Abstract
Apart from serving as a parameter in describing the evolution of a system, time appears also as an observable property of a system in experiments where one measures ‘the time of occurrence’ of an event associated with the system. However, while the observables normally encountered in quantum theory (and characterized by self-adjoint operators or projection-valued measures) correspond to instantaneous measurements, a time of occurrence measurement involves continuous observations being performed on the system to monitor when the event occurs. It is argued that a time of occurrence observable should be represented by a positive-operator-valued measure on the interval over which the experiment is carried out. It is shown that while the requirement of time-translation invariance and the spectral condition rule out the possibility of a self-adjoint time operator (Pauli’s theorem), they do allow for time of occurrence observables to be represented by suitable positive-operator-valued measures. It is also shown that the uncertainty in the time of occurrence of an event satisfies the time-energy uncertainty relation as a consequence of the time-translation invariance, only if the time of occurrence experiment is performed on the entire time axis.
Similar content being viewed by others
References
Aharanov Y and Bohm D 1961Phys. Rev. 122 1649
Aharanov Y and Bohm D 1964Phys. Rev. B134 1417
Akhiezer N I and Glazman I M 1961Theory of linear operators in Hilbert space Vol. II (New York: Ungar)
Allock G R 1969Ann. Phys. 53 253
Amrein W O, Jauch J M and Sinha K B 1977Scattering theory in quantum mechanics (New York: Benjamin)
Bauer M and Mello P A 1978Ann. Phys. 111 38
Benioff P 1972J. Math. Phys. 13 1347
Bohr N 1928Nature (London) 121 580
Borchers H J 1967Commun. Math. Phys. 4 315
Cooper J L B 1947Ann. Math. 48 827
Davies E B 1969Commun. Math. Phys. 15 277
Davies E B 1975Helv. Phys. Acta 48 365
Davies E B 1976Quantum theory of open systems (New York: Academic Press)
Emch G G 1972Algebraic methods in statistical mechanics and quantum field theory (New York: Wiley Interscience)
Engelmann F and Fick E 1959Suppl. Nuovo. Cim. 12 63
Engelmann F and Fick E 1964Z. Phys. 178 551
Fock V A 1962Sov. Phys. JETP 15 784
Fock V A 1966Sov. Phys. Usp. 8 628
Foias C, Geher L and Nagy BSz 1960Acta Sci. Math. (Szeged) 21 78
Friedman C N 1976Ann. Phys. 98 87
Gnanapragasam B and Srinivas M D 1979Pramana 12 699
Haba Z and Novicki A A 1976Phys. Rev. D13 523
Hegerfeldt G C 1974Phys. Rev. D10 3320
Hegerfeldt G C and Ruijsenaars 1980 Remarks on causality localization and spreading of wave packets (Princeton Univ. Preprint)
Heisenberg W 1927Z. Phys. 43 172
Helstrom C W 1974Int. J. Theor. Phys. 11 357
Holevo A S 1978Rep. Math. Phys. 13 379
Houtappel R M F, VanDam H and Wigner E P 1965Rev. Mod. Phys. 37 595
Jammer M 1974The philosophy of quantum mechanics (New York: John Wiley)
Jauch J M 1968Foundations of quantum mechanics (Reading: Addison-Wesley)
Kijowski J 1974Rep. Math. Phys. 6 361
Kraus K 1980 A note on zeno’s paradox in quantum theory (Univ. of Texas at Austin Preprint).
Landau L D and Peirls R F 1931Z. Phys. 69 56
Mackey G W 1949Duke Math. J. 16 313
Mandelstam L and Tamm I G 1945J. Phys. USSR 9 249
Misra B and Sudarshan E C G 1977J. Math. Phys. 18 756
Mollow B R 1968Phys. Rev. 168 1896
Olkhovsky N S, Recami E and Gerasimchuk A I 1974Nuovo. Cim. A22 263
Paul H 1962Ann. Phys. 9 252
Pauli W 1933 Die allgemeinen Prinzipien der Wellenmechanik inHandbuch der Physik eds. Geiger H and Scheel K Vol. 24 part 1 (Berlin: Springer Verlag)
Perez J F and Wilde I F 1977Phys. Rev. D16 315
Putnam C R 1967Commutation properties of Hilbert space operators and related topics (Berlin: Springer)
Razavy M 1967Ann. J. Phys. 35 955
Razavy M 1969Nuovo. Cim. B63 271
Scully M O and Lamb W E 1969Phys. Rev. 179 368
Skagerstam B S 1976Int. J. Theor. Phys. 15 213
Srinivas M D 1975J. Math. Phys. 16 1672
Srinivas M D and Davies E B 1981Opt. Acta (in press)
Streater R F and Wightman A S 1964PCT spin statistics and all that (New York: Benjamin)
Von Neumann J 1931Ann. Math. 104 570
Von Neumann J 1955Mathematical foundations of quantum mechanics (Princeton: University Press)
Wightman A S 1962Rev. Mod. Phys. 34 845
Wigner E P 1972 inAspects of quantum theory, eds A Salam and E P Wigner (London: Cambridge Univ. Press)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Srinivas, M.D., Vijayalakshmi, R. The ‘time of occurrence’ in quantum mechanics. Pramana - J. Phys. 16, 173–199 (1981). https://doi.org/10.1007/BF02848181
Issue Date:
DOI: https://doi.org/10.1007/BF02848181