Momentum Uncertainty Research Articles

Probability density of relativistic spinless particles

In this paper, a new conserved current for Klein-Gordon equation is derived. It is shown, for $1+1$-dimensions, the first component of this current is non-negative and reduces to $|\phi|^2$ in non-relativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from $|\phi|^2$ only when the uncertainty in momentum is greater than $m_0c$.

Black Hole Thermodynamics and Generalized Uncertainty Principle with Higher Order Terms in Momentum Uncertainty

We study the modification of thermodynamic properties of Schwarzschild and Reissner-Nordström black hole in the framework of generalized uncertainty principle with correction terms up to fourth order in momentum uncertainty. The mass-temperature relation and the heat capacity for these black holes have been investigated. These have been used to obtain the critical and remnant masses. The entropy expression using this generalized uncertainty principle reveals the area law up to leading order logarithmic corrections and subleading corrections of the form 1/An. The mass output and radiation rate using Stefan-Boltzmann law have been computed which show deviations from the standard case and the case with the simplest form for the generalized uncertainty principle.

Currency Carry, Momentum, and US Monetary Policy Uncertainty

Returns to currency carry and momentum are compensations for the risk of global interest rate uncertainty (IRU), with risk exposures explaining 92% of cross-sectional return variations. Higher carry and momentum signal increase investor's expectations for future excess returns and stimulate more risk-taking, yet the global IRU adversely affects investor's risk-bearing capacity and triggers losses of carry and momentum strategies. Expected returns estimated from those risk exposures predict carry and momentum returns, with large economic and statistical significance. Consistently, I find the risk of global IRU predicts contemporaneous and subsequent worsening of intermediary's financial constraints, and it is also priced in momentum returns across other asset classes. The explanatory power is robust to using currency-level tests and is not driven by existing measures of uncertainty or risk aversion.

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.

Environment-induced uncertainties on moving mirrors in quantum critical theories via holography

Environment effects on a n-dimensional mirror from the strongly coupled d-dimensional quantum critical fields with a dynamic exponent z in weakly squeezed states are studied by the holographic approach. The dual description is a n+1-dimensional probe brane moving in the d+1-dimensional asymptotic Lifshitz geometry with gravitational wave perturbations. Using the holographic influence functional method, we find that the large coupling constant of the fields reduces the position uncertainty of the mirror, but enhances the momentum uncertainty. As such, the product of the position and momentum uncertainties is independent of the coupling constant. The proper choices of the phase of the squeezing parameter might reduce the uncertainties, nevertheless large values of its amplitude always lead to the larger uncertainties due to the fact that more quanta are excited as compared with the corresponding normal vacuum and thermal states. In the squeezed vacuum state, the position and momentum of the mirror gain maximum uncertainties from the field at the dynamic exponent z=n+2 when the same squeezed mode is considered. As for the squeezed thermal state, the contributions of thermal fluctuations to the uncertainties decrease as the temperature increases in the case 1<z<n+2, whereas for z>n+2 the contributions increase as the temperature increases. These results are in sharp contrast with those in the environments of the relativistic free field. Some possible observable effects are discussed.

Radial position-momentum uncertainties for the infinite circular well and Fisher entropy

We show how the product of the radial position and momentum uncertainties can be obtained analytically for the infinite circular well potential. Some interesting features are found. First, the uncertainty Δr increases with the radius R and the quantum number n, the n-th root of the Bessel function. The variation of the Δr is almost independent of the quantum number n for n>4 and it will arrive to a constant for a large n, say n>4. Second, we find that the relative dispersion Δr/〈r〉 is independent of the radius R. Moreover, the relative dispersion increases with the quantum number n but decreases with the azimuthal quantum number m. Third, the momentum uncertainty Δp decreases with the radius R and increases with the quantum numbers m>1 and n. Fourth, the product ΔrΔpr of the position-momentum uncertainty relations is independent of the radius R and increases with the quantum numbers m and n. Finally, we present the analytical expression for the Fisher entropy. Notice that the Fisher entropy decreases with the radius R and it increases with the quantum numbers m>0 and n. Also, we find that the Cramer–Rao uncertainty relation is satisfied and it increases with the quantum numbers m>0 and n, too.

Criterion for determining resolving power in the optical near field

The resolution limit, i.e., the lower limit to the size of a luminous object that can be determined accurately by ordinary optical microscopy, was determined over a century ago on the basis of the imprecision that is imparted to the details of an image by diffraction. Such criteria assume one is making observations at distances greater than the wavelength of the light emitted or scattered by the luminous object, i.e., in the so-called far field. Although near-field microscopy is an established imaging technique, the analogous resolution criterion pertinent to the near field, in which the distance between the observer or the observing instrument and the light emitter is smaller than the wavelength, has not yet been quantitatively enunciated. Here I propose two ansatze pertinent to optical resolution in the near field. The first, based on classical physics, concludes that the minimal lateral near-field resolution limit is equal to the (perpendicular) distance between the near-field probe and the object. The second is a quantum limitation based on the uncertainty principle, which shows that the momentum uncertainty arising from atomic vibrations results in a minimal resolvable size, which in most instances is of the order of 10 pm.

