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

Abstract

For quantum systems, we expect to see the classical behaviour at the limit of large quantum numbers. Hence, we apply Bohmian approach for describing the evolution of Earth around the Sun. We obtain possible trajectories of the Earth system with different initial conditions which converge to a certain stable orbit, known as the Kepler orbit, after a given time. The trajectories are resulted from the guiding equation $p=\nabla S$ in the Bohmian mechanics, which relates the momentum of the system to the phase part of the wave function. Except at some special situations, Bohmian trajectories are not Newtonian in character. We show that the classic behaviour of the Earth can be interpreted as the consequence of the guiding equation at the limit of large quantum numbers.

Appendix A

The differential equation (18) can be solved in any time domain in which $\xi$ is considered as a constant. One can write this equation as

$$\begin{aligned} \frac{\mathrm{d}r}{\mathrm{d}t}=\xi \sin \theta \sqrt{\frac{2\mu }{r(t)}-\frac{\mu }{a}}, \end{aligned}$$

(A.1)

where both $\theta$ and r are time-dependent, but have no dependency on each other. For $\theta$ we have from (9)

$$\begin{aligned} \sin \theta =\sqrt{1-\frac{Z_h^2}{r^2(t)}}. \end{aligned}$$

(A.2)

During the Earth’s rotation around the Sun, the time variations of r are insignificant. With regard to (A.2) and noticing that the constant $Z_h$ has the same order of magnitude as r, one can neglect the time dependency of $\theta$ too. Therefore, to solve eq. (A.1), we can consider the term $\xi \sin \theta$ as a constant.

Define the variable q as

$$\begin{aligned} q=r-r_\mathrm{eq}, \end{aligned}$$

(A.3)

where $r_\mathrm{eq}$ represents the equilibrium distance, and eq. (A.1) yields

$$\begin{aligned} \frac{\mathrm{d}q}{\mathrm{d}t}=\xi \sin \theta \sqrt{\frac{2\mu }{r_\mathrm{eq}(\frac{q}{r_\mathrm{eq}}+1)}-\frac{\mu }{a}}. \end{aligned}$$

(A.4)

Hence, the variable q represents the deviation of r from the equilibrium distance $r_\mathrm{eq}\simeq a$. The term $q/r_\mathrm{eq}$ is small, so that one can expand $(1+q/r_\mathrm{eq})^{-1}$ to obtain

$$\begin{aligned} \frac{\mathrm{d}q}{\mathrm{d}t}=\xi \sin \theta \sqrt{\frac{\mu }{r_\mathrm{eq}}\left( 2-\frac{q}{r_\mathrm{eq}}\right) -\frac{\mu }{a}}. \end{aligned}$$

(A.5)

By expanding the radical term, one finally gets

$$\begin{aligned} \frac{\mathrm{d}q}{\mathrm{d}t}=\xi \sin \theta \left[ B-\frac{\mu }{2Br_\mathrm{eq}^2}q-\frac{\mu ^2}{8B^3r_\mathrm{eq}^4}q^2\right] , \end{aligned}$$

(A.6)

where B is already defined in eq. (22). Equation (A.6) is a linear differential equation which can be solved easily. Consequently, eq. (20) is obtained as an answer from (A.6), followed by relations (21) and (22) as definitions in (20).