Minimum uncertainty in quantum mechanical cooperative phenomena

Abstract

Simple unified expressions are found for the minimum uncertainty in quantum mechanical cooperative phenomena. Minimum uncertainty states are defined here as states which minimize the weighted sum of squared uncertainties of individual physical variables. Jackiw's method is used to show that the variational problems are equivalent to minimum eigenvalue problems. When Hamiltonians have the same form as the eigenvalue operators, the ground states are the minimum uncertainty states. In typical examples such as laser, nonlinear optical phenomena, superfluidity, superconductivity, charge density wave, and ferromagnetism, the mean field approximations for the cooperative phenomena yield Hamiltonians with this property. This fact indicates a relationship between the quantum mechanical cooperative phenomena in the ideal limits and the minimum uncertainty.