Hearing shapes of few electrons quantum drums: A configuration–interaction study

The – highly remarkable – existence of non-congruent yet vibrationally isospectral shapes has been first proved theoretically and then also tested experimentally – by using electromagnetic waves in cavities, vibrating smectic films or electrons in nanostructures. In this context, we address the question whether isospectrality holds if two or more electrons interact electrostatically, using the accurate configuration–interaction method, in a discrete representation of the Bilby and Hawk shapes. Isospectral pairs offer an unique possibility to test how identical sets of single-particle energies may combine differently in the few-electrons eigenmodes, due to different wave functions spatial distributions. Our results point towards the break down of isospectrality in the presence of interactions. Thus one should be able to ”hear” the shapes of few electrons quantum drums. Interestingly however, for the analyzed two and three electrons cases, there exists an interaction strength (which can be tuned by changing the size of the shapes), for which the ground states energies of Bilby and Hawk coincide, but not the excited states as well. Wigner localization is studied and shown to occur at about the same size for both Bilby and Hawk shapes. Next, an exercise is proposed to use the two-electrons charge density of the Bilby and Hawk ground states in the phase extraction scheme as proposed by Moon et al. (2008). Results show that out-of-phase regions appear if the linear size of the shapes exceeds the Bohr radius as occupation of higher Slater determinants becomes significant.