The paper explores the quantum mechanical implications of a minimal length uncertainty relation, which is a key concept in quantum gravity and string theory. The authors derive a generalized uncertainty relation that describes a nonzero minimal uncertainty in position measurements, denoted as $\Delta x_0$. This relation is formulated in one dimension and extends to non-relativistic quantum mechanics, focusing on the case where there is no minimal uncertainty in momentum. The paper constructs a Hilbert space representation for this generalized uncertainty relation, using a generalized Bargmann-Fock space. This representation allows for the exploration of the physical implications of a minimal length in a more manageable way compared to the more complex case with minimal uncertainties in both position and momentum. Key findings include: 1. **Minimal Uncertainty in Position**: The introduction of a minimal uncertainty $\Delta x_0$ implies that there are no physical states with zero uncertainty in position, unlike in ordinary quantum mechanics. 2. ** Representation on Momentum Space**: The position operator is no longer essentially self-adjoint but is symmetric, and the momentum operator remains essentially self-adjoint. The representation is defined on momentum space wave functions. 3. **Maximal Localization States**: States with maximal localization around a position $\xi$ have a well-defined uncertainty in position $\Delta x_0$ and are proper physical states with finite energy. 4. **Quasi-Position Wave Functions**: These states can be used to define quasi-position wave functions, which have a direct physical interpretation and are useful for describing the localization of particles. 5. **Harmonic Oscillator Example**: The energy spectrum and eigenfunctions of the harmonic oscillator are derived, showing how the introduction of a minimal length affects the behavior of wave functions and energy levels. The authors also discuss the potential applications of these findings, including their use in quantum field theory, new regularization methods, and the description of non-pointlike particles such as strings and mesons. The paper concludes with an outlook on further research directions, emphasizing the profound implications of minimal length uncertainties in quantum mechanics.The paper explores the quantum mechanical implications of a minimal length uncertainty relation, which is a key concept in quantum gravity and string theory. The authors derive a generalized uncertainty relation that describes a nonzero minimal uncertainty in position measurements, denoted as $\Delta x_0$. This relation is formulated in one dimension and extends to non-relativistic quantum mechanics, focusing on the case where there is no minimal uncertainty in momentum. The paper constructs a Hilbert space representation for this generalized uncertainty relation, using a generalized Bargmann-Fock space. This representation allows for the exploration of the physical implications of a minimal length in a more manageable way compared to the more complex case with minimal uncertainties in both position and momentum. Key findings include: 1. **Minimal Uncertainty in Position**: The introduction of a minimal uncertainty $\Delta x_0$ implies that there are no physical states with zero uncertainty in position, unlike in ordinary quantum mechanics. 2. ** Representation on Momentum Space**: The position operator is no longer essentially self-adjoint but is symmetric, and the momentum operator remains essentially self-adjoint. The representation is defined on momentum space wave functions. 3. **Maximal Localization States**: States with maximal localization around a position $\xi$ have a well-defined uncertainty in position $\Delta x_0$ and are proper physical states with finite energy. 4. **Quasi-Position Wave Functions**: These states can be used to define quasi-position wave functions, which have a direct physical interpretation and are useful for describing the localization of particles. 5. **Harmonic Oscillator Example**: The energy spectrum and eigenfunctions of the harmonic oscillator are derived, showing how the introduction of a minimal length affects the behavior of wave functions and energy levels. The authors also discuss the potential applications of these findings, including their use in quantum field theory, new regularization methods, and the description of non-pointlike particles such as strings and mesons. The paper concludes with an outlook on further research directions, emphasizing the profound implications of minimal length uncertainties in quantum mechanics.