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.