Position in Minimal Length Quantum Mechanics

Several approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems.

Theory and experiment for resource-efficient joint weak-measurement

Incompatible observables underlie pillars of quantum physics such as contextuality and entanglement. The Heisenberg uncertainty principle is a fundamental limitation on the measurement of the product of incompatible observables, a 'joint' measurement. However, recently a method using weak measurement has experimentally demonstrated joint measurement. This method [Lundeen, J. S., and Bamber, C. Phys. Rev. Lett. 108, 070402, 2012] delivers the standard expectation value of the product of observables, even if they are incompatible. A drawback of this method is that it requires coupling each observable to a distinct degree of freedom (DOF), i.e., a disjoint Hilbert space. Typically, this 'read-out' system is an unused internal DOF of the measured particle. Unfortunately, one quickly runs out of internal DOFs, which limits the number of observables and types of measurements one can make. To address this limitation, we propose and experimentally demonstrate a technique to perform a joint weak-measurement of two incompatible observables using only one DOF as a read-out system. We apply our scheme to directly measure the density matrix of photon polarization states.

Uncertainty principles for the windowed offset linear canonical transform

The windowed offset linear canonical transform (WOLCT) can be identified as a generalization of the windowed linear canonical transform (WLCT). In this paper, we generalize several different uncertainty principles for the WOLCT, including Heisenberg uncertainty principle, Hardy’s uncertainty principle, Donoho–Stark’s uncertainty principle and Nazarov’s uncertainty principle. Finally, as application analogues of the Poisson summation formula and sampling formulas are given.

Study on energy and information-entropic measures of Hulthén potential in D dimension by an intuitive approximation to centrifugal term

Energy spectrum as well as various information theoretic measures are considered for Hulthén potential in D dimension. For a given ℓ≠0 state, analytic expressions are derived, following a simple intuitive approximation for accurate representation of centrifugal term, within the conventional Nikiforov-Uvarov method. This is derived from a linear combination of two widely used Greene-Aldrich and Pekeris-type approximations. Energy, wave function, normalization constant, expectation value in r and p space, Heisenberg uncertainty relation, entropic moment of order α¯, Shannon entropy, Rényi entropy, disequilibrium, majorization as well as four selected complexity measures like LMC (López-Ruiz, Mancini, Calbert), shape Rényi complexity, Generalized Rényi complexity and Rényi complexity ratio are offered for different screening parameters (δ). The effective potential is described quite satisfactorily throughout the whole domain. Obtained results are compared with theoretical energies available in literature, which shows excellent agreement. Performance of six different approximations to centrifugal term is critically discussed. An approximate analytical expression for critical screening for a specific state in arbitrary dimension is offered. Additionally, some inter-dimensional degeneracy occurring in two states, at different dimension for a particular δ is also uncovered. PACS: 02.60.-x, 03.65.Ca, 03.65.Ge, 03.65.-w Keywords: Hulthén potential, Rényi complexity ratio, Statistical complexity, Majorization, Pekeris approximation, Greene-Aldrich approximation.

Generalized uncertainty principle: from the harmonic oscillator to a QFT toy model

Several models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg uncertainty principle into the generalized uncertainty principle. In this work, we study the implications of a polynomial generalized uncertainty principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way. Furthermore, a new set of ladder operators is derived which factorize the Hamiltonian exactly. The above formalism is finally exploited to construct a quantum field theoretic toy model based on the generalized uncertainty principle.

An analysis on Quantum mechanical Stability of Regular Polygons on a Point Base Using Heisenberg Uncertainty Principle

It is a well-known fact in physics that classical mechanics describes the macro-world, and quantum mechanics describes the atomic and sub-atomic world. However, principles of quantum mechanics, such as Heisenberg’s Uncertainty Principle, can create visible real-life effects. One of the most commonly known of those effects is the stability problem, whereby a one-dimensional point base object in a gravity environment cannot remain stable beyond a time frame. This paper expands the stability question from 1- dimensional rod to 2-dimensional highly symmetrical structures, such as an even-sided polygon. Using principles of classical mechanics, and Heisenberg’s uncertainty principle, a stability equation is derived. The stability problem is discussed both quantitatively as well as qualitatively. Using the graphical analysis of the result, the relation between stability time and the number of sides of polygon is determined. In an environment with gravity forces only existing, it is determined that stability increases with the number of sides of a polygon. Using the equation to find results for circles, it was found that a circle has the highest degree of stability. These results and the numerical calculation can be utilized for architectural purposes and high-precision experiments. The result is also helpful for minimizing the perception that quantum mechanical effects have no visible effects other than in the atomic, and subatomic world. Keywords: Quantum mechanics, Heisenberg Uncertainty principle, degree of stability, polygon, the highest degree of stability

Full monitoring of ensemble trajectories with 10 dB-sub-Heisenberg imprecision

The change of a quantum state can generally only be fully monitored through simultaneous measurements of two non-commuting observables $$\hat{X}$$ X ̂ and $$\hat{Y}$$ Y ̂ spanning a phase space. A measurement device that is coupled to the thermal environment provides at a time a pair of values that have a minimal uncertainty product set by the Heisenberg uncertainty relation, which limits the precision of the monitoring. Here, we report on an optical ensemble measurement setup that is able to monitor the time-dependent change of the quantum state’s displacement in phase space ($$\langle \hat{X}(t)\rangle ;\langle \hat{Y}(t)\rangle$$ ⟨ X ̂ ( t ) ⟩ ; ⟨ Y ̂ ( t ) ⟩ ) with an imprecision 10 dB below the Heisenberg uncertainty limit. Our setup provides pairs of values (X(ti); Y(ti)) from simultaneous measurements at subsequent times ti. The measurement references are not coupled to the thermal environment but are established by an entangled quantum state. Our achievement of a tenfold reduced quantum imprecision in monitoring arbitrary time-dependent displacements supports the potential of the quantum technology required for entanglement-enhanced metrology and sensing as well as measurement-based quantum computing.

Exploring electroencephalography with a model inspired by quantum mechanics

An outstanding issue in cognitive neuroscience concerns how the brain is organized across different conditions. For instance, during the resting-state condition, the brain can be clustered into reliable and reproducible networks (e.g., sensory, default, executive networks). Interestingly, the same networks emerge during active conditions in response to various tasks. If similar patterns of neural activity have been found across diverse conditions, and therefore, different underlying processes and experiences of the environment, is the brain organized by a fundamental organizational principle? To test this, we applied mathematical formalisms borrowed from quantum mechanisms to model electroencephalogram (EEG) data. We uncovered a tendency for EEG signals to be localized in anterior regions of the brain during “rest”, and more uniformly distributed while engaged in a task (i.e., watching a movie). Moreover, we found analogous values to the Heisenberg uncertainty principle, suggesting a common underlying architecture of human brain activity in resting and task conditions. This underlying architecture manifests itself in the novel constant KBrain, which is extracted from the brain state with the least uncertainty. We would like to state that we are using the mathematics of quantum mechanics, but not claiming that the brain behaves as a quantum object.