Heisenberg uncertainty relations in the non-Hamiltonian quantum statistical mechanics

Aniello Fedullo

Ijaz A. Rauf

The Heisenberg uncertainty postulates are a convenient way to describe the qualitative properties of quantum mechanics. The response from Heisenberg’s uncertainty principle is that one can create states that require a certain amount of momentum and position localization. The numerical quantities used by the old Newtonian mechanics, position, momentum, and energy are mere “shadows” of what nature describes: operators. The energy value contradicts the Newtonian notion of any energy level being necessary for a harmonic oscillator. The basic postulates of classical statistical mechanics are now applied to quantum statistical mechanics. In classical mechanics, the variables characterize a microstate that evolves according to the Hamiltonian equations. If the standard deviation relative to what we are interested in is very high, it is probably safe to use the expected value as a classical value. A quantum field theory is a relativistic invariant quantum theory where there is a particular form of quantity operators connected with the classical fields.

Andrzej Kossakowski

Dominic C. Horsman

The status of the uncertainty relations varies between the different interpretations of quantum mechanics. The aim of the current paper is to explore their meanings within a certain neo-Everettian many worlds interpretation. We will also look at questions that have been linked with the uncertainty relations since Heisenberg's uncertainty principle: those of joint and repeated measurement of non-commuting (or otherwise `incompatible') observables. This will have implications beyond the uncertainty relations, as we will see the fundamentally different way in which statistical statements are interpreted in the neo-Everett theory that we use.

Tao Wei, You-Yang Xu

Abstract In quantum mechanics the simultaneous measurement of non-commuter variables has attracted widespread attention. Here we propose a scheme through simultaneously weak measurement to achieve the approximated weak values of two non-commuting variables. It can be observed that the measurement error is better than the Heisenberg uncertainty relation. Furthermore, the effects of free Hamiltonian and higher-order expansion of the time evolution operator are considered.

Dorje C. Brody, Lane P. Hughston