Generalized uncertainty relations

TL;DR

This paper introduces a generalized form of quantum uncertainty relations using two vector states, offering increased flexibility and potential solutions to limitations of traditional UR, with applications that improve technical aspects and provide new insights.

Contribution

It proposes a new generalized uncertainty relation framework based on two vector states, expanding the applicability and interpretability of UR in quantum mechanics.

Findings

Enhanced flexibility in uncertainty relations through dual states

Potential to address domain issues in traditional UR

Applications lead to technical improvements and new insights

Abstract

The standard uncertainty relations (UR) in quantum mechanics are typically used for unbounded operators (like the canonical pair). This implies the need for the control of the domain problems. On the other hand, the use of (possibly bounded) functions of basic observables usually leads to more complex and less readily interpretable relations. Also, UR may turn trivial for certain states if the commutator of observables is not proportional to a positive operator. In this letter we consider a generalization of standard UR resulting from the use of two, instead of one, vector states. The possibility to link these states to each other in various ways adds additional flexibility to UR, which may compensate some of the above mentioned drawbacks. We discuss applications of the general scheme, leading not only to technical improvements, but also to interesting new insight.