Kirk T. McDonald
It has been suggested that the uncertainty in the measurement of a particle's momentum could be made arbitrarily small by observing the particle at two ends of an arbitrarily long flight path. However, consideration of the nature of the detection process shows that the usual limits of the uncertainty principle hold independent of the length of the flight path.
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We have examined the head-on collision of two electrons in approximation of coherent states. We have shown that the character of collision depends mainly on ratio of initial relative electron's momentum to momentum uncertainty of electrons. When this ratio becomes greater then 1, the Coulomb interaction does not practically influence the scattering.
We consider a Generalized Uncertainty Principle (GUP) framework which predicts a maximal uncertainty in momentum and minimal uncertainties both in position and momentum. We apply supersymmetric quantum mechanics method and the shape invariance condition to obtain the exact harmonic oscillator eigenvalues in this GUP context. We find the supersymmetric partner Hamiltonians and show that the harmonic oscillator belongs to a hierarchy of Hamiltonians with a shift in momentum representation and different masses and frequencies. We also study the effect of a uniform electric field on the harmonic o…
We present a hitherto unknown fundamental limitation to a basic measurement: that of the position of a quantum object when the total momentum of the object and apparatus is conserved. This result extends the famous Wigner-Araki-Yanase (WAY) theorem, and shows that accurate position measurements are only practically feasible if there is a large momentum uncertainty in the apparatus.
We characterize all position and momentum observables on R and on R^3. We study some of their operational properties and discuss their covariant joint observables.
We give two simple Kochen-Specker arguments for complementary between the position and momentum components of spinless particles, arguments that are identical in structure to those given by Peres and Mermin for spin-1/2 particles.
We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form $(ΔP)(ΔQ)\geq C\hbar$, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision $ΔQ$ of the position measurement, and the perturbation $ΔP$ of the conjugate variable introduced by such a measurement. We also determine the best constant $C$, which is attained for a unique p…
We report on a momentum-position realization of the EPR paradox using direct detection in the near and far fields of the photons emitted by collinear type-II phase-matched parametric downconversion. Using this approach we achieved a measured two-photon momentum-position variance product of $0.01\hbar^2$, which dramatically violates the bounds for the EPR and separability criteria.
Perivolaropoulos has recently proposed a position-deformed Heisenberg algebra which includes a maximal length [Phys.Rev.95, 103523 (2017)]. He has shown that this length scale naturally emerges in the context of cosmological particle's horizon or cosmic topology. Following this work, we propose a new deformed algebra and derive the maximal length uncertainty and its corresponding minimal momentum uncertainty from the generalized uncertainty principle. We also construct the corresponding Fourier transform and its inverse representations. Finally, we propose n-dimensional representation of this …
This paper presents the uncertainty related to position and momentum localization of a quantum state in terms of entropic uncertainty relations. We slightly improve the inequality given in [Phys. Rev. A 74, 052101 (2006)] and introduce a new entropic measure with corresponding uncertainty relation.
We calculate the uncertainties in the position and momentum of a particle in the 1D potential V(x)=F|x|, F>0, when the position and momentum operators obey the deformed commutation relation [x,p]=i\hbar(1+βp^2), β>0. As in the harmonic oscillator case, which was investigated in a previous publication, the Hamiltonian H_1 = p^2/2m + F|x| admits discrete positive energy eigenstates for both positive and negative mass. The uncertainties for the positive mass states behave as Δx ~ 1/Δp as in the β=0 limit. For the negative mass states, however, in contrast to the harmonic oscillator case where we …
We demonstrate experimentally that it is possible to prepare and detect photon pairs created by spontaneous parametric down-conversion which exhibit simultaneous position-momentum and polarization correlations that are adequate to implement a four-dimensional key distribution protocol.
