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Averages of observables on Gamow states

Year 2022, , 51 - 65, 30.06.2022

Manuel Gadella , Carlos San Mıllan

https://doi.org/10.33769/aupse.1100013

Abstract

We propose a formulation of Gamow states, which is the part of unstable quantum states that decays exponentially, with two advantages in relation with the usual formulation of the same concept using Gamow vectors. The first advantage is that this formulation shows that Gamow states cannot be pure states, so that they may have a non-zero entropy. The second is the
possibility of correctly defining averages of observables on Gamow states.

Keywords

Quantum resonances Gamow states algebraic formulation of states and observables

Supporting Institution

Ministry of Science of Spain

Project Number

PID2020-113406GB-I00

References

  • Bohm, A.,Quantum Mechanics: Foundations and Applications, 3rd Ed. Springer Verlag, New York and Berlin, 2001.
  • Newton, R. G., Scattering Theory of Waves and Particles, 2rd Ed. Springer Verlag, Berlin and Heidelberg, 1982.
  • Nussenzveig, H. M., Causality and Dispersion Relations, Academic, New York and London, 1972.
  • Kukulin, V. I., Krasnopolski, V. M., Horacek, J., Theory of Resonances, Principles and Applications, Academia, Prag, 1989.
  • Fonda, L., Ghirardi, G. C., Rimini, A ., Decay theory of unstable quantum systems. Rep Progr. Phys., 41 (1978), 587-631, https://doi.org/10.1088/0034-4885/41/4/003.
  • Khalfin, L. A., Contribution to the decay theory of a quasi stationary state, Sov. Phys., JETP USSR, 6 (6) (1958), 1053-1063.
  • Fischer, M. C., Gutierrez-Medina, B., Raizen, M. G., Observation of the quantum Zeno and anti-Zeno effects in an unstable system, Phys. Rev. Lett., 87 (4) (2001), 040402, https://doi.org/10.1103/PhysRevLett.87.040402.
  • Rothe, C., Hintschich, S. I., Monkman, A. P., Violation of the exponential decay law at long times, Phys. Rev. Lett., 96 (16) (2006), 163601, https://doi.org/10.1103/PhysRevLett.96.163601.
  • Nakanishi, N., A note on the physical state of unstable particles, Progr. Theor. Phys., 21 (1) (1959), 216-217, https://doi.org/10.1143/PTP.21.216.
  • Albeverio, S., Kurasov, P., Singular Perturbations of Differential Operators, Lecture Note Series, vol. 271, Cambridge, UK, London Mathematical society.
  • Losada, M., Fortin, S., Gadella, M. Holik, F., Dynamics of algebras in quantum unstable systems, Int. J. Mod. Phys. A, 33 (2018), 1850109, https://doi.org/10.1142/S0217751X18501099.
  • Gelfand, I. M., Shilov, G. E., Generalized Functions, Vol. II, Academic Press, New York, 1964.
  • Horvath, J., Topological Vector Spaces and Distributions, Addison-Wesley, London, 1966.
  • Bohm, A., Rigged Hilbert space and the mathematical description of physical systems, Boulder Lecture Notes in Theoretical Physics, Vol. 9A, Gordon and Breach Science Publishers, New York, 1967, 255-317.
