Position-momentum Uncertainty Research Articles

Theoretical Quantum Mass

In the present theoretical work, attempts have been made to find the least mass, which quantifies the discrete mass, in terms of the reduced Planck constant and cosmological constant in the Heisenberg position-momentum uncertainty relation as well as by employing the time period of oscillating universes in mass-time inequality. It has been estimated and reported that the least mass exists in the range 5.98 × 10-74 < m < 2 × 10-68 kg, which quantifies discrete masses in the universe of size < 8.64 G ly.

The uncertainty principle and quantum wave functions that are their own Fourier transforms

We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses.

Uncertainty relations for the relativistic Jackiw-Nair anyon: A first-principles derivation

In this paper we have explicitly computed the position-position and position-momentum (Heisenberg) uncertainty relations for the model of relativistic particles with arbitrary spin, proposed by Jackiw and Nair (Jackiw R. and Nair V. P. Phys. Rev. D, 43 (1991) 1933) as a model for anyon, in a purely quantum mechanical framework. This supports (via Schwarz inequality) the conjecture that anyons live in a 2-dimensional non-commutative space. We have computed the non-trivial uncertainty relation between anyon coordinates, , using the recently constructed anyon wave function (Majhi J. et al., Phys. Rev. Lett., 123 (2019) 164801), in the framework of Bialynicki-Birula I. and Bialynicka-Birula Z., New J. Phys., 21 (2019) 07306. We also compute the Heisenberg (position-momentum) uncertainty relation for anyons. Lastly we show that the identical formalism when applied to electrons, yield a trivial position uncertainty relation, consistent with their living in a 3-dimensional commutative space.

Quantum uncertainty: what to teach?

We present a new secondary school teaching method of quantum uncertainties of two-state systems. Intending to be a material teachable in schools, only two-state systems described by real numbers can be considered. An elementary argumentation based on school statistics leads to the identification of the uncertainty of a physical quantity in such systems with the standard deviation of two random variables. We provide a qualitative picture on the state-dependence of the uncertainty, leading to a pictorial representation in the form of four petals of a flower. When considering the product of uncertainty of two essentially different physical quantities we conclude that the general feature: “if the measurement of one of the quantities is certain, the other remains uncertain”, cannot be faithfully expressed by means of an inequality, the product has no lower bound different from zero. The application of techniques used by school materials for teaching quantum physics leads to an exact formula for the state-dependence of the uncertainty valid in any two-state system described by real numbers, in full harmony with the qualitative picture. We compare the two-state case with the celebrated Heisenberg position-momentum uncertainty relation and show that these are both specific facets, but only the Heisenberg relation can be expressed by an inequality. The latter hardly provides any hint on the uncertainties of physical quantities in two-state systems. We conclude that the two-state approach is worth teaching in schools also in relation to the uncertainty relation, even if the Heisenberg relation is not part of the curriculum.

Exploiting the Momentum Distribution in Atomically Confined Plasmonic Fields by Inelastic Scatterings

The utilization of atomically confined plasmonic fields has revolutionized the imaging technique. According to the fundamental position-momentum uncertainty principle, such a narrow spatial distribution certainly leads to a broad momentum distribution in the fields, which has however been overlooked. Here we propose a novel exploitation for the momentum distribution by adaptively satisfying the conservation law of momentum in inelastic Raman scatterings in periodic systems, providing a unique optical means of directly measuring the whole phonon dispersions. The proposed technique is particularly useful for measuring phonon dispersions of low-dimensional hydrogen-rich materials, which are completely inaccessible via other techniques. The numerical results for a single all-trans polyacetylene chain demonstrate that all phonon dispersion branches can be conclusively measured from their Raman images for the first time. Our findings highlight a unique advantage of the emerging momentum-based nanophotonics and open the door for exploiting highly confined plasmonic fields in another dimension.