Coherence and squeezing along quantum trajectories

We perform a detailed analysis of the behavior of coherent and squeezed states undergoing time evolution. We calculate time dependence of expectation values of position and momentum in coherent and squeezed states (which can be interpreted as quantum trajectories in coherent and squeezed states) and examine how coherence and squeezing is affected during time development, calculating time dependence of position and momentum uncertainty. We focus our investigations on two quantum systems. First we consider quantum linear system with Hamiltonian quadratic in $q$ and $p$ variables. As the second system we consider the simplest quantum nonlinear system with Hamiltonian quartic in $q$ and $p$ variables. We calculate the explicit formulas for the time development of expectation values and uncertainties of position and momentum in an initial coherent state.

Solar system and binary pulsars tests of the minimal momentum uncertainty principle

In order to explore a minimal momentum uncertainty principle in Heisenberg's uncertainty principle, the minimal momentum uncertainty principle (MMUP) was proposed. The correction on the canonical commutation relation of MMUP can be parametrized by α and α′. In the macroscopic scale, MMUP could change bodies' orbital motion, which is different from the results of general relativity (GR). In the present work, we mainly focus on the influence of MMUP on the tests of the Solar System and the binary pulsars. In the Solar System, the perihelion shift of a planet has been remodeled under MMUP by the data of INPOP10a, EPM2011 and INPOP15a. In five systems of binary pulsars, the upper bounds of α and α′ are obtained by using the data of the secular periastron precessions. These five systems are PSR B1913+16, PSR B1534+12, PSR J1756-2251, PSR B2127+11C and PSR J0737-3039. Few studies about the tests have been done before and the effects of MMUP on these systems are investigated for the first time. They show that the upper bounds of α and α′ constrained by planets of the Solar System and the binary pulsars are , tighter than the previous result by at least 2 orders of magnitude. It is expected that the bounds on MMUP will be further improved by tremendous advances in techniques for deep space exploration and by new tracking data sets.

Thermodynamics of a Non-Stationary Black Hole Based on Generalized Uncertainty Principle

In the general theory of relativity (GTR), black holes are defined as objects with very strong gravitational fields even light can not escape. Therefore, according to GTR black hole can be viewed as a non-thermodynamic object. The worldview of a black hole began to change since Hawking involves quantum field theory to study black holes and found that black holes have temperatures that analogous to black body radiation. In the theory of quantum gravity there is a term of the minimum length of an object known as the Planck length that demands a revision of Heisenberg's uncertainty principle into a Generalized Uncertainty Principle (GUP). Based on the relationship between the momentum uncertainty and the characteristic energy of the photons emitted by a black hole, the temperature and entropy of the non-stationary black hole (Vaidya-Bonner black hole) were calculated. The non-stationary black hole was chosen because it more realistic than static black holes to describe radiation phenomena. Because the black hole is dynamic then thermodynamics studies are conducted on both black hole horizons: the apparent horizon and its event horizon. The results showed that the dominant correction term of the temperature and entropy of the Vaidya-Bonner black hole are logarithmic.

Stationary Phonon Squeezing by Optical Polaron Excitation.

We demonstrate that a stationary squeezed phonon state can be prepared by a pulsed optical excitation of a semiconductor quantum well. Unlike previously discussed scenarios for generating squeezed phonons, the corresponding uncertainties become stationary after the excitation and do not oscillate in time. The effect is caused by two-phonon correlations within the excited polaron. We demonstrate by quantum kinetic simulations and by a perturbation analysis that the energetically lowest polaron state comprises two-phonon correlations which, after the pulse, result in an uncertainty of the lattice momentum that is continuously lower than in the ground state of the semiconductor. The simulations show the dynamics of the polaron formation process and the resulting time-dependent lattice uncertainties.

Correction of harmonic motion and Kepler orbit based on the minimal momentum uncertainty

In this paper we consider the deformed Heisenberg uncertainty principle with the minimal uncertainty in momentum which is called a minimal momentum uncertainty principle (MMUP). We consider MMUP in D-dimension and its classical analogue. Using these we investigate the MMUP effect for the harmonic motion and Kepler orbit.

Weak values obtained from mass-energy equivalence

Quantum weak measurement, measuring some observable quantities within the selected subensemble of the entire quantum ensemble, can produce many interesting results such as the superluminal phenomena. An outcome of such a measurement is the weak value which has been applied to amplify some weak signals of quantum interactions in lots of previous references. Here, we apply the weak measurement to the system of relativistic cold atoms. According to mass-energy equivalence, the internal energy of an atom will contribute its rest mass and consequently the external momentum of center of mass. This implies a weak coupling between the internal and external degrees of freedom of atoms moving in the free space. After a duration of this coupling, a weak value can be obtained by post-selecting an internal state of atoms. We show that the weak value can change the momentum uncertainty of atoms and consequently help us to experimentally measure the weak effects arising from mass-energy equivalence.

Position–Momentum Uncertainty Relation for an Open Macroscopic Quantum System

In this study, we explore the validity of the original Heisenberg position- momentum uncertainty relation for a macroscopic harmonic oscillator interacting with environmental micro particles. Our results show that, in the quasi-classical situation, the original uncertainty relation does not hold when the number of particles in the environment is small. Nonetheless, increasing the environmental degrees of freedom resolves the violation in the region of our investigation.

Exact solutions and coherent states for a generalized potential

Exact solutions of the vibrational Schrödinger equation for a generalized potential energy function hbox {V(R)}=hbox {C}_{0}(mathrm{{R}-mathrm {R}}_{mathrm{e}})^{2}/[hbox {aR},+,(mathrm{{b}-mathrm {a}})hbox {R}_{mathrm{e}}]^{2} are obtained. It includes those of Dunham, Ogilvie and Simons–Parr–Finlan potentials as special cases corresponding to b = 1, a = 0, 1/2, 1, respectively. The analytical wave functions derived are useful to test the quality of numerical methods or to perform perturbative or variational calculations for the problems that cannot be solved exactly. Coherent states for generalized potential, which minimize the position–momentum uncertainty relation are also constructed.

A class of solutions under GUP for a harmonic oscillator and a particle in a box

The existence of a minimal observable length is the common prediction of different theories of quantum gravity such as string theory, loop quantum gravity and black hole physics. The ordinary Heisenberg uncertainty principle (HUP) is not compatible with this prediction, so the HUP was generalized to the Generalized Uncertainty Principle (GUP) in literature. The consequences of this generalization are the generalized commutation relations between operators and so the generalized Hamiltonian for systems. In this paper, we represent the solutions of a harmonic oscillator and a particle in a box in the sense of a generalized commutation relation. We show that using of the Hellmann-Feynman (HF) theorem leads to the approximate energy spectrum of a harmonic oscillator and the exact energy spectrum of a particle in a box. In these cases, we do not solve the corresponding generalized Schro¨ dinger equation directly, but the HF theorem exhibit the effect of GUP on the energy spectrum. Also we obtain the uncertainties in both position and momentum for a harmonic oscillator and derive the GUP.

Electromagnetic field fluctuations and uncertainty relation for general quantum superpositions of coherent states

Quantum statistical properties of general superpositions of coherent states of the type have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude and phase superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.

Quantum information entropies for the $$\ell $$ ℓ -state Pöschl–Teller-type potential

In this study, we obtained the position–momentum uncertainties and some uncertainty relations for the Poschl–Teller-type potential for any $$\ell $$ . The radial expectation values of $$r^{-2}$$ , $$r^{2}$$ and $$p^{2}$$ are obtained from which the Heisenberg Uncertainty principle holds for the potential model under consideration. The Fisher information is then obtained and it is observed that the Fisher-information-based uncertainty relation and the Cramer–Rao inequality hold for this even power potential. Some numerical and graphical results are displayed.

Uncertainty product of an out-of-equilibrium many-particle system

In the present work we show, analytically and numerically, that the variance of many-particle operators and their uncertainty product for an out-of-equilibrium Bose-Einstein condensate (BEC) can deviate from the outcome of the time-dependent Gross-Pitaevskii dynamics, even in the limit of infinite number of particles and at constant interaction parameter when the system becomes 100% condensed. We demonstrate our finding on the dynamics of the center-of-mass position--momentum uncertainty product of a freely expanding as well as of a trapped BEC. This time-dependent many-body phenomenon is explained by the existence of time-dependent correlations which manifest themselves in the system's reduced two-body density matrix used to evaluate the uncertainty product. Our work demonstrates that one has to use a many-body propagation theory to describe an out-of-equilibrium BEC, even in the infinite particle limit.

Thermodynamics of Black Holes and the Symmetric Generalized Uncertainty Principle

In this paper, we have investigated the thermodynamics of Schwarzschild black holes using the symmetric generalized uncertainty principle which contains correction terms involving momentum and position uncertainty. We obtain the mass-temperature relation and the heat capacity of the black hole using which we compute the critical and remnant masses. The entropy is found to satisfy the area law upto leading order corrections from the symmetric generalized uncertainty principle.