the Hydrogen-like energy of the Earth is related to its classical one [1]. He showed if the Earth-Sun Hamiltonian is assumed similar to the Hydrogen atom, regardless of the additional term K, the energy levels could be calculated as E n = −G 2 m 3 e m 2 s /2 2 n 2 .…”

“…Thereupon our first assumption here is that the energy of the Earth wave function is in a form similar to the same system with Hydrogen-like Hamiltonian. (For Hydrogen-like Earth-Sun system see [1]). The first assumption is an appropriate statement due to the fact that our explanation should be in a complete agreement with the classical results.…”

“…In the most general sense, the correspondence principle determines the guideline of scientific theories. However, not only the latest achievements require to make predictions that earlier studies were inadequate to address, but also they need to confirm the correct predictions of the previous theories [1]. In this issue, Bohmian mechanics which comes up with trajectories for quantum systems can be an appropriate option for the development of other studies describing the micro-world.…”