Quantum support to BoHua Sun's conjecture

TL;DR

This paper explores the application of BoHua Sun's generalized Kepler's third law to quantum systems of self-gravitating particles, suggesting a potential bridge between classical orbital laws and quantum mechanics.

Contribution

It demonstrates that Sun's conjecture can be extended to quantum N-body systems with appropriate period selection, broadening its scope beyond classical mechanics.

Findings

Sun's law applies to quantum self-gravitating systems with suitable period choices

The generalized Kepler's third law can describe quantum N-body dynamics

Potential link between classical orbital laws and quantum gravitational systems

Abstract

A generalization of the Kepler's third law has been proposed by BoHua Sun for $N$-body periodic orbits in a Newtonian gravitation field. In this paper, it is shown that this formula can apply for a quantum system of $N$ self-gravitating identical particles, for a good choice of the period for the quantum motion.

Equations

$$\bar{T}=\frac{T}{L_f}$$

$$T_{\ast }=\bar{T}|E{|}^{3/2},$$

$$T_{\ast }=\frac{\pi }{\sqrt{2}}G{\left(\frac{{\sum }_{i=1}^N{\sum }_{j=i+1}^N(m_i{m}_j{)}^3}{{\sum }_{i=1}^N{m}_i}\right)}^{1/2}.$$

$$T_{\ast }=\frac{\pi }{2}G{m}^{5/2}\sqrt{N-1}.$$

$$E_q=-\frac{N^2(N-1{)}^3}{16}\frac{G^2{m}^5}{Q^2{\hslash }^2},$$

$$Q=\sum _{i=1}^{N-1}(2n_i+l_i)+(N-1)\frac{D}{2}$$

$$r_0=\frac{2^{3/2}}{N^{1/2}(N-1{)}^{3/2}}\frac{Q^2{\hslash }^2}{G{m}^3}.$$

$$T_q=\frac{2\pi d_0}{v_0}=2\pi \frac{r_0}{N}\frac{m}{p_0}.$$

$$T_q|E_q{|}^{3/2}=\frac{\pi }{4}G{m}^{5/2}N(N-1{)}^{3/2}$$

$$T_{\ast q}=\frac{2T_q}{N(N-1)}|E_q{|}^{3/2}$$

$$H=\frac{{\mathbf{p}}^2}{2\mu }-\frac{\kappa }{r},$$

$$T_{\ast q}=\frac{\pi }{\sqrt{2}}{\mu }^{1/2}\kappa ,$$

Full text

The Kepler’s third law is certainly a milestone in the development of mechanics. But this peculiar relation between the period and the size of the orbit (or the period and the energy of the orbit) seemed only relevant for two-body systems. Nevertheless, the computation of a great number of Newtonian periodic planar collisionless three-body orbits suggests the existence of a quasi Kepler’s third law li17 . A definition of an average period

[TABLE]

must be chosen, with $T$ the period of the orbit and $L_{f}$ the length of the so-called “free group element” of the orbit. This corresponds to a method to classify the topology of the orbit around the three two-body collision points suv14 . The scale-invariant average period

[TABLE]

where $E$ is the total energy (kinetic and potential) of the orbit is found by numerical computation approximately equal to a universal constant for three identical bodies li17 . Using arguments based on dimensional analysis, BoHua Sun suggested then a generalization of the Kepler’s third law for $N$-body periodic orbits sun18 :

[TABLE]

This formula reduces to the usual Kepler’s third law for $N=2$, and gives results compatible with computations based on a great number of periodic planar collisionless orbits for $N=3$ li17 ; li17b ; li18 . More precisely, the agreement is good when one of the mass is much larger than the two others, and poor when one of the mass is much smaller than the two others. In particular, for identical bodies, formula (3) reduces to

[TABLE]