Characterization of non-classical particle propagation using superpositions of position and momentum
研究通过Sagnac干涉仪生成位置和动量叠加的光子，利用不同测量设置获取横向分布，展示干涉效应如何导致位置和动量的局部化，并违反牛顿第一定律。
Wigner function is a quasi-distribution that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between Wigner function and appropriate measurements involving the system position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system position and momentum. We show that one can relate Wigner function to Kirkwood joint quasi-distribution of position and momentum, the latter, in turn, being a par…
Commutator-based entropic uncertainty relations in multidimensional position and momentum spaces are derived, twofold generalizing previous entropic uncertainty relations for one-mode states. They provide optimal lower bounds and imply the multidimensional variance-based uncertainty principle. The article concludes with an open conjecture.
We study the minimum volume ellipsoid estimator associates to a cloud of points in phase space. Using as a natural measure of uncertainty the symplectic capacity of the covariance ellipsoid we find that classical uncertainties obey relations similar to those found in non-standard quantum mechanics.
It is shown that the Fourier transformation that relates position and momentum representations of quantum mechanics can be understood as a consequence of a symmetry principle that establishes the equivalence of being and becoming in the description of reality. There are however other transformation compatible with the same principle that could lead to different formalisms of quantum mechanics.
This paper has been withdrawn by the author, due to the fact that a similar paper (quant-ph/9904075) has been published.
For a quantum particle with a single degree of freedom, we derive preparational sum and product uncertainty relations satisfied by $N$ linear combinations of position and momentum observables. The state-independent bounds depend on their degree of incompatibility defined by the area of a parallelogram in an $N$-dimensional coefficient space. Maximal incompatibility occurs if the observables give rise to regular polygons in phase space. We also conjecture a Hirschman-type uncertainty relation for N observables linear in position and momentum, generalizing the original relation which lower-bound…
Position and momentum observables are considered and their correlation is studied for the simplest quantum system of a free particle moving in one dimension. The algebra and the eigenvalue problem for the correlation observable is presented and its possible relevance for the solution of the Pauli problem is analysed. The correlation provides a simple explanation of the shrinking and spreading of wave packets in an interpretation of quantum mechanics based in an ontology suggested by quantum field theory. Several properties and speculations concerning position-momentum correlations are mentione…
It seems that the problem of finding a suitable position operator for photon has been solved in a recently published work which is based on a new commutation relation between position and momentum operators of massless particles[1]. Although the authors of [1] have presented a classical reasoning in favour of the new commutator,here we are going to find this new commutator based on a quantum mechanical argument.
We illustrate the use of the statistical method of moments for determining the position and momentum distributions of a quantum object from the statistics of a single measurement. The method is used for three different, though related, models; the sequential measurement model, the Arthurs-Kelly model and the eight-port homodyne detection model. In each case, the method of moments gives the position and momentum distribution for a large class of initial states, the relevant condition being the exponential boundedness of the distributions.
Controlling and measuring a superposition of position and momentum
本文通过狭缝、透镜和干涉仪制备光子的位置与动量叠加态，实验观测到位置与动量间的量子干涉，并验证其概率结果比经典下界低5.9%。
We show that it is, in principle, possible to perform local realism violating experiments of the Hardy type in which only position and momentum measurements are made on two particles emanating from a common source. In the optical domain, homodyne detection of the in-phase and out-of-phase amplitude components of an electromagnetic field is analogous to position and momentum measurement. Hence, local realism violations of the Hardy type are possible in optical systems employing only homodyne detection.
We analyse the correlation properties of light in non-linear wave-guides and transmission lines, predict the position-momentum realization of EPR paradox for photon pairs in Kerr-type non-linear photonic circuits, and we show how two-photon entangled states can be generated and detected.
We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by [x,p]=i\hbar(1+βp^2). This deformed commutation relation leads to the minimal length uncertainty relation Δx > (\hbar/2)(1/Δp +βΔp), which implies that Δx ~ 1/Δp at small Δp while Δx ~ Δp at large Δp. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m>0), derived in Ref. [1], only populate the Δx ~ 1/Δp branch.…
Position-Momenta Uncertainties in Classical Systems
研究设计了一种热浴以维持角动量守恒，发现经典粒子的位置-动量不确定性与平均角动量成正比，且在中心势中比例常数为1/2。
In this paper, we consider an interpretation of the Husimi function as the probability distribution of a successive measurement, which is clearly separated into measurements of the position and the momentum. We also show this successive measurement can be easily realized in the situation close to the experiment, and this measurement corresponds to one of Aharanov's weak measurement.
