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A Microcanonical Comparison of Richardsons Equations With The BCS Mean Field Approximation

From, David LU (2016) PHYM01 20151
Mathematical Physics
Department of Physics
Abstract
The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation... (More)
The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered. (Less)
Popular Abstract
In quantum mechanics when one wants to model a short distance attractive force the most common approach is to use the pairing model. The assumption of this model is that the particles pair up and are able to reduce their energy in this way. By pairing it is meant that the particles occupy each others time reversed states. By assuming this pairing force to be equally strong at all energy levels the particles can occupy one obtains the reduced pairing model. Now in order to solve this exactly one has to solve the Richardson equations. These are very difficult equations and involves terms diverging to infinities at certain pairing strengths. The large numbers makes it hard for a computer to process the equations correctly. Another simpler way... (More)
In quantum mechanics when one wants to model a short distance attractive force the most common approach is to use the pairing model. The assumption of this model is that the particles pair up and are able to reduce their energy in this way. By pairing it is meant that the particles occupy each others time reversed states. By assuming this pairing force to be equally strong at all energy levels the particles can occupy one obtains the reduced pairing model. Now in order to solve this exactly one has to solve the Richardson equations. These are very difficult equations and involves terms diverging to infinities at certain pairing strengths. The large numbers makes it hard for a computer to process the equations correctly. Another simpler way of getting results is to use the BCS approximation. This approximation is done by replacing the forces on the particles exerted by the other particles by their mean. The result is that obtaining the total energy of a state is reduced to solving for two new parameters; the gap and the Fermi energy.

The gap is interpreted as the energy of exciting a particle past the Fermi energy, which is usually located between the highest occupied level in the ground state and the level above it. In the exact solution, the Richardson equations, the gap is approximated by the mean energy of adding a particle in the lowest unoccupied energy level and removing a particle from the topmost occupied level. The comparison spans a couple of ground states with low particle numbers. The gaps of several ground states are found to be similar in both methods but deviating more and more from each other, albeit very slowly, as the strength of the pairing interaction is increased.

Furthermore the excitation spectra of a couple of different systems of energy levels at certain numbers of particles were compared. It was found that if the gap in the BCS approximation is re-evaluated for each state most of the excitation spectra is very close to the spectra obtained for the Richardson equations. However low lying states seem consistently to be overestimated by the BCS approximation. Due to the advantage of computing the spectra quickly it is sometimes possible to use the same gap, e.g. that of the ground state, in the BCS approximation for all states. That approach is found to give unreasonable results when applied to the systems considered. The reason for this is that the gap apparently decreases the more excitations that are made. The gap is even zero for many states much longer than the ground state. (Less)
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author
From, David LU
supervisor
organization
course
PHYM01 20151
year
type
H2 - Master's Degree (Two Years)
subject
language
English
id
8885020
date added to LUP
2016-07-26 20:15:58
date last changed
2016-07-26 20:15:58
@misc{8885020,
  abstract     = {The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.},
  author       = {From, David},
  language     = {eng},
  note         = {Student Paper},
  title        = {A Microcanonical Comparison of Richardsons Equations With The BCS Mean Field Approximation},
  year         = {2016},
}