Path Integral Action for a Resonant Detector of Gravitational Waves in the Generalized Uncertainty Principle Framework

The Heisenberg uncertainty principle is modified by the introduction of an observer-independent minimal length. In this work, we have considered the resonant gravitational wave detector in the modified uncertainty principle framework, where we have used the position momentum uncertainty relation with a quadratic order correction only. We have then used the path integral approach to calculate an action for the bar detector in the presence of a gravitational wave and then derived the Lagrangian of the system, leading to the equation of motion for the configuration-space position coordinate in one dimension. We then find a perturbative solution for the coordinate of the detector for a circularly polarized gravitational wave, leading to a classical solution of the same for the given initial conditions. Using this classical form of the coordinate of the detector, we finally obtain the classical form of the on-shell action describing the harmonic oscillator–gravitational wave system. Finally, we have obtained the free particle propagator containing the quantum fluctuation term considering gravitational wave interaction.

High Dimensional Atomic States of Hydrogenic Type: Heisenberg-like and Entropic Uncertainty Measures.

High dimensional atomic states play a relevant role in a broad range of quantum fields, ranging from atomic and molecular physics to quantum technologies. The D-dimensional hydrogenic system (i.e., a negatively-charged particle moving around a positively charged core under a Coulomb-like potential) is the main prototype of the physics of multidimensional quantum systems. In this work, we review the leading terms of the Heisenberg-like (radial expectation values) and entropy-like (Rényi, Shannon) uncertainty measures of this system at the limit of high D. They are given in a simple compact way in terms of the space dimensionality, the Coulomb strength and the state’s hyperquantum numbers. The associated multidimensional position–momentum uncertainty relations are also revised and compared with those of other relevant systems.

A complete set of Lorentz-invariant wave packets and modified uncertainty relation

We define a set of fully Lorentz-invariant wave packets and show that it spans the corresponding one-particle Hilbert subspace, and hence the whole Fock space as well, with a manifestly Lorentz-invariant completeness relation (resolution of identity). The position–momentum uncertainty relation for this Lorentz-invariant wave packet deviates from the ordinary Heisenberg uncertainty principle, and reduces to it in the non-relativistic limit.

On the Alleged Extra-Structures of Quantum Mechanics

I argue that a particle ontology naturally emerges from the basic dynamical equations of non-relativistic quantum mechanics, when the quantum continuity equation is realistically interpreted. This was recognized by J.J. Sakurai in his famous textbook “Modern Quantum Mechanics”, and then dismissed on the basis of the Heisenberg position–momentum uncertainty principle. In this paper, I show that the reasons of this rejection are based on a misunderstanding of the physical import of the uncertainty principle. As a consequence, a particle ontology can be derived from the quantum formalism without the need of additional ad hoc assumptions, and therefore it cannot be regarded as “extra-structure”.

A justification of the invariance of the speed of light by quantum theoretical considerations

phenomena can often be analysed in a visually clear way by thought experiments. In the present work a new thought experiment is analysed, which describes the emission of a photon by a special mechanism. The analysis leads to the finding, that there is a natural connection between the energy-time uncertainty and the Heisenberg position-momentum uncertainty principle. A deeper analysis of this thought experiment, especially with regard to a justification of the Doppler effect from the uncertainty principle, leads to the conclusion, that the speed of light is invariant. Therefore it seems reasonable to assume, that the origin of relativistic laws lies in non-relativistic quantum mechanics.

Uncertainty Relations: Curiosities and Inconsistencies

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 ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥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 (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.

Remarks on the uncertainty relations

We analyze general uncertainty relations and we show that there can exist such pairs of non-commuting observables [Formula: see text] and [Formula: see text] and such vectors that the lower bound for the product of standard deviations [Formula: see text] and [Formula: see text] calculated for these vectors is zero: [Formula: see text]. We also show that for some pairs of non-commuting observables the sets of vectors for which [Formula: see text] can be complete (total). The Heisenberg, [Formula: see text], and Mandelstam–Tamm (MT), [Formula: see text], time–energy uncertainty relations ([Formula: see text] is the characteristic time for the observable [Formula: see text]) are analyzed too. We show that the interpretation [Formula: see text] for eigenvectors of a Hamiltonian [Formula: see text] 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 [Formula: see text] and [Formula: see text].

Microscale Generation of Entangled Photons without Momentum Conservation.

