Localization–delocalization of a particle in a quantum corral in presence of a constant magnetic field

We obtained the energy and wave functions of a particle in a quantum corral subjected to a constant magnetic field, as a function of the radius of the quantum corral $$R_\mathrm{c}$$ and the intensity of the magnetic field $$b^2$$. We also computed the standard deviation and the Shannon information entropies as a function of $$R_\mathrm{c}$$ and $$b^2$$, which in turn are compared to determine their effectiveness in measuring particle (de)localization. For a fixed magnitude of the magnetic field $$b^2$$, the Shannon entropy of all states diminishes as the confinement radius $$R_\mathrm{c}$$ decreases revealing an extensive localization. For a fixed value of $$R_\mathrm{c}$$, the Shannon entropy of the states (0, 0) and (0, 1) decreases monotonically as the magnetic field $$b^2$$ grows, whereas for the states (1, 0), (2, 0), (1, 1) and (2, 1), the Shannon entropy grows slowly, reaching a maximum (delocalization), and then diminishes as $$b^2$$ increases. The expectation value of $$\left\langle r\right\rangle $$ for a fixed value $$R_\mathrm{c}$$, for the states (0, 0) and (0, 1), decreases monotonically as $$b^2$$ increases, whereas for the states (1, 0), (2, 0), (1, 1) and (2, 1) increases and after reaching a maximum, it decreases as $$b^2$$ grows. This behavior is counter-intuitive because the particle is forecasted to be closer to the origin as the magnetic field grows