Although macroscopic and microscopic worlds are very different, strong connections exist between classical and quantum theories. The most famous one is certainly the Ehrenfest theorem, showing that expectation values obey Newton’s second law. Moreover, three fundamental theorems of quantum mechanics (Hellmann-Feynman theorem, virial theorem and comparison theorem) have classical counterparts sema18a . So, a natural question arises: Is the Kepler’s third law can also be relevant for quantum $N$-body systems? Such systems are not easy to treat, but the problem is less difficult for identical particles. In this case, the necessity to (anti)symmetrize the wave-function puts strong constraints on the solution, which can help the computation. The envelope theory is an efficient and convenient tool to obtain approximate solutions for quantum systems made of $N$ identical particles hall80 ; sema13a ; sema18b . With this method, the quantum energy of $N$ self-gravitating particles with a mass $m$ in $D$-dimensional space is given by sema15a

[TABLE]

where

[TABLE]

is a global quantum number (corresponding to $N-1$ identical harmonic oscillators). The equivalent of the quantity $\bar{T}$ for this quantum system is not easy to define, since the dynamics is described by a wave-function in the configuration space and not by motions of particles through the real space. Nevertheless, the equations of the envelope theory can be interpreted within a semi-classical framework sema13a ; sema18b in the following way: the $N$ particles are in circular motion with the same momentum $p_{0}=Q\,\hbar/r_{0}$ (and thus the same speed $v_{0}=p_{0}/m$) at a distance $d_{0}=r_{0}/N$ from the center of mass, each particle being at an angular distance of $2\,\pi/N$ from its neighbors. The distance $r_{0}$ for self-gravitating particles is given by sema15a

[TABLE]

Within this picture, the quantum period $T_{q}$ can be estimated by the time taken by a particle to run a complete turn

[TABLE]

Formulas (5) and (9) give

[TABLE]

This quantity is $N(N-1)/2$ times $T_{*}$ given by (4). $N(N-1)/2$ is the number of pairs but also the number of two-body collision points in the $N$-body systems. So, if we assume that the period must be rescaled by this number, the quantum scale-invariant average period is given by

[TABLE]

for $N$ identical particles. Then, $T_{*q}$ and $T_{*}$ give both the same result.

One can ask how is it possible that an approximate quantum formula gives the same result than a classical (assumed) exact relation? First, the quantum character of formulas (5) and (7) is carried by the quantity $Q\,\hbar$, which disappears in the computation of $T_{*q}$. Second, the main source of error in the calculations from the envelope theory is the strong degeneracy (inherent to the method) due to the quantum number $Q$. The results can be greatly improved by modifying the structure of $Q$, either by a fit on numerical accurate computations sema15a or by the use of a complementary method sema15b . As mentioned above, this quantum number disappears in the computation of $T_{*q}$.

It is interesting to look at the quantum two-body Coulomb system for which exact results are well-known. If the Hamiltonian is written

[TABLE]

and if we take $d_{0}=\langle 1/r\rangle^{-1}$ and $v_{0}=\sqrt{\langle\bm{p}^{2}\rangle}/\mu$, then

[TABLE]

with obviously only one two-body collision point. If $\mu=m_{1}\,m_{2}/(m_{1}+m_{2})$ and $\kappa=G\,m_{1}\,m_{2}$, (3) is recovered with $N=2$. With a good and reasonable choice for the period, classical and quantum results are again identical.

Let us mention that a linear dependence seems to exist between $T\,|E|^{3/2}$ and a number representing the orbit’s topological complexity for collisionless periodic three-body orbits in the Coulomb potential with one positively charged particle and two negatively charged particles sind18 . The existence of attractive and repulsive interactions in this system makes it different from the ones presented above. Nevertheless, the dependence found could be an indication of some universal relation for Coulombic systems.

I suspect that the identical classical and quantum results obtained for a generalization of the Kepler’s third law for $N$-body systems is more than a simple happy coincidence, even if the “quantum period” must be well chosen. But this problem certainly deserves more studies. A possible way is to search for regularities in Newtonian mechanics for more than three orbiting bodies. Another way is to generalize the envelope theory for different particles and look for new invariants.