Position-momentum Uncertainty Research Articles

Position-momentum entropic uncertainty relation and complementarity in single-slit and double-slit experiments

The exact analytical values of the position and momentum information entropies are calculated for the single-slit and double-slit diffraction experiments. In both cases, the product of the exponentials of the entropies is strictly greater than the lower bound $\ensuremath{\pi}e\ensuremath{\Elzxh}$ given by the optimal entropic uncertainty relation for position and momentum, which implies that the single-slit and double-slit configurations are not minimum-uncertainty states. The results obtained show that the position-momentum entropic uncertainty relation provides a rigorous quantitative expression for the uncertainty principle in these experiments, unlike the Heisenberg inequality for standard deviations. However, it is also shown that, in the double-slit experiment, wave-particle duality cannot be derived from the entropic uncertainty relation.

Time and energy operators in quantum mechanics

Starting from a time operator, the form of the so-called energy operator that is conjugate to the time operator is derived in time representation by analyzing the properties of time translation. This analysis also establishes the commutator between the time and the energy operators. It is seen from the analysis that the energy operator has nothing to do conceptually with the Hamiltonian operator of a system, so that the time operator is not conjugate to the Hamiltonian. Furthermore, it is shown that the Hermiticity of the energy operator requires introducing time-integrate inner product. The time-energy commutator and the time-integral inner product put the time-energy uncertainty relation on the same footing as the position-momentum uncertainty relation.

The role of atomic recoil in micromaser ‘Welcher Weg’ detection

In recent work involving micromaser ‘which-path’ detectors it was shown that the loss of interference in a two slit experiment is based completely on complementarity. The position-momentum uncertainty relation plays no essential role. Storey et al. have argued that the loss of interference is due to atomic recoil originating in the emission and absorption of microwave photons. This was refuted by Englert et al., who showed that the kicks due to these microwave photons were, to a good approximation, negligible. But the argument continues in a recent Nature article, and that is the reason for this detailed calculation. The result proves that the correlation between detector and measured system, i.e., between micromaser cavities and atom, is the only reason for the loss of interference. Transverse momentum kicks play no role.

Path detection and the uncertainty principle

QUANTUM mechanics predicts that any detector capable of determining the path taken by a particle through a double slit will destroy the interference. This follows from the principle of complementarity formulated by Niels Bohr: simultaneous observation of wave and particle behaviour is prohibited. But such a description makes no reference to the physical mechanism by which the interference is lost. In the best studied welcher Weg (‘which path’) detection schemes1,2, interference is lost by the transfer of momentum to the particle whose path is being determined, the extent of momentum transfer satisfying the position–momentum uncertainty relation. This has prompted the question as to whether complementarity is always enforced in welcher Weg schemes by momentum transfer. Scully et al3. have recently responded in the negative, suggesting that complementarity must be accepted as an independent component of quantum mechanics, rather than as simply a consequence of the uncertainty principle. But we show here that, in any path detection scheme involving a fixed double slit, the amount of momentum transferred to the particle by a perfectly efficient detector (one capable of resolving the path unambiguously) is related to the slit separation in accordance with the uncertainty principle. If less momentum than this is transferred, interference is not completely destroyed and the path detector cannot be perfectly efficient.

Force of radiation on a two-level atom near a small dielectric.

We calculate the force of radiation on a two-level atom near a small dielectric such as a nanometer-scale fiber probe. Our results show that a spontaneously emitted photon may have large momentum by the position-momentum uncertainty principle.

On the maximal acceleration and the maximal energy loss

It has been shown that the maximal acceleration and the maximal energy loss follow from the minimum position-momentum uncertainty principle. It turns out that the maximal energy loss goes as the square of the maximal acceleration.

On the asymptotic solutions of Belavkin's stochastic wave equation

A posterior dynamics of an observed quantum particle in a quadratic potential is studied. It is proved that Belavkin's stochastic wave equation for the quantum particle with continuously observed 1) position, 2) momentum, 3) position and momentum is uniquely relaxing to the corresponding Gaussian wave packet for any initial square-integrable wave function. A comparison of the asymptotic position-momentum uncertainties of these measurements is given.

Quantum optical tests of complementarity

Simultaneous observation of wave and particle behaviour is prohibited, usually by the position–momentum uncertainty relation. New detectors, constructed with the aid of modern quantum optics, provide a way around this obstacle in atom interferometers, and allow the investigation of other mechanisms that enforce complementarity.

Quantum mechanics made transparent

This article is a ‘‘sampler,’’ which shows how quantum mechanics may be presented to students in a way that makes apparent how natural quantum mechanics is as a description of the world. The mathematical machinery of Hilbert space, the idea of representing observables by operators, the Schrödinger equation, and the position-momentum uncertainty relation all follow from natural assumptions that students can readily accept. The basic ideas of quantum mechanics are developed from intuitive first principles to the point where one can connect with more traditional treatments of quantum mechanics.

Gravity from strings

Superstrings being consistent theories that include gravity have to produce classical gravity within limits provided by unambiguous quantum effects. Through the study of a hard scattering process — clearly provided by string theory — it is shown that infinite genus calculations give indeed rise to a classical limit, when “large” distances are explored, as well as quantum effects. These are dominant at distances of the order of the string length (related to the Planck langth) but may extend much beyond that region hinting indeed to gravitational instabilities. Below the string length even space-time loses meaning as a classical concept. A new position-momentum uncertainty relation is produced that assigns to the string length the meaning of a minimum observable distance.

