Critical look at the time-energy uncertainty relations (1908.05273v1)

Abstract: The Heisenberg and Mandelstam-Tamm time-energy uncertainty relations are analyzed. The conlusion resulting from this analysis is that within the Quantum Mechanics of Schr\"{o}dinger and von Neumann, the status of these relations can not be considered as the same as the status of the position-momentum uncertainty relations, which are rigorous. The conclusion is that the time--energy uncertainty relations can not be considered as universally valid.

• The paper critically examines the Heisenberg and Mandelstam-Tamm time-energy uncertainty relations, asserting they lack universal validity compared to position-momentum relations.
• It highlights that unlike position-momentum, these relations stem from heuristic arguments or rely on assumptions that may fail in specific quantum states, such as stationary states.
• The analysis emphasizes the necessity for careful, non-universal application of these relations, especially in theoretical physics contexts where their limitations are crucial.

Critical Examination of Time–Energy Uncertainty Relations

The paper authored by K. Urbanowski presents a detailed analysis of the time–energy uncertainty relations, specifically targeting two widely recognized formulations: the Heisenberg and the Mandelstam–Tamm relations. Through rigorous examination, the paper postulates the critical assertion that these time–energy uncertainty relations cannot be regarded with the same level of universal validity as the position–momentum uncertainty relations.

Summary and Analysis

The foundational concept analyzed in this paper is the Heisenberg Uncertainty Principle, primarily known for its stringent mathematical formulation in the context of position–momentum, denoted by the relation $\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$. This relationship is a cornerstone of quantum mechanics, derived from the geometry of Hilbert space and the non-commutative nature of quantum operators.

However, Heisenberg also introduced an analogous relation for time and energy, initially based on heuristic arguments. This time–energy relation, often expressed as $\Delta t \cdot \Delta E \geq \frac{\hbar}{2}$, lacks the rigorous mathematical derivation available for the position–momentum uncertainty relations. Urbanowski's paper thoroughly scrutinizes efforts to establish rigor in this context, notably focusing on the Mandelstam–Tamm variant, which forms the basis of many physical theories and assertions in quantum mechanics.

Theoretical Foundations and Implications

Within the Mandelstam–Tamm framework, the paper investigates the assumption that operators $A$ (representing observables) and $H$ (the Hamiltonian) follow $[A, H] \neq 0$, which is instrumental in deriving time–energy relations. Urbanowski challenges the universality of such relations by demonstrating that under specific scenarios, such as stationary states or eigenvectors of $H$, the derived inequalities break down or reduce to trivial identities. This insight elucidates significant limitations in applying these uncertainty relations universally across all quantum mechanical systems.

Critical Considerations

The critique emphasizes the necessity for careful application of these relations, particularly in contexts where the Hamiltonian exhibits a discrete spectrum with stationary states. The analysis is meticulous in pointing out where conventional derivations assume non-zero operation of commutators among state vectors that may not universally hold.

The theoretical takeaway from Urbanowski's paper urges a reevaluation of how these principles are applied within quantum mechanics, advocating for a cautious approach in areas like astrophysics and cosmology where such relations often underpin theoretical predictions.

Future Directions

The exploration prompts further paper into establishing a more robust theoretical foundation or alternative frameworks that accurately reflect the temporal evolution and energy dispersion of quantum systems. As quantum computing and information theories advance, a more profound understanding of the limitations and potential extensions of the time–energy uncertainty relations can provide invaluable insights.

Conclusion

Urbanowski's analysis compellingly argues for the selective and informed application of time–energy uncertainty relations. By elucidating the inherent assumptions and potential misapplications, this paper provides a valuable pivot for future research endeavors aiming to resolve the nuances and establish a more universally validated expression of these pivotal quantum mechanical relations.