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Symmetry Analysis of Three Dimensional Time Independent Schrodinger-Newton Equation

Emanullah Hızel, Nurettin Cenk Turgay, Bahri Guldoùgan

Published Jan 1, 2008• 1 citation
OpenAlex

Abstract

Schrödinger-Newton (SN) equations are modifications of the Schrödinger equation proposed by Roger Penrose involved in a gravitationally-induced wave function collapse scheme. In this paper, the three dimensional time independent Schrödinger-Newton (SN) equation is analyzed. The Lie group formalism is applied to deduce symmetries of the time indepen-dent SN equation for three dimensional case. One and two dimensional optimal systems and reduced equations are obtained.

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Schrödinger-Newton (SN) equations are modifications of the Schrödinger equation proposed by Roger Penrose involved in a gravitationally-induced wave function collapse scheme. In this paper, the three dimensional time independent Schrödinger-Newton (SN) equation is analyzed. The Lie group formalism is applied to deduce symmetries of the time indepen-dent SN equation for three dimensional case. One and two dimensional optimal systems and reduced equations are obtained.
The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of ∣ψ∣2|\psi|^2∣ψ∣2, where ψ\psiψ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the (n+1)(n+1)(n+1)th state has nnn zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for nnn quadruples of complex eigenvalues for the (n+1)(n+1)(n+1)th state, where a quadruple consists of {λ,λˉ,−λ,−λˉ}\{\lambda,\bar{\lambda},-\lambda,-\bar{\lambda}\}{λ,λˉ,−λ,−λˉ}. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.
Lie symmetry group methods provide a useful tool for the analysis of differential equations in a variety of areas in physics and applied mathematics. The nature of symmetry is that it provides information on properties which remain invariant under transformation. In differential equations this invariance provides a route toward complete integrations, reductions, linearisations and analytical solutions which can evade standard techniques of analysis. In this thesis we study two problems in quantum mechanics from a symmetry perspective: We consider for pedagogical purposes the linear time dependent Schrodinger equation in a potential and provide a symmetry analysis of the resulting equations. Thereafter, as an original contribution, we study the group theoretic properties of the density matrix equation for the quantum Brownian motion of a free particle interacting with a bath of harmonic oscillators. We provide a number of canonical reductions of the system to equations of reduced dimensionality as well as several complete integrations.
There are certain prime features of physical systems—notably rotation and its associated angular momentum—that simply do not appear in a one-dimensional world. This chapter introduces the three-dimensional Schrodinger equation and applies it to the analysis of a particle in a three-dimensional box and to the spherically symmetric wave functions of hydrogen. The time-independent Schrodinger equation can be viewed in a way that gives it a family resemblance to many other differential equations of physics, and which expresses its structure in very general terms. A computer solution uses dimensionless variables based on natural units. The Schrodinger theory does not have the limitations of the Bohr theory. Schrodinger theory incorporates the well-established wave nature of matter and can be generalized to analyze systems for which the Bohr theory has proven inadequate: multi-electron atoms, chemical properties of atoms, molecular and solid-state physics and chemistry, as well as more subtle effects in hydrogen itself.
Time-independent and time-dependent forms of Schrodinger equations are presented in one and three dimensions along with their interpretations. Also, the time-independent version is shown to follow from the time-dependent version. For better conceptualization of wave motion, the general equation of wave motion is discussed and solved, explaining the physical significance of variables involved. Born interpretation of the wave function as a measure of the probability of finding the particle in a region of space at a given instant is described. The requirements to be fulfilled by the wave function for representing a physical world system are outlined.
The Schrodinger-Newton (S-N) equation—a non-linear modification of quantum mechanics where the wave function experiences its own gravitational self-energy—has long been studied as a deterministic model for gravitational wave function collapse. This work provides a focused analysis of its phenomenology, explicitly treating it as a deterministic alternative to stochastic collapse models like Continuous Spontaneous Localization (CSL). We clarify the emergence of classical particle trajectories within this framework and derive its unique experimental signature: a mass- and separation-dependent shift in matter-wave interference fringes, distinct from the mere visibility loss predicted by decoherence or stochastic models. We provide a detailed roadmap for testing this prediction against competing approaches, critically assessing the experimental challenges and parameter constraints. Our work reframes the S-N equation as a concrete, falsifiable phenomenological model with a clear discriminant for future matter-wave experiments.
It has been suggested by Diósi and Penrose that the occurrence of quantum state reduction in macroscopic objects is related to a manifestation of gravitational effects in quantum mechanics. Although within Penrose's framework the dynamics of the quantum state reduction is not prescribed, it was suggested that the so-called Schrödinger–Newton equation can be used to at least identify the resulting classical end states. Here we analyse the extent to which the Schrödinger–Newton equation can be used as a model to generate a full, time-dependent description of the quantum state reduction process. We find that when supplied with an imaginary gravitational potential, the Schrödinger–Newton equation offers a rationalization for some of the hitherto unexplained characteristics of quantum state reduction. The description remains incomplete however, because it is unclear how to fully recover Born's rule.