  • Bohm, A., The Rigged Hilbert Space and Quantum Mechanics, Lecture Notes in Physics 78, Springer, New York, 1978.
  • Roberts, J. E., Rigged Hilbert spaces in quantum mechanics, Commun. Math. Phys., 2 (1966), 98-119, https://doi.org/10.1007/BF01645448.
  • Antoine, J. P., Dirac formalism and symmetry problems in quantum mechanics. I. General Dirac formalism. J. Math. Phys., 10 (1969), 53-69, https://doi.org/10.1063/1.1664761.
  • Melsheimer, O., Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory. J. Math. Phys., 15 (1973), 902-916, https://doi.org/10.1063/1.1666769.
  • Bellomonte, G., Trapani, C. Rigged Hilbert spaces and contractive families of Hilbert spaces. Monatshefte Math. 164 (2011), 271-285, https://doi.org/10.1007/s00605-010-0249-1.
  • Bellomonte, G., di Bella, S.,Trapani, C., Operators in rigged Hilbert spaces: some spectral properties. J. Math. Anal. Appl., 411 (2014), 931-946, https://doi.org/10.1016/j.jmaa.2013.10.025.
  • Chiba, H., A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions. Adv. Math., 273 (2015), 324-379, https://doi.org/10.1016/j.aim.2015.01.001.
  • Chiba, H., A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions II: Applications to Schrodinger operators. Kyushu J. Math., 72 (2018), 375-405, https://doi.org/10.2206/kyushujm.72.375.
  • Gadella, M., Gomez, F., A unified mathematical formalism for the Dirac formulation of quantum mechanics. Found. Phys., 32 (2002), 815-869, https://doi.org/10.1023/A:1016069311589.
  • Gadella, M., Gomez, F., On the mathematical basis of the Dirac formulation of Quantum Mechanics. Int. J. Theor. Phys., 42 (2003), 2225-2254, https://doi.org/10.1023/B:IJTP.0000005956.11617.e9.
  • Gadella, M., Gomez-Cubillo, F., Eigenfunction Expansions and Transformation Theory. Acta Appl. Math., 109 (2010), 721-742, https://doi.org/10.1007/s10440-008-9342-z.
  • Celeghini, E., Gadella, M., del Olmo, M. A., Applications of rigged Hilbert spaces in quantum mechanics and signal processing, J. Math. Phys., 57 (2016), 072105, https://doi.org/10.1063/1.4958725.
  • Celeghini, E., Gadella, M., del Olmo, M. A., Spherical harmonics and rigged Hilbert spaces, J. Math. Phys., 59 (5) (2018), 053502, https://doi.org/10.1063/1.5026740.
  • Celeghini, E., Gadella, M., del Olmo, M. A., Zernike functions, rigged Hilbert spaces and potential applications, J. Math. Phys., 60 (2019), 083508, https://doi.org/10.1063/1.5093488.
  • Celeghini, E., Gadella, M., del Olmo, M. A., Groups, Jacobi Functions and rigged Hilbert spaces, J. Math. Phys., 61 (2020), 033508, https://doi.org/10.1063/1.5138238.
  • Bohm, A., Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics, J. Math. Phys., 22 (12) (1980), 2813-2823, https://doi.org/10.1063/1.524871.
  • Bohm, A., Gadella, M., Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture Notes in Physics, 348. Springer Verlag, Berlin 1989, https://doi.org/10.1007/3-540-51916-5.
