Mechanical analogy between the second-order Schrödinger equation without potential for the case of a particle in an ideal infinite well with the fourth-order Schrödinger equation in connection with the potential manifestation of negative mass in Bose–Einstein condensates and exciton–polaritons in cavity

Abstract

Building on Schrödinger's original formulation of quantum mechanics from 1926, which initially involved a fourth-order differential equation, this article explores the mechanical analogy between this first Schrödinger equation without potential V and the dynamic behavior of vibrating elastic structures in the specific case of a particle in a potential well. Revisiting this fourth-order approach, we find a mathematical equivalence with the modern second-order Schrödinger equation which is strictly equivalent to the initial fourth-order Schrödinger equation in this specific case, while also revealing the possibility of solutions positive and negative masses. These results resonate with recent experimental observations on Bose–Einstein condensates and spin–orbit coupled exciton–polaritons. Following this research, it seems that negative mass effects should appear in the particular case of particles in a potential well situation close to this specific case like a Bose–Einstein condensate at a temperature close to 0 or another quantum entity in a cavity well.