We report, for the first time, the observation of spontaneous parametric down-conversion (SPDC) free of phase matching (momentum conservation). We alleviate the need to conserve momentum by exploiting the position-momentum uncertainty relation and using a planar geometry source, a 6 μm thick layer of lithium niobate. Nonphase-matched SPDC opens up a new platform on which to investigate fundamental quantum effects but it also has practical applications. The ultrasmall thickness leads to a frequency spectrum an order of magnitude broader than that of phase-matched SPDC. The strong two-photon correlations are still preserved due to energy conservation. This results in ultrashort temporal correlation widths and huge frequency entanglement. The studies we make here can be considered as the initial steps into the emerging field of nonlinear quantum optics on the microscale and nanoscale.

Critical Look at the Time--Energy Uncertainty Relations

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.

On non-linear Schrödinger equations for open quantum systems

Recently two generalized nonlinear Schr\"{o}dinger equations have been proposed by Chavanis [Eur. Phys. J. Plus 132 (2017) 286] by applying Nottale's theory of scale relativity relying on a fractal space-time to describe dissipation in quantum systems. Several existing nonlinear equations are then derived and discussed in this context leading to a continuity equation with an extra source/sink term which violates Ehrenfest theorem. An extension to describe stochastic dynamics is also carried out by including thermal fluctuations or noise of the environment. These two generalized nonlinear equations are analyzed within the Bohmian mechanics framework to describe the corresponding dissipative and stochastic dynamics in terms of quantum trajectories. Several applications of this second generalized equation which can be considered as a generalized Kostin equation have been carried out. The first application consists of the study of the position-momentum uncertainty principle in a dissiaptive dynamics. After, the so-called Brownian-Bohmian motion is investigated by calculating classical and quantum diffusion coefficients. And as a third example, transmission through a transient (time dependent) parabolic repeller is studied where the interesting phenomenon of early arrival is observed even in the stochastic dynamics although the magnitude of early arrival is reduced by friction.

Atom interferometers with weak-measurement path detectors and their quantum mechanical analysis**Project supported by the National Key Research and Development Program of China (Grant No. 2018YFA 0306200), the National Natural Science Foundation of China (Grant No. 11434017), and the Guangdong Innovative and Entrepreneurial Research Team Program, China (Grant No. 2016ZT06C594).

According to the orthodox interpretation of quantum physics, wave-particle duality (WPD) is the intrinsic property of all massive microscopic particles. All gedanken or realistic experiments based on atom interferometers (AI) have so far upheld the principle of WPD, either by the mechanism of the Heisenberg’s position-momentum uncertainty relation or by quantum entanglement. In this paper, we propose and make a systematic quantum mechanical analysis of several schemes of weak-measurement atom interferometer (WM-AI) and compare them with the historical schemes of strong-measurement atom interferometer (SM-AI), such as Einstein’s recoiling slit and Feynman’s light microscope. As the critical part of these WM-AI setups, a weak-measurement path detector (WM-PD) deliberately interacting with the atomic internal electronic quantum states is designed and used to probe the which-path information of the atom, while only inducing negligible perturbation of the atomic center-of-mass motion. Another instrument that is used to directly interact with the atomic center-of-mass while being insensitive to the internal electronic quantum states is used to monitor the atomic center-of-mass interference pattern. Two typical schemes of WM-PD are considered. The first is the micromaser-cavity path detector, which allows us to probe the spontaneously emitted microwave photon from the incoming Rydberg atom in its excited electronic state and record unanimously the which-path information of the atom. The second is the optical-lattice Bragg-grating path detector, which can split the incoming atom beam into two different directions as determined by the internal electronic state and thus encode the which-path information of the atom into the internal states of the atom. We have used standard quantum mechanics to analyze the evolution of the atomic center-of-mass and internal electronic state wave function by directly solving Schrödinger’s equation for the composite atom-electron-photon system in these WM-AIs . We have also compared our analysis with the theoretical and experimental studies that have been presented in the previous literature. The results show that the two sets of instruments can work separately, collectively, and without mutual exclusion to enable simultaneous observation of both wave and particle nature of the atoms to a much higher level than the historical SM-AIs, while avoiding degradation from Heisenberg’s uncertainty relation and quantum entanglement. We have further investigated the space–time evolution of the internal electronic quantum state, as well as the combined atom–detector system and identified the microscopic origin and role of quantum entanglement, as emphasized in numerous previous studies. Based on these physics insights and theoretical analyses, we have proposed several new WM-AI schemes that can help to elucidate the puzzling physics of the WPD of the atoms. The principle of WM-AI scheme and quantum mechanical analyses made in this work can be directly extended to examine the principle of WPD for other massive particles.