  • Civitarese, O., Gadella, M., Physical and Mathematical Aspects of Gamow States, Phys. Rep., 396 (2004), 41-113, https://doi.org/10.1016/j.physrep.2004.03.00.
  • Berggren, T., On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes, Nucl. Phys. A, 109 (2) (1968), 265-287, https://doi.org/10.1016/0375-9474(68)90593-9.
  • Berggren, T., Expectation value of an operator in a resonant state, Phys. Lett. B, 373 (1-3) (1996), 1-4, https://doi.org/10.1016/0370-2693(96)00132-3
  • Civitarese, O., Gadella, M., Id Betan, R., On the mean value of the energy for resonance states, Nucl. Phys. A, 660 (1999), 255-266, https://doi.org/10.1016/S0375-9474(99)00405-4.
  • Civitarese, O., Gadella, M., On the concept of entropy for quantum decaying systems, Found. Phys., 43 (2013), 1275-1294, https://doi.org/10.1007/s10701-013-9746-0.
  • Civitarese, O., Gadella, M., The definition of entropy for quantum unstable systems: A view-point based on the properties of Gamow states, Entropy, 20 (4) (2018), 231, https://doi.org/10.1007/s10701-014-9860-7.
  • Amrein, W. O., Jauch, J. M., Sinha, K. B., Scattering Theory in Quantum Mechanics. Physical Principles and Mathematical Methods, 917 Bejamin, Reading, Massachusetts, USA, 1977.
  • Reed, M., Simon, B., Scattering Theory, Academic, New York, 1979.
  • Gelfand, I. M., Vilenkin, N. Ya., Generalized Functions. Applications of Harmonic Analysis, Academic, New York, 1970.
  • Maurin, K., Generalized eigenfunction expansions and unitary representations of topological groups, Polish Scientific Publishers, Warszawa, 1968.
  • Castagnino, M., Gadella, M., Id Betan, R., Laura, R., Gamow functionals on operator algebras, J. Phys. A: Math. Gen. 34 (2001), 10067-10083
  • Gadella, M., et al., To appear in Entropy.
  • Segal, I. E., Postulates for General Quantum Mechanics, Annal. Math., 48 (1947), 930-948, https://doi.org/10.2307/1969387.
  • Brateli, O., Robinson, B., Operator Algebras and Quantum Statistical Mechanics, Vol I and II, Springer, New York, 1979.
  • Reed, M., Simon, B., Functional Analysis, Academic Press, New York, 1972.
  • Antoniou, I. E., Laura, R., Suchanecki, Z., Tasaki, S., Intrinsic irreversibility of quantum systems with diagonal singularity, Phys. A, 241 (1997), 737-772, https://doi.org/10.1016/S0378-4371(97)00167-2.
  • van Hove, L., The approach to equilibrium in quantum statistics, Physica, 23 (1957), 441-480, https://doi.org/10.1016/S0031-8914(57)92891-4.
  • van Hove, L., The ergodic behaviour of quantum many-body systems, Physica, 25 (1959), 268-276, https://doi.org/10.1016/S0031-8914(59)93062-9.
  • Castagnino, M., Gadella, M., Id Betan, R., Laura, R., The Gamow functional, Phys. Lett. A, 282 (2001), 245-250, https://doi.org/10.48550/arXiv.quant-ph/0209146.
  • Bohm, A., Bryant, P. W., Uncu, H., Wickramasekara, S., Schleich, W. P., The beginning of time observed in quantum jumps, Fort. Phys., 65 (2017), 1700015, http://dx.doi.org/10.1002/prop.201700015.

Year 2022, , 51 - 65, 30.06.2022

Manuel Gadella , Carlos San Mıllan

https://doi.org/10.33769/aupse.1100013

Abstract

Project Number

PID2020-113406GB-I00

References

  • Bohm, A.,Quantum Mechanics: Foundations and Applications, 3rd Ed. Springer Verlag, New York and Berlin, 2001.
  • Newton, R. G., Scattering Theory of Waves and Particles, 2rd Ed. Springer Verlag, Berlin and Heidelberg, 1982.
  • Nussenzveig, H. M., Causality and Dispersion Relations, Academic, New York and London, 1972.
  • Kukulin, V. I., Krasnopolski, V. M., Horacek, J., Theory of Resonances, Principles and Applications, Academia, Prag, 1989.
  • Fonda, L., Ghirardi, G. C., Rimini, A ., Decay theory of unstable quantum systems. Rep Progr. Phys., 41 (1978), 587-631, https://doi.org/10.1088/0034-4885/41/4/003.
  • Khalfin, L. A., Contribution to the decay theory of a quasi stationary state, Sov. Phys., JETP USSR, 6 (6) (1958), 1053-1063.
  • Fischer, M. C., Gutierrez-Medina, B., Raizen, M. G., Observation of the quantum Zeno and anti-Zeno effects in an unstable system, Phys. Rev. Lett., 87 (4) (2001), 040402, https://doi.org/10.1103/PhysRevLett.87.040402.
  • Rothe, C., Hintschich, S. I., Monkman, A. P., Violation of the exponential decay law at long times, Phys. Rev. Lett., 96 (16) (2006), 163601, https://doi.org/10.1103/PhysRevLett.96.163601.
  • Nakanishi, N., A note on the physical state of unstable particles, Progr. Theor. Phys., 21 (1) (1959), 216-217, https://doi.org/10.1143/PTP.21.216.
  • Albeverio, S., Kurasov, P., Singular Perturbations of Differential Operators, Lecture Note Series, vol. 271, Cambridge, UK, London Mathematical society.
  • Losada, M., Fortin, S., Gadella, M. Holik, F., Dynamics of algebras in quantum unstable systems, Int. J. Mod. Phys. A, 33 (2018), 1850109, https://doi.org/10.1142/S0217751X18501099.
  • Gelfand, I. M., Shilov, G. E., Generalized Functions, Vol. II, Academic Press, New York, 1964.
  • Horvath, J., Topological Vector Spaces and Distributions, Addison-Wesley, London, 1966.
  • Bohm, A., Rigged Hilbert space and the mathematical description of physical systems, Boulder Lecture Notes in Theoretical Physics, Vol. 9A, Gordon and Breach Science Publishers, New York, 1967, 255-317.
  • Bohm, A., The Rigged Hilbert Space and Quantum Mechanics, Lecture Notes in Physics 78, Springer, New York, 1978.
  • Roberts, J. E., Rigged Hilbert spaces in quantum mechanics, Commun. Math. Phys., 2 (1966), 98-119, https://doi.org/10.1007/BF01645448.
  • Antoine, J. P., Dirac formalism and symmetry problems in quantum mechanics. I. General Dirac formalism. J. Math. Phys., 10 (1969), 53-69, https://doi.org/10.1063/1.1664761.
  • Melsheimer, O., Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory. J. Math. Phys., 15 (1973), 902-916, https://doi.org/10.1063/1.1666769.
  • Bellomonte, G., Trapani, C. Rigged Hilbert spaces and contractive families of Hilbert spaces. Monatshefte Math. 164 (2011), 271-285, https://doi.org/10.1007/s00605-010-0249-1.
  • Bellomonte, G., di Bella, S.,Trapani, C., Operators in rigged Hilbert spaces: some spectral properties. J. Math. Anal. Appl., 411 (2014), 931-946, https://doi.org/10.1016/j.jmaa.2013.10.025.
  • Chiba, H., A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions. Adv. Math., 273 (2015), 324-379, https://doi.org/10.1016/j.aim.2015.01.001.
  • Chiba, H., A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions II: Applications to Schrodinger operators. Kyushu J. Math., 72 (2018), 375-405, https://doi.org/10.2206/kyushujm.72.375.
  • Gadella, M., Gomez, F., A unified mathematical formalism for the Dirac formulation of quantum mechanics. Found. Phys., 32 (2002), 815-869, https://doi.org/10.1023/A:1016069311589.
  • Gadella, M., Gomez, F., On the mathematical basis of the Dirac formulation of Quantum Mechanics. Int. J. Theor. Phys., 42 (2003), 2225-2254, https://doi.org/10.1023/B:IJTP.0000005956.11617.e9.
  • Gadella, M., Gomez-Cubillo, F., Eigenfunction Expansions and Transformation Theory. Acta Appl. Math., 109 (2010), 721-742, https://doi.org/10.1007/s10440-008-9342-z.
  • Celeghini, E., Gadella, M., del Olmo, M. A., Applications of rigged Hilbert spaces in quantum mechanics and signal processing, J. Math. Phys., 57 (2016), 072105, https://doi.org/10.1063/1.4958725.
  • Celeghini, E., Gadella, M., del Olmo, M. A., Spherical harmonics and rigged Hilbert spaces, J. Math. Phys., 59 (5) (2018), 053502, https://doi.org/10.1063/1.5026740.
  • Celeghini, E., Gadella, M., del Olmo, M. A., Zernike functions, rigged Hilbert spaces and potential applications, J. Math. Phys., 60 (2019), 083508, https://doi.org/10.1063/1.5093488.
  • Celeghini, E., Gadella, M., del Olmo, M. A., Groups, Jacobi Functions and rigged Hilbert spaces, J. Math. Phys., 61 (2020), 033508, https://doi.org/10.1063/1.5138238.
  • Bohm, A., Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics, J. Math. Phys., 22 (12) (1980), 2813-2823, https://doi.org/10.1063/1.524871.
  • Bohm, A., Gadella, M., Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture