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Quantum Mechanics of the Solar System

L. Acedo, Matemática Multidisciplinar

Published Jan 1, 2014• 1 citation
OpenAlex

Abstract

According to the correspondence principle, as formulated by Bohr, both in the old and the modern quantum theory, the classical limit should be recovered for large values of the quantum numbers in any quantum system. However, this classical limit of quantum theory is not so straightforward as in the interface of other generalizations of classical mechanics and other domains. In particular, relativistic kinematics and mechanics reduce to Newtonian equations by simple algebra in the case of bodies moving with small velocities compared to the speed of light in vacuum. In this paper we consider the correspondence limit to the two-body problem in gravitational physics, the limit in which both the principal and the angular quantum numbers, N, L are very large. In this limit, we compare with the classical elliptical orbits and we find that the macroscopic coherent quantum states correspond to the statistical average of every classical state compatible with conservation laws for the total energy and angular momentum. We also consider the perturbed Kepler problem with a central perturbation force proportional to the inverse of the cube of the distance to the central body. The exact solution for the quantum eigenstates shows that the first order perturbation to the energy eigenvalues are obtained classically as the temporal orbital average of the perturbation potential.

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According to the correspondence principle, as formulated by Bohr, both in the old and the modern quantum theory, the classical limit should be recovered for large values of the quantum numbers in any quantum system. However, this classical limit of quantum theory is not so straightforward as in the interface of other generalizations of classical mechanics and other domains. In particular, relativistic kinematics and mechanics reduce to Newtonian equations by simple algebra in the case of bodies moving with small velocities compared to the speed of light in vacuum. In this paper we consider the correspondence limit to the two-body problem in gravitational physics, the limit in which both the principal and the angular quantum numbers, N, L are very large. In this limit, we compare with the classical elliptical orbits and we find that the macroscopic coherent quantum states correspond to the statistical average of every classical state compatible with conservation laws for the total energy and angular momentum. We also consider the perturbed Kepler problem with a central perturbation force proportional to the inverse of the cube of the distance to the central body. The exact solution for the quantum eigenstates shows that the first order perturbation to the energy eigenvalues are obtained classically as the temporal orbital average of the perturbation potential.
Within a stochastic picture quantum systems can be described in terms of kinematic and dynamic equations where the particles follow a conservative diffusion process. Pavon has shown that these equations and the Schrödinger equation can be derived from a quantum Hamilton principle, generalizing the Hamilton principle from classical mechanics. As shown recently by Köppe et al for the one-dimensional case, the reformulation of the quantum Hamilton principle as a stochastic optimal control problem leads to quantum Hamilton equations of motion which can be seen as a non relativistic generalization of Hamilton's equations of motion to the quantum world. In this report we will extend this to higher dimensional problems and derive, by incorporating the concept of supersymmetric quantum mechanics, a general scheme that allows the determination of all bound states within the stochastic framework. Physical principles from classical mechanics like the decoupling of the center-of-mass motion in multi-particle systems or the solution of the Kepler problem, as a special case of the two-body problem, using space-time symmetries, are translated to this formulation of quantum mechanics. We will present numerical results for the ground state and lower excited states of the hydrogen atom in Cartesian coordinates, and, additionally, we will show that for spherically symmetric potentials the solution of the two-body problem can be reduced to a system of equations concerning only the radial part.
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 m l 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 m l . 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.
In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws, also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit is to: (i) approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue problem, (ii) define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.
Traditionally, Newtonian mechanics is recognized as an approximate theory which is said to arise from quantum mechanics (apparently a more detailed theory) in the large energy level limit (correspondence principle). We argue that both quantum mechanics and Newtonian mechanics are both approximate and both complement each other at any energy level. In the Newtonian case, quantum effects are so small that they are not visible, but in the quantum case, Newtonian effects are visible, we argue. We first note that exp(-iEt+ipx), the free particle wavefunction (complex probability) does not exist without the Newtonian expression x=vt. x=vt implies it is possible for dx->0, dt->0, while hbar/p and hbar/E show that there are limitations with respect to interactions. The only way to ascertain x=vt is through measurement and if these measurements have dt=hbar/E and dx=hbar/p uncertainties, one cannot have dx->0, dt->0 resolution. Nevertheless, the quantum free particle does move as x=vt (as long as one respects the uncertainties of dt,dx).The quantum exp(-iEt+ipx) does not exist without the Newtonian x=vt. In a single particle quantum bound state (time-independent), we suggest there are significant approximations. First, a conservation of energy equation exists in Newtonian physics, namely pp/2m + V(x) = E ((1)). This equation does not contain time and does not explicitly show x(t), but that does not mean that such motion does not physically exist. The quantum bound state solution follows from ((1)) which is an equation which does not display full information, even in the classical case, and so this loss of information carries down to the quantum solution. In particular, in the quantum case one solves a time-independent problem and so uses a(p)exp(ipx) + a(-p)exp(-ipx) with a(p)=a(-p) or a(p)= -a(-p). These conditions are equivalent to insisting that the local momentum average at each x (-i W* grad W + i W grad W) is 0. The fact that one is solving a quantum bound state using only energy considerations means that one averages out time which means averaging momentum. Thus, W*(x)W(x) describes a resolution spatial pattern for interaction, but W = W(x) exp(-iEt+ip x) where p=0 at each x. In other words, just because one does not see forward/backward motion in a particular quantum solution (based on energy alone) does not mean that it does not exist and that it cannot be measured, we suggest. For high energy levels, one should be able to measure such motion. The factor exp(-iEt+ipx) is a further approximation of the bound state which may be used if one wishes to describe a particle of mo in a potential V(x) created by a particle of mass M as an M+m+E rest mass system. Thus, one may see that various levels of approximation are used in the same quantum bound state solution. Furthermore, classical and Newtonian mechanics are still linked at any energy level because the quantum average KE (in the nonrelativistic case) must equal the classical kinetic energy. Thus, the notion of x= vrms(x) t is present and if hbar/(m vrms) << dx (where dx <<L with L being the system size), then one again has exp(-ipx+i KE t) and x=vrms(x) within each tiny dx, as an approximation.