It is suggested that a measurement of the products of photoemission by alkali atoms excited after extraction from a trap, might, using the EPR strategy, show a significant violation of the momentum-position uncertainty relation. If this failed, as is quite likely, possible causes, such as retroactive propagation of influences and retrodiction failure, could be tested on the proposed apparatus.
We propose a scheme for preparing an EPR state in position and momentum of a pair of distantly-separated trapped atoms. The scheme utilizes the entangled light fields output from a nondegenerate optical parametric amplifier. Quantum state exchange between these fields and the motional states of the trapped atoms is accomplished via interactions in cavity QED.
A very simple illustration of the Bell-Kochen-Specker contradiction is presented using continuous observables in infinite dimensional Hilbert space. It is shown that the assumption of the \emph{existence} of putative values for position and momentum observables for one single particle is incompatible with quantum mechanics.
The uncertainty relation for angle and angular momentum has a lower bound which depends on the form of the state. Surprisingly, this lower bound can be very large. We derive the states which have the lowest possible uncertainty product for a given uncertainty in the angle or in the angular momentum. We show that, if the given angle uncertainty is close to its maximum value, the lowest possible uncertainty product tends to infinity.
Although non-commutativity of a certain set of quantum operators (e.g., creation/annihilation operators and Pauli spin operators) has been shown experimentally in recent years, the commu- tation relation for the position and the momentum operators has not been directly demonstrated to date. In this paper, we propose and analyze an experimental scheme for directly observing the non-commutativity of the position and the momentum operators using single-photon quantum in- terference. While the scheme is studied for the single-photon state as the input quantum state, the analysis applies equally to…
It is shown that, given a reasonable continuity assumption regarding possessed values, it is possible to construct a Kochen-Specker obstruction for any coordinate and its conjugate momentum, demonstrating that at most one of these two quantities can have a noncontextual value.
Using a position operator obtained for spin 1 particles by the present author and Wigner, we obtain a quantum relativistic result for the hidden momentum force experienced by particles with structure. In particular, our result applies to the hidden magnetic forces manifest in some problems of electromagnetism. We also discuss spin and orbital angular momentum operators, as well as Wigner rotation.
Multi-parameter estimation is necessary for force sensing due to simultaneous and nontrivial small changes of position and momentum. Designing quantum probes that allow simultaneous estimation of all parameters is therefore an important task. The optimal methods for estimation of the conjugate changes of position and momentum of quantum harmonic oscillator employ probes in entangled or quantum non-Gaussian states. We show that the same results can be obtained in a significantly more feasible fashion by employing independent sets of differently squeezed Gaussian states classically correlated to…
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.
In this paper, an experiment of quantum diffraction of position-momentum entangled photons from a straight sharp edge is presented. Path of a single photon of an entangled pair is partially blocked by a sharp edge whereas the other photon is detected at a stationary location without revealing the which-path information of the other photon. Quantum diffraction pattern of the sharp edge is revealed only in the correlated conditional detection of spatially separated photons and no diffraction pattern is formed in local detections of individual photons. Theoretical analysis of the quantum diffract…
Quantum Cryptography Using Momentum and Position Variables in a Simple Optical Arrangement
本文提出在由激光器、狭缝和透镜构成的简单光学装置中，利用位置与动量量子态实现BB84协议变体，从而验证了该量子密钥分发方案的可行性。
In a recent, modified double-pinhole diffraction experiment the existence of an interference pattern was established indirectly along with a near-perfect imaging of the double pinhole. Our theoretical analysis shows that the experiment constitutes a preparation of a quantum state that is, to a good approximation, a joint eigenstate of commuting functions of position and momentum. Gaining information about the momentum distribution by means of the particular experimental setup is thus possible with negligible impact on the position distribution. Furthermore, we construct explicitly a class of s…
We point out two interesting features of position-momentum uncertainty product: $U=Δx Δp$. We show that two special (non-differentiable) eigenstates of the Schr{ö}dinger operator with the Dirac Delta potential $[V(x)=-V_0 δ(x)],V_0>0$, also satisfy the Heisenberg's uncertainty principle by yielding $U> \frac{\hbar}{2}$. One of these eigenstates is a zero-energy and zero-curvature bound state.
