This paper presents a detailed quantum mechanical analysis of the minimal length uncertainty relation, which arises in quantum gravity and string theory. The authors propose a generalized quantum theoretical framework that incorporates a nonzero minimal uncertainty in position measurements, which is interpreted as a minimal observable length. They explore the implications of this minimal uncertainty in non-relativistic quantum mechanics and show that it leads to a modified uncertainty relation, where the product of position and momentum uncertainties is bounded from below by a term that depends on the momentum uncertainty. This results in a nonzero minimal uncertainty in position, which cannot be arbitrarily reduced. The authors construct a Hilbert space representation of this commutation relation, showing that the position operator is no longer essentially self-adjoint but only symmetric. This leads to a modified representation of the Heisenberg algebra, where the position and momentum operators are defined on a generalized Bargmann-Fock space. The authors also derive the implications of this minimal uncertainty for the harmonic oscillator, showing that the energy levels of the system depend on the quantum number and the minimal uncertainty parameter. They further generalize the formalism to n dimensions, showing that the minimal uncertainty in position leads to a noncommutative geometry of space. The paper concludes with an outlook on the implications of the minimal length uncertainty relation for quantum field theory and the possibility of using it to describe nonpointlike particles.This paper presents a detailed quantum mechanical analysis of the minimal length uncertainty relation, which arises in quantum gravity and string theory. The authors propose a generalized quantum theoretical framework that incorporates a nonzero minimal uncertainty in position measurements, which is interpreted as a minimal observable length. They explore the implications of this minimal uncertainty in non-relativistic quantum mechanics and show that it leads to a modified uncertainty relation, where the product of position and momentum uncertainties is bounded from below by a term that depends on the momentum uncertainty. This results in a nonzero minimal uncertainty in position, which cannot be arbitrarily reduced. The authors construct a Hilbert space representation of this commutation relation, showing that the position operator is no longer essentially self-adjoint but only symmetric. This leads to a modified representation of the Heisenberg algebra, where the position and momentum operators are defined on a generalized Bargmann-Fock space. The authors also derive the implications of this minimal uncertainty for the harmonic oscillator, showing that the energy levels of the system depend on the quantum number and the minimal uncertainty parameter. They further generalize the formalism to n dimensions, showing that the minimal uncertainty in position leads to a noncommutative geometry of space. The paper concludes with an outlook on the implications of the minimal length uncertainty relation for quantum field theory and the possibility of using it to describe nonpointlike particles.