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Physical Review Letters

Hierarchy of Local Minimum Solutions of Heisenberg's Uncertainty Principle

David K. Hoffman, Donald Jack Kouri

Published Dec 18, 2000• 20 citations
Publisher

Abstract

We derive a new hierarchy of local minimum Heisenberg-uncertainty states by introducing a superposition of "small waves" onto some initial state. Our objective is to increase the resolution in one observable, with the least decrease in the resolution in the conjugate observable. This leads to a constrained minimization which in a well-defined sense yields the best possible way of achieving this goal. The results are relevant to many topics (e.g., quantum optics and control, Bose-Einstein condensation, path integration, etc.).

DOI
10.1103/physrevlett.85.5263

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We derive a new hierarchy of local minimum Heisenberg-uncertainty states by introducing a superposition of "small waves" onto some initial state. Our objective is to increase the resolution in one observable, with the least decrease in the resolution in the conjugate observable. This leads to a constrained minimization which in a well-defined sense yields the best possible way of achieving this goal. The results are relevant to many topics (e.g., quantum optics and control, Bose-Einstein condensation, path integration, etc.).
Local minimum solutions to Heisenberg's uncertainty principle are studied in detail. Of particular concern is obtaining an optimal procedure to squeeze the uncertainty in one canonical operator while introducing, in a well-defined mathematical sense, the least possible increase in the uncertainty of the relevant conjugate operator. The result is a hierarchy of ``minimum uncertainty $(\ensuremath{\mu}\ensuremath{-})$ wavelets,'' which are intimately associated with the ``Hermite distributed approximating functionals'' [D. K. Hoffman and D. J. Kouri, Phys. Rev. Lett. 85, 5263 (2000)]. The $\ensuremath{\mu}$ wavelets can be used with any quantum state to squeeze the uncertainty of a particular observable. Several specific examples are given and some possible applications are discussed.
We ask which is the best strategy to reveal uncertainty relations between complementary observables of a continuous variable system for coarse-grained measurements. This leads to the derivation of new uncertainty relations for coarse-grained measurements that are always valid, even for detectors with low precision. These relations should be particularly relevant in experimental demonstrations of squeezing in quantum optics, quantum state reconstruction, and the development of trustworthy entanglement criteria.
In (1), which deals with entropic uncertainty relationships, it is pointed out that x-variances of quantum bound states may possibly behave in a counterintuitive manner. In particular an example is given of two small boxes attached to the ends of a long pipe centered at zero so the average x is 0. If a particle has a probability of .5 to be in one box or the other a certain variance is found. If the particle can also be anywhere along the pipe or in the boxes the authors of (1) argue there is more uncertainty yet the variance decreases. Mathematically this occurs because there is now probability associated with small x values. We argue this is only counterintuitive if one considers probability distributions which decrease monotonically such as a Gaussian. In such a case, the variance increases if the particle is spread over a larger range. Such, however, does not seem to be the case if one has increased resolution which seems to be the case for quantum bound states such as a particle in a box with infinite potential walls or a quantum oscillator. Increased resolution occurs with increasing energy, but by the Heisenberg uncertainty principle the variance of p and x (or at least their product) increases. Increased resolution which seems to be linked to more order (less uncertainty) as argued in (1) can lead to an increased variance if probability only exists at distances far from the origin (i.e. high x) as in the closed two boxes example. As a result, the idea of uncertainty in the expression Heisenberg’s uncertainty principle seems to be based on the idea of how far the particle may be from the average value. As noted in (1) this means a large variance, but could represent high resolution i.e. a lack of uncertainty in finding the particle. We suggest there are two different kinds of uncertainty, Heisenberg’s uncertainty linked to the average and a resolution uncertainty. We examine the cases of a particle in a box and and the quantum oscillator.
We propose a geometric framework in which the apparent indeterminacy of quantum mechanics arises naturally from the projection of higher-dimensional objects onto lower-dimensional observer spaces. Using cardinality arguments from set theory, we establish a strict hierarchy: the set of all k-dimensional shapes embeddable in 3-dimensional space has cardinality ℵ(k+1), strictly increasing with k. Combined with a cardinality-existence correspondence principle, this implies the physical existence of infinitely many spatial dimensions. We then develop a dimensional projection framework in which quantum phenomena — superposition, wave-particle duality, wave function collapse, entanglement, tunneling, and spin — are reinterpreted as geometric consequences of a 3-dimensional observer perceiving higher-dimensional objects. The Heisenberg uncertainty relations are shown to be observer-relative rather than fundamental limits of nature. The incompatibility between general relativity and quantum mechanics is argued to stem from Einstein's implicit assumption that observer dimensionality is absolute. We propose that treating observer dimensionality as a relative quantity — analogously to how Einstein relativised time and space — may provide a path toward unification. Keywords: quantum mechanics, dimensional projection, cardinality, Heisenberg uncertainty, wave-particle duality, quantum gravity, higher dimensions, measurement problem