, Volume 23, Issue 7, pp 759–786 | Cite as

How Long Does a Quantum Particle or Wave Stay in a Given Region of Space?

  • S. Anantha RamakrishnaEmail author
  • Arun M. Jayannavar
General Article


The delay time associated with a scattering process is one of the most important dynamical aspects in quantum mechanics. A common measure of this is the Wigner delay time based on the group velocity description of a wave packet, which may easily indicate superluminal or even negative times of interaction that are unacceptable. Many other measures such as dwell times have been proposed, but also suffer from serious deficiencies, particularly for evanescent waves. One important way of realising timescales that are causally connected to the spatial region of interest relies on utilising the dynamical evolution of extra degrees of freedom, called quantum clocks, such as the precession of the spin of an electron in an applied magnetic field or the coherent decay or growth of light in an absorptive or amplifying medium placed within the region of interest. Here we provide a review the several approaches developed to answer the basic question “how much time does a quantum particle (or wave) spend in a specified region of space?” While a unique answer still evades us, important progress has been made in understanding the timescales and obtaining positive definite times of interaction by noting that all such clocks are affected by spurious scattering concomitant with the very clock potentials, however, weak they be, and by eliminating the spurious scattering.


Quantum clock phase velocity Wigner delay time Smith dwell time Larmour precession soujourn times quantum systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Suggested Reading

  1. [1]
    A F J Siegert, On the First Passage Time Probability Problem, Phys. Rev., 81, 617–23, 1951.CrossRefGoogle Scholar
  2. [2]
    V Ranjith and N Kumar, 1D Tight-Binding Models Render Quantum First Passage Time ‘Speakable’, Int. J. Theor. Phys., 54, 4204, 2015.CrossRefGoogle Scholar
  3. [3]
    J G Muga, A Ruschhaupt, A Campo (eds.), Time in Quantum Mechanics, Vol.2, Springer, Berlin, 2010.Google Scholar
  4. [4]
    R Landauer and Th Martin, Barrier Interaction Time in Tunneling, Rev. Mod. Phys., 66, 217, 1994.CrossRefGoogle Scholar
  5. [5]
    M Büttiker, Larmor Precession and the Traversal Time for Tunneling, Phys. Rev. B., 27, 6178–6188, 1983.CrossRefGoogle Scholar
  6. [6]
    S A Ramakrishna and N Kumar, Correcting the Quantum Clock: Conditional Sojourn Times, Europhys. Lett., 60, 491–497, 2002.CrossRefGoogle Scholar
  7. [7]
    M Born and E Wolf, Principles of Optics, 6th Ed., Pergammon Press, Oxford, 1989.Google Scholar
  8. [8]
    S A Ramakrishna and T M Grzegorczyk, Physics and Applications of Negative Refractive Index Materials, CRC Press, Boca Raton, 2009.Google Scholar
  9. [9]
    Z Dutton, N S Ginsberg, C Slowe, and L V Hau, Europhysics News, March–April Issue, 33–39, 2004; DOI Scholar
  10. [10]
    F T Smith, Lifetime Matrix in Collision Theory, Phys. Rev., 118, 349, 1960.CrossRefGoogle Scholar
  11. [11]
    L D Landau, V M Lischitz and L P Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed., Butterworth-Heinemann (Elsevier), New Delhi, 2005.Google Scholar
  12. [12]
    J Peatross, S A Glasgow and M Ware, Average Energy Flow of Optical Pulses in Dispersive Media, Phys. Rev. Lett., 84, 2370, 2000.CrossRefGoogle Scholar
  13. [13]
    L Nanda, A Basu and S A Ramakrishna, Delay Times and Detector Times for Optical Pulses Traversing Plasmas and Negative Refractive Media, Phys. Rev., 74, 036601, 2006.Google Scholar
  14. [14]
    T E Hartman, Tunneling of a Wave Packet, J. Appl.Phys., 33, 3427, 1962.CrossRefGoogle Scholar
  15. [15]
    H Salecker and E P Wigner, Quantum Limitations of the Measurement of Space-time Distances, Phys. Rev., 109, 571–577, 1958.CrossRefGoogle Scholar
  16. [16]
    M Büttiker and R Landauer, Traversal Time for Tunneling, Phys. Rev. Lett., Vol.49, p.1739, 1982.CrossRefGoogle Scholar
  17. [17]
    A M Jayannavar, A Note on Traversal Time for Tunneling, Pramana–J. Phys., Vol.29, p.341, 1987.CrossRefGoogle Scholar
  18. [18]
    M Büttiker in a private communication to N Kumar, 2000.Google Scholar
  19. [19]
    G Dayal and S A Ramakrishna, Design of Highly Absorbing Metamaterials for Infrared Frequencies, Opt. Express., Vol.20, pp.17503–17508, 2012.CrossRefGoogle Scholar
  20. [20]
    C Benjamin, A M Jayannavar, Wave Attenuation to Clock Sojourn Times, Solid State Commun., Vol.121, pp.591–595, 2002.CrossRefGoogle Scholar
  21. [21]
    N Kumar, Quantum First-passage Problem, Pramana–J. Phys., 25, 363, 1985.CrossRefGoogle Scholar
  22. [22]
    S Dhar, S Dasgupta and A Dhar, Quantum Time of Arrival Distribution in a Simple Lattice Model, J. Phys. A: Math. Theor., Vol.48, p.115304, 2015.CrossRefGoogle Scholar
  23. [23]
    S Dhar, S Dasgupta, A Dhar, and D Sen, Detection of a Quantum Particle on a Lattice Under Repeated Projective Measurements, Phys. Rev., Vol.91, p.062115, 2015.CrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of PhysicsIITKanpurIndia
  2. 2.Institute of PhysicsSachivalaya MargBhubaneswarIndia

Personalised recommendations