Maximal acceleration and the time-energy uncertainty relation

In 1981, Caianiello showed that maximal acceleration arises when quantum considerations, namely the position-momentum uncertainty relations, are incorporated into the geometry of a particle in eight-dimensional phase space. The time-energy uncertainty relation was later used by Caianiello in an alternative derivation of maximal acceleration. The alternative derivation, given in 1984, was subsequently criticized by Sharma and Srirankanathan. This criticism is shown here to be unfounded. Caianiello’s 1984 derivation is also analyzed. It is found that, while maximal acceleration follows from an equation due to Landau and Lifshitz, this equation cannot be treated as a consequence of the time-energy uncertainty relation.

A possible evidence for a violation of Heisenberg's position-momentum uncertainty relation

In the light of recent works by Fitchard and Mackinnon, experimental evidence by Nuttall and Gallon is interpreted as an indication of a violation of Heisenberg's uncertainty relation. Possible loopholes and their avoidance in future experiments are discussed.

Can spacetime be probed below the string size?

Strings may be explored, through a scattering process at planckian energies, down to distances of the order of the string size. We show that this is possible if the energy is not so extreme to cause a gravitational instability and when the scattering angle approaches some critical value from below. Above this angle, the distance starts increasing, thus departing from the usual position-momentum uncertainty relation, and in no instance is the resolution smaller than the string length. This suggests that below the Planck scale the very concept of space-time changes meaning.

Numerical simulations in stochastic mechanics.

The stochastic differential equation of Nelson's stochastic mechanics is integrated numerically for several simple quantum systems. The calculations are performed with use of Helfand and Greenside's method and pseudorandom numbers. The resulting trajectories are analyzed both individually and collectively to yield insight into momentum, uncertainty principles, interference, tunneling, quantum chaos, and common models of diatomic molecules from the stochastic quantization point of view. In addition to confirming Shucker's momentum theorem, these simulations illustrate, within the context of stochastic mechanics, the position-momentum and time-energy uncertainty relations, the two-slit diffraction pattern, exponential decay of an unstable system, and the greater degree of anticorrelation in a valence-bond model as compared with a molecular-orbital model of ${\mathrm{H}}_{2}$. The attempt to find exponential divergence of initially nearby trajectories, potentially useful as a criterion for quantum chaos, in a periodically forced oscillator is inconclusive. A way of computing excited energies from the ground-state motion is presented. In all of these studies the use of particle trajectories allows a more insightful interpretation of physical phenomena than is possible within traditional wave mechanics.

Uncertainty relations in stochastic mechanics

Position–momentum uncertainty in Nelson’s stochastic mechanics [Phys. Rev. 150, 1079 (1966)] has previously been investigated by de la Peña-Auerbach and Cetto [Phys. Lett. A 39, 65 (1972)]. In this paper their result is generalized, and full equivalence between the uncertainty relations in stochastic mechanics and conventional quantum mechanics is established. Force–momentum uncertainty is also considered.

A Note on Quantum Theory, Complementarity, and Uncertainty

Uncertainty relations and complementarity of canonically conjugate position and momentum observables in quantum theory are discussed with respect to some general coupling properties of a function and its Fourier transform. The question of joint localization of a particle on bounded position and momentum value sets and the relevance of this question to the interpretation of position-momentum uncertainty relations is surveyed. In particular, it is argued that the Heisenberg interpretation of the uncertainty relations can consistently be carried through in a natural extension of the usual Hilbert space frame of the quantum theory.

Time-energy uncertainty relation and Lorentz covariance

The uncertainty relations applicable to space and time separations between the quarks in a hadron are discussed. It is pointed out that the time-energy uncertainty relation between the time and energy separations is the same as the relationship between the widths and lifetimes of unstable states. It is then shown that this relation can be combined with Heisenberg’s position-momentum uncertainty relation to give the uncertainty principle in a covariant form. It is pointed out that this effect manifests itself in Feynman’s parton picture.

Momentum from sample paths in stochastic mechanics

According to Shucker’s analysis, quantum-mechanical momentum can be read from the asymptotic behaviour of Nelson’s sample paths. Here we use the position-momentum uncertainty product as a measure of the rate of approach to the large time asymptotics.

Position-Momentum Uncertainty Relations in Stochastic Mechanics

A simple inequality relating the root mean square deviations of position and osmotic velocity for a diffusion process is presented and, in the framework of Nelson's stochastic mechanics, is related to the Heisenberg position-momentum uncertainty relations.

A modified set of Feynman postulates in quantum mechanics

Certain modifications, by way of improvement, are proposed for the Feynman postulates in quantum mechanics. These modifications incorporate a criterion for the applicability of the principle of superposition. It is shown that the modified postulates, together with certain assumptions regarding the trajectory of a particle, lead to an expression for the position-momentum uncertainty relationship which is broadly in agreement with the conventional expression. The time-energy uncertainty relationship is, however, found to have a likely place only in the relativistic theory. A criterion, in the form of a ratio involving the linear dimensions of the particle, is obtained for the validity of the classical mechanics approximation. The modified postulates are suggested to favor the statistical interpretation of quantum mechanics over the Copenhagen interpretation.