Harrison, R. (2001) A numerical study of the Schrödinger-Newton equations. PhD thesis, University of Oxford.
The Schrödinger-Newton (S-N) equations were proposed by Penrose  as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of , where 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  and Jones et al . The ground state which has the lowest energy has no zeros. The higher states are such that the th state has 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 quadruples of complex eigenvalues for the th state, where a quadruple consists of . 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.
|Item Type:||Thesis (PhD)|
|Subjects:||O - Z > Partial differential equations|
O - Z > Quantum theory
H - N > Numerical analysis
|Research Groups:||Mathematical Physics Group|
Oxford Centre for Industrial and Applied Mathematics
|Deposited By:||Eprints Administrator|
|Deposited On:||10 Mar 2004|
|Last Modified:||20 Jul 2009 14:18|
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