Dynamics of a deformed atom cavity field system in presence of a Kerr-like medium

In this paper, we investigate the interaction between a vee-type three-level atom and a single mode of the electromagnetic field in the presence of a nonlinear Kerr-like medium and an intensity-dependent coupling. We have elucidated the system by a nonlinear Hamiltonian constructed from the standard Jaynes–Cummings model by deforming the field operators and adding some nonlinear terms. Using the initial conditions that the atom is prepared in an excited state and the field is in a coherent state, the state vector of the entire system is determined analytically. The time evolution of nonclassical properties such as Mandel Q, quantum entanglement and position-momentum uncertainty relation (squeezing) of the field are investigated. The quasiprobability distributions are also computed for the resultant state. The effects of the detuning parameters, generalized Kerr medium and intensity-dependent coupling on the previous nonclassicality signs are analysed, in detail.

Entropic uncertainty relation and revival structure of quantum wave packets

We consider a wave packet in the one-dimensional infinite square well formed by a combination of ground state and the first few excited states, and identify three different methods to study its time development. The first method proceeds by monitoring the temporal evolution of the so-called autocorrelation function while the second one relies on the behavior of standard deviation based Heisenberg’s position-momentum uncertainty relation as a function of time. The third method of our interest makes use of the time variation of the Shannon entropy sum or the so-called entropic uncertainty relation. The chosen form of the wave packet allowed us to study its revivals, super-revivals and fractional revivals by using analytic methods, the inevitable numerical routine being invoked at the last stage of the game. We found that, in general, the uncertainty-based approaches furnish more detailed information on the revival structure of wave packets than those provided by the autocorrelation function. The results of the entropic uncertainty relation, however, appear to be more accurate than the corresponding predictions of the standard deviation based uncertainty relation.

Radial position–momentum uncertainties for the infinite spherical well and the Fisher entropy

We present the analytical expression of the product of the radial position and momentum uncertainties and for the infinite spherical well. We find a few interesting features. First, the uncertainty increases with the radius R and the quantum number n, the nth root of the spherical Bessel function, but finally arrives at a constant for a large n and decreases with the angular momentum number l. It is seen that the becomes imaginary, which arises from the fact that the moving particle is shifted to axis y suddenly from the original axis x as the quantum number n increases. Furthermore, the becomes zero when the n increases for a given l. This means that the particle is spiraling around a circle whose radius r < R changes between a varying radius and a constant but with an increasing radius r as the n increases. Finally, the particle moves around a circle with a maximum radius R. Second, the relative dispersion is independent of the radius R, and it increases with the quantum number n but decreases with the quantum number l. Third, the radial momentum uncertainty decreases with the radius R and increases with the quantum numbers l > 0 and n. We notice that there exists a turning point for the uncertainty when l = 1 and n > 1. This also leads to the product . Fourth, the product is independent of the radius R and increases with the quantum numbers l > 0 and n. Finally, we obtain the analytical expression of the Fisher entropy and notice that it decreases with the radius R but increases with the quantum numbers l and n. We also find that the Cramer–Rao uncertainty relation is satisfied, and it increases with the quantum numbers l and n.

Dynamics of entanglement and uncertainty relation in coupled harmonic oscillator system: exact results

The dynamics of entanglement and uncertainty relation is explored by solving the time-dependent Schrodinger equation for coupled harmonic oscillator system analytically when the angular frequencies and coupling constant are arbitrarily time dependent. We derive the spectral and Schmidt decompositions for vacuum solution. Using the decompositions, we derive the analytical expressions for von Neumann and Renyi entropies. Making use of Wigner distribution function defined in phase space, we derive the time dependence of position–momentum uncertainty relations. To show the dynamics of entanglement and uncertainty relation graphically, we introduce two toy models and one realistic quenched model. While the dynamics can be conjectured by simple consideration in the toy models, the dynamics in the realistic quenched model is somewhat different from that in the toy models. In particular, the dynamics of entanglement exhibits similar pattern to dynamics of uncertainty parameter in the realistic quenched model.