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Register a new userCritical look at the time-energy uncertainty relations
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Added by
Krzysztof Urbanowski
Publication date
2019
fields
Physics
and research's language is
English
Authors
K. Urbanowski
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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.
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Uncertainty relations play a crucial role in quantum mechanics. A well-defined method exists for deriving such uncertainties for pairs of observables. It does not include, however, an important family of fundamental relations: the time-energy uncertainty relations. As a result, different approaches have been used for obtaining them in diversified scenarios. The one of interest here revolves around the idea of the existence or inexistence of a minimum duration for an energy measurement with a certain precision. In our study, we use the Page and Wooters timeless framework to investigate how energy measurements modify the relative flow of time between internal and external clocks. This provides a unified framework for discussing the topic, recovering previous results and leading to new ones. We also show that the evolution of the external clock with respect to the internal one is non-unitary.
Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $Delta A$ and $Delta B$ calculated for these vectors is zero: $Delta A,cdot,Delta B geq 0$. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices $(2times 2)$ and $(3 times 3)$ and the position--momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in $cal PT$--symmetric quantum theory and the problems associated with it are also studied.
New uncertainty relations for n observables are established. The relations take the invariant form of inequalities between the characteristic coefficients of order r, r = 1,2,...,n, of the uncertainty matrix and the matrix of mean commutators of the observables. It is shown that the second and the third order characteristic inequalities for the three generators of SU(1,1) and SU(2) are minimized in the corresponding group-related coherent states with maximal symmetry.
Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order $alpha$ rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.
We analyze general uncertainty relations and we show that there can exist such pairs of non--commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $Delta A$ and $Delta B$ calculated for these vectors is zero: $Delta A,cdot,Delta B geq 0$. We show also that for some pairs of non--commuting observables the sets of vectors for which $Delta A,cdot,Delta B geq 0$ can be complete (total). The Heisenberg, $Delta t ,cdot, Delta E geq hbar/2$, and Mandelstam--Tamm (MT), $ tau_{A},cdot ,Delta E geq hbar/2$, time--energy uncertainty relations ($tau_{A}$ is the characteristic time for the observable $A$) are analyzed too. We show that the interpretation $tau_{A} = infty$ for eigenvectors of a Hamiltonian $H$ does not follow from the rigorous analysis of MT relation. We show also that contrary to the position--momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of $A$ and $H$.
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