"Time operator": the challenge persists

Mielnik, Bogdan; Torres-Vega, Gabino

doi:10.48550/arxiv.1112.4198

Cited by 6 publications

(8 citation statements)

References 40 publications

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“…The measurement of the arrival time of a quantum mechanical particle in a given detection region is a longstanding and fundamental problem in quantum mechanics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In spite of much effort, the construction of a time operator has been found to be controversial [18]. One possible approach that one could take to find the time of arrival is the following.…”

Section: Introductionmentioning

confidence: 99%

Detection of a quantum particle on a lattice under repeated projective measurements

et al. 2015

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We consider a quantum particle, moving on a lattice with a tight-binding Hamiltonian, which is subjected to measurements to detect it's arrival at a particular chosen set of sites. The projective measurements are made at regular time intervals τ , and we consider the evolution of the wave function till the time a detection occurs. We study the probabilities of its first detection at some time and conversely the probability of it not being detected (i.e., surviving) up to that time. We propose a general perturbative approach for understanding the dynamics which maps the evolution operator, consisting of unitary transformations followed by projections, to one described by a nonHermitian Hamiltonian. For some examples, of a particle moving on one and two-dimensional lattices with one or more detection sites, we use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results. A mean field model with hopping between all pairs of sites and detection at one site is solved exactly. For the one-and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle. Finally, we show an interesting and non-trivial connection between the dynamics of the particle in our model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites.

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Section: Introductionmentioning

confidence: 99%

Detection of a quantum particle on a lattice under repeated projective measurements

et al. 2015

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“…Yet, to restrict the problem to the 'left component' is not the same as to consider the 'right movers' (and vice versa). Moreover, the obtained probability distribution excluded the interference of both (left and right) components at the detection point against the basic principles of quantum theory (the difficulty discovered by Leavens [83,84,85,86,87,88,89]). In spite of subsequent works on POV measures and some new but still partial results [90,91,92], it turns obvious that the idea of time-operator has too many gaps to grant the time-energy uncertainty (except if some additional limitations of quantum measurements proved real [93]).…”

Section: The Quasistatic Approximationmentioning

confidence: 80%

Quantum operations: technical or fundamental challenge?

Mielnik¹

2013

J. Phys. A: Math. Theor.

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A class of unitary operations generated by idealized, semiclassical fields is studied. The operations implemented by sharp potential kicks are revisited and the possibility of performing them by softly varying external fields is examined. The possibility of using the ion traps as 'operation factories' transforming quantum states is discussed. The non-perturbative algorithms indicate that the results of abstract δ-pulses of oscillator potentials can become real. Some of them, if empirically achieved, could be essential to examine certain atypical quantum ideas. In particular, simple dynamical manipulations might contribute to the Aharonov-Bohm criticism of the time-energy uncertainty principle, and some others, to verify the existence of fundamental precision limits of the position measurements or the reality of 'non-commutative geometries'.

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“…Moreover, we refer to some quite recent papers ( [14], [15], [16]) and an old work by W. Pauli (Ref. [17]) on time-operator.…”

Section: Conclusion Prospects For Further Research and Some Remarksmentioning

confidence: 99%

“…B. Mielnik and G. Torres-Vega pose a possibility to construct such an operator using POV (Positive Operator Valued) measure (see Ref. [16]). J. Kijowski criticizes (see Ref.…”

Section: Conclusion Prospects For Further Research and Some Remarksmentioning

confidence: 99%