Symmetries of Schrödinger's cats and a formal analogy with the Pauli exclusion principle for “jumping off cats”

Abstract

We consider the dynamics of the simplest Schrödinger's cat composed by a superposition of two canonical coherent states of the Schrödinger quantum harmonic oscillator. We demonstrate that the interchange of the “alive” and “dead” constituents in the whole Schrödinger's cat wave function opens the possibility to reveal one interesting, but obviously, only formal analogy with the Pauli exclusion principle. We show analytically and computationally that the symmetric cat states will occupy the ground state of the harmonic oscillator, whereas the antisymmetric cat states, vice versa, will never occupy the ground state when these states closely approach in space each other. The antisymmetric overlapping in space and time “alive” and “dead” Schrödinger cat states simply jump onto the first excited energy level of the quantum harmonic oscillator. This effect could be considered as the formal analogy of the Pauli exclusion principle applied to the antisymmetric “jumping off” Schrödinger cats. As the name suggests, the main feature of the “jumping off” Schrödinger cat consists in the transformation of the probability distribution (both in ordinary coordinate space and in the momentum space as well) in such a way that antisymmetric “alive” and “dead” cats are found to be in the first excited energy state of the quantum harmonic oscillator. We conclude that the wave and particle-like analogies in the dynamics of the symmetric and antisymmetric Schrödinger's cat states confirm at least formally the famous Pauli statements that “a symmetric solution can never develop into an antisymmetric one, and vice versa”, and that “for particles with antisymmetric states it can, therefore, never happen that two particles find themselves in the same state”