Using the quantum transition path time probability distribution we show that time averaging of weak values leads to unexpected results. We prove a weak value time-energy uncertainty principle and time-energy commutation relation. We also find that time averaging allows one to predict in advance the momentum of a particle at a post selected point in space with accuracy greater than the limit of ℏ/2 as dictated by the uncertainty principle. This comes at a cost—it is not possible at the same time to predict when the particle will arrive at the post selected point. A specific example is provided for one dimensional scattering from a square barrier.
Using the quantum transition path time probability distribution we show that time averaging of weak values leads to unexpected results. We prove a weak value time-energy uncertainty principle and time-energy commutation relation. We also find that time averaging allows one to predict in advance the momentum of a particle at a post selected point in space with accuracy greater than the limit of ℏ/2 as dictated by the uncertainty principle. This comes at a cost—it is not possible at the same time to predict when the particle will arrive at the post selected point. A specific example is provided for one dimensional scattering from a square barrier.
Time averaging of weak values using the quantum transition path time probability distribution enables us to establish a general uncertainty relation for the weak values of two not necessarily Hermitian operators. This new relation is a weak value analog of the Schr\"odinger strong value uncertainty relation. It leads to the conclusion that it is possible to determine with high accuracy the simultaneous mean weak values of noncommuting operators by judicious choice of the pre- and postselected states even when the postselected state is not an eigenfunction of one of the respective operators. When the time fluctuations of the two weak values are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision even though their corresponding operators do not commute. To exemplify these properties we consider specific weak value uncertainty relations for the time-energy, coordinate-momentum, and coordinate--kinetic-energy pairs. In addition we analyze spin operators and the Stern-Gerlach experiment in weak and strong inhomogeneous magnetic fields. This classic case leads to anomalous spin values when the field is weak. The weak value uncertainty relation implies that anomalous spin values are associated with large variances so that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in scattering experiments.
A quantum transition can be seen as a result of interference between various pathways (e.g., Feynman paths), which can be labeled by a variable f. An attempt to determine the value of f without destroying the coherence between the pathways produces a weak value of f. We show f to be an average obtained with an amplitude distribution which can, in general, take negative values, which, in accordance with the uncertainty principle, need not contain information about the actual range of f which contributes to the transition. It is also demonstrated that the moments of such alternating distributions have a number of unusual properties which may lead to a misinterpretation of the weak-measurement results. We provide a detailed analysis of weak measurements with and without post-selection. Examples include the double-slit diffraction experiment, weak von Neumann and von Neumann--like measurements, traversal time for an elastic collision, phase time, and local angular momentum.
Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.
It is often said that measuring a system's position must disturb the complementary property, momentum, by some minimum amount due to the Heisenberg uncertainty principle. Using a 'weak-measurement', this disturbance can be reduced. One might expect this comes at the cost of also reducing the measurement's precision. However, it was recently demonstrated that a sequence consisting of a weak position measurement followed by a regular momentum measurement can probe a quantum system at a single point, with zero width, in position-momentum space. Here, we study this 'joint weak-measurement' and reconcile its compatibility with the uncertainty principle. While a single trial probes the system with a resolution that can saturate Heisenberg's limit, we show that averaging over many trials can be used to surpass this limit. The weak-measurement does not trade away precision, but rather another type of uncertainty called 'predictability' which quantifies the certainty of retrodicting the measurement's outcome.
Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.
It is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic. Here, it is shown that the problem is due to a confusion between the position coordinates of a point particle (a material system) and the coordinates of a point in space: The time coordinate should be put on a par with the space coordinates, not with the canonical position coordinates of a material system. Whereas quantum mechanics incorporates a Heisenberg uncertainty relation between the canonical position coordinates and their conjugate momenta, there is no reason why a Heisenberg relation should hold between the space coordinates and the canonical momenta, or between the time coordinate and the energy of the system. However, uncertainty relations of a different kind exist between the space coordinates and the total momentum of the system and between the time coordinate and the total energy. These relations are completely similar and may be taken together to form a relativistically covariant set of uncertainty relations. The relation between the time coordinate and the energy implies the well-known relation between the lifetime of a state and its energy spread.