#### Alladi Ramakrishnan Hall

#### Quantum First-passage time rendered -speakable : A 1d curious case

#### Ranjith V

##### University of Cagliari, Cagliari, Italy

*"Classically, for any and every event happening in space-time, there exists -- at least in principle -- a 'first time' at which the event happens. However, the very idea of a First Passage Time (or even its probability density in time) has been an ill-posed problem in the domain of quantum mechanics. *

The reasons can be summarily seen in the fact that the quantum probabilities in general do not satisfy the Kolmogorov sum rule: the probabilities for entering and non-entering of Feynman paths into a given region of space-time do not in general add up to unity, much owing to the interference of alternative paths.

In the present work, it is pointed out that a special case exists (within the standard quantum framework), where the concept of first-passage-time becomes 'speakable'. This happens when there is one and only one available path (i.e., door-way) to mediate the (first) passage – no alternative path to interfere with. The 1d tight binding Hamiltonian systems fall under this special category. The associated first passage time distributions are obtained analytically using a recipe originally devised for classical (stochastic) mechanics by Schroedinger in 1915."

The talk is be based on the work done with Late. Prof Narendra Kumar at RRI , Bengaluru:

"1D Tight-Binding Models Render Quantum First Passage Time 'Speakable', V Ranjith and N Kumar, Int. J. Theor. Phys. 53.12, (2014).

References:

1) "Quantum first-passage problem", Pramana, 25.4: 363-367, (1985)

2) "Pitfalls of Path Integrals: Amplitudes for Spacetime Regions and the

Quantum Zeno Effect" , Halliwell, J.J., Yearsley, J.M. , Phys. Rev. D 86, 024016 , (2012)

3) "Extending Schrödinger’s first-passage-time probability to quantum mechanics", Lumpkin, O., Phys. Rev. A 51, 2758–61 (1995)"

4) "First-passage time: Lattice versus continuum" , Sharma, K., Kumar N. Phys. Rev. E 86, 032104 (2012).

Done