We propose a trade-off between the Lipschitz constants of the position and momentum probability distributions for arbitrary quantum states. We refer to the trade-off as a quantum reciprocity relation. The Lipschitz constant of a function may be considered to quantify the extent of fluctuations of that function, and is, in general, independent of its spread. The spreads of the position and momentum distributions are used to obtain the celebrated Heisenberg quantum uncertainty relations. We find that the product of the Lipschitz constants of position and momentum probability distributions is bou…
We consider two entropic uncertainty relations of position and momentum recently discussed in literature. By a suitable rescaling of one of them, we obtain a smooth interpolation of both for high-resolution and low-resolution measurements respectively. Because our interpolation has never been mentioned in literature before, we propose it as a candidate for an improved entropic uncertainty relation of position and momentum. Up to now, the author has neither been able to falsify nor prove the new inequality. In our opinion it is a challenge to do either one.
Localised solution of stochastic Schrödinger equation describing simultaneous continuous measurement of position and momentum
本文提出描述同时连续测量位置与动量的随机 Schrödinger 方程，求解得到满足不确定性原理的局域化状态，表现为中心在相空间随机运动的固定 Gaussian 波包，并证明其局域稳定性。
Monochromatic light is prepared by a single slit of spatial width Δx. The diffraction pattern is considered to obtain the corresponding probability density of the momentum. Let P be the probability weight to measure a momentum in a finite window of width Δp. For small Δp the asymptotic law P ~ ΔxΔp/h is verified, while h is Planck's constant of action.
The measurement of the position-momentum EPR correlations of a two-photon state is important for many quantum information applications ranging from quantum key distribution to coincidence imaging. However, all the existing techniques for measuring the position-momentum EPR correlations involve coincidence detection and thus suffer from issues that result in less accurate measurements. In this letter, we propose and demonstrate an experimental scheme that does not require coincidence detection for measuring the EPR correlations. Our technique works for two-photon states that are pure, irrespect…
The position-momentum uncertainty-like inequality based on moments of arbitrary order for d-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the Renyi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in d-dimensional physics: the hydrogenic and oscillator-like systems.
We study a possible realization of the position- and momentum-correlated atomic pairs that are confined to adjacent sites of two mutually shifted optical lattices and are entangled via laser-induced dipole-dipole interactions. The Einstein-Podolsky-Rosen (EPR) ``paradox'' [Phys. Rev. 47, 777 (1935)] with translational variables is then modified by lattice-diffraction effects. This ``paradox'' can be verified to a high degree of accuracy in this scheme.
We present a treatment of the double-slit interference of matter-waves represented by Gaussian wavepackets. The interference pattern is modelled with Green's function propagator which emphasizes the coordinate correlations and phases. We explore the connection between phases and position-momentum correlations in the intensity, visibility and predictability of the wavepackets interference. This formulation will indicate some aspects that can be useful for theoretical and experimental treatment of particles, atoms or molecules interferometry.
In this study, the information-theoretic measures in both the position and momentum spaces for the pseudoharmonic potential using Fisher information, Shannon entropy, Renyi entropy, Tsallis entropy and Onicescu information energy are investigated analytically and numerically. The results obtained are applied to some diatomic molecules. The Renyi and Tsallis entropies are analytically obtained in position space using Srivastava-Niukkanen linearization formula in terms of the Lauricella hypergeometric function. Also they are obtained in the momentum space in terms of the multivariate Bell polyno…
Cooling methods and particle slowers as well as accelerators are basic tools for fundamental research and applications in different fields and systems. We put forward a generic mechanism to scale the momentum of a particle, regardless of its initial position and momentum, by means of a transient harmonic potential. The design of the time-dependent frequency makes use of a linear invariant and inverse techniques drawn from ''shortcuts to adiabaticity''. The timing of the process may be decided beforehand and its influence on the system evolution and final features is analyzed. We address quantu…
We investigate the utility of Einstein-Podolsky-Rosen correlations of the position and momentum of photon pairs from parametric down-conversion in the implementation of a secure quantum key distribution protocol. We show that security is guaranteed by the entanglement between downconverted pairs, and can be checked by either direct comparison of Alice and Bob's measurement results or evaluation of an inequality of the sort proposed by Mancini et al. (Phys. Rev. Lett. 88, 120401 (2002)).
We consider particles prepared by the von Neumann-Lüders projection. For those particles the standard deviation of the momentum is discussed. We show that infinite standard deviations are not exceptions but rather typical. A necessary and sufficient condition for finite standard deviations is given. Finally, a new uncertainty relation is derived and it is shown that the latter cannot be improved.
Recently it was shown in [New J. Phys. 8, 330 (2006)] that the three dimensional Heisenberg uncertainty principle might be signifficantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a new lower bound which involves the mean value and the variance of the $\hat{L}^2$ operator.
In this paper, we study the dynamical properties of two coupled quantum harmonic oscillators coupled with bosonic non-Markovian environment both in position and momentum. We deduce the exact analytical master equation using Quantum State Diffusion method and give the quantum trajectory description when the control is added to the system by applying interaction between two harmonic oscillators. With numerical simulation, we compare the evolution of entanglement under different controlling effects. At last, we use nonlinear Quantum State Diffusion method to strengthen our above results by gettin…
We investigate the time evolution of momentum and position uncertainties for wave packets of arbitrary shape in at most quadratic potentials. We consider all possible cases of potentials and initial conditions. Doing so we see that the mixed uncertainty and for Gaussian wave packets moreover the chirp are convenient and important tools to identify the characteristic features of uncertainty dynamics. Special attention is given to the spreading and narrowing of wave packets.
The distribution of measured values for maximally accurate, unbiased simultaneous measurements of position and momentum is investigated. It is shown, that if the measurement is retrodictively optimal, then the distribution of results is given by the initial state Husimi function (or Q-representation). If the measurement is predictively optimal, then the distribution of results is related to the final state anti-Husimi function (or P-representation). The significance of this universal property for the interpretation of the Husimi function is discussed.
The aim of Part II of this paper is to try to describe wave functions on q-deformed versions of position and momentum space. This task is done within the framework developed in Part I of the paper. In order to make Part II self-contained the most important results of Part I are reviewed. Then it is shown that q-deformed exponentials and q-deformed delta functions play the role of momentum and position eigenfunctions, respectively. Their completeness and orthonormality relations are derived. For both bases of eigenfunctions matrix elements of position and momentum operators are calculated. A q-…
Optimal simultaneous control of position and momentum can be achieved by maximizing the probabilities of finding their experimentally observed values within two well-defined intervals. The assumption that particles move along straight lines in free space can then be tested by deriving a lower limit for the probability of finding the particle in a corresponding spatial interval at any intermediate time t. Here, it is shown that this lower limit can be violated by quantum superpositions of states confined within the respective position and momentum intervals. These violations of the particle pro…
Inspired by an old idea of von Neumann, we seek a pair of commuting operators X,P which are, in a specific sense, "close" to the canonical non-commuting position and momentum operators, x,p. The construction of such operators is related to the problem of finding complete sets of orthonormal phase space localized states, a problem severely constrained by the Balian-Low theorem. Here these constraints are avoided by restricting attention to situations in which the density matrix is reasonably decohered (i.e., spread out in phase space). Commuting position and momentum operators are argued to be …
A challenge of modern physics is to investigate the quantum behavior of a bulk material object, for instance a mechanical oscillator. We have earlier demonstrated that by coupling a mechanical oscillator to the energy levels of embedded rare-earth ion dopants, it is possible to prepare such a resonator in a low phonon number state. Here, we describe how to extend this protocol in order to prepare momentum- and position squeezed states, and we analyze how the obtainable degree of squeezing depends on the initial conditions and on the coupling of the oscillator to its thermal environment.
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