Quantum breathing mode for electrons with 1/r2 interaction.

and where vc is the cyclotron frequency. The exact spectrum of the interacting electron system therefore contains an infinite ladder of energy levels at integer multiples of 2\V . However, these authors do not explain why the unphysical 1/r interaction is special, apart from the mathematical fact that it permits a separation of the many-particle Schrödinger equation in the hyperradial and hyperangular coordinates employed. Furthermore, the physical nature of the breathing mode is not fully explained. Given the simplicity of the result ~1!, it is natural to ask whether there is a more direct way of obtaining it. The purpose of this paper is to point out that the quantum breathing mode excitation spectrum can also be obtained directly from the behavior of the Hamiltonian under a scale transformation, and in a manner that makes evident the special property of the inverse-square interaction for the quantum breathing mode. We shall work in the symmetric gauge and write the Hamiltonian as