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Annals of Physics

Behavior of single-particle states and their eigenvalues near zero energy

Hodgson. Atomic Energy Research Establishment

Published Jun 1, 1977• 0 citations
Publisher
DOI
10.1016/0003-4916(77)90250-0

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The energy variance extrapolation method consists of relating the approximate energies in many-body calculations to the corresponding energy variances and inferring eigenvalues by extrapolating to zero variance. The method needs a fast evaluation of the energy variances. For many-body methods that expand the nuclear wavefunctions in terms of deformed Slater determinants, the best available method for the evaluation of energy variances scales with the sixth power of the number of single-particle states. We propose a new method which depends on the number of single-particle orbits and the number of particles rather than the number of single-particle states. We discuss as an example the case of 4 He using the chiral N3LO interaction in a basis consisting up to 184 single-particle states.
The existence of particle outflow in the decay of many-body system leads to a special description with two sets of state eigenvectors and corresponding complex eigenvalues. Using such eigenvectors the general time behavior of the decaying system is obtained. As an example, we consider the problem of a particle decay of the many-body system to demonstrate the difference between this problem and the problem of a single particle in potential well. We also consider the case of particle scattering by an optical potential and show that the existence of outflow also leads to two sets of state eigenvectors, but with real energy eigenvalues.
We study the evolution of the energy and magnetic moment of a quantum charged particle placed in a homogeneous magnetic field, when this field changes its sign adiabatically. We show that after a single magnetic field passage through zero value, the famous adiabatic invariant ratio of energy to frequency is reestablished again, but with a proportionality coefficient higher than in the initial state. The concrete value of this proportionality coefficient depends on the power index of the frequency dependence on time near zero point. In particular, the adiabatic ratio of the initial ground state (with zero radial and angular quantum numbers) triplicates if the frequency tends to zero linearly as a function of time. If the Larmor frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of the time intervals between zero points, so that the mean energy behavior can be quasi-stochastic after many passages through zero value. The original Born-Fock adiabatic theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many instantaneous Fock states, whose weights do not depend on time in the new adiabatic regime.