Abstract

In this chapter we discuss relations between main operators of quantum mechanics, that is, relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups as well as their consequences. Since in most uncertainty relations and in these operators the appearing weights are radially symmetric, it turns out that these relations can be extended to also hold on general homogeneous groups. In particular, we obtain both isotropic and anisotropic uncertainty principles in a refined form, where the radial derivative operators are used instead of the elliptic or hypoelliptic differential operators.

Highlights

  • In this chapter we aim at presenting an independent treatment of inequalities following from certain identities involving the appearing operators

  • Wolfgang Pauli and Hermann Weyl provided the mathematical aspects of uncertainty relations involving position and momentum operators, but the first rigorous proof was given by Earle Kennard [Ken27]

  • The first equality in (9.1) gives a relation between the position and momentum operator; it will be clear from Example 9.1.3 that it is satisfied by the classical position and momentum operators of the Euclidean quantum mechanics

Read more

Summary

Chapter 9

In this chapter we discuss relations between main operators of quantum mechanics, that is, relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups as well as their consequences. Since in most uncertainty relations and in these operators the appearing weights are radially symmetric, it turns out that these relations can be extended to hold on general homogeneous groups. Throughout this book, most of the inequalities imply the corresponding uncertainty principles. In this chapter we aim at presenting an independent treatment of inequalities following from certain identities involving the appearing operators. In this respect such uncertainty relations can be sometimes obtained independently from Hardy inequalities in alternative ways, see, e.g., Ciatti, Ricci and Sundari [CRS07]. Wolfgang Pauli and Hermann Weyl provided the mathematical aspects of uncertainty relations involving position and momentum operators, but the first rigorous proof was given by Earle Kennard [Ken27].

Chapter 9. Uncertainty Relations on Homogeneous Groups
Abstract position and momentum operators
Definition and assumptions
Examples
Further position-momentum identities
Heisenberg–Kennard and Pythagorean inequalities
Euler–Coulomb relations
Heisenberg–Pauli–Weyl uncertainty principle
Radial dilations – Coulomb relations
Further weighted uncertainty type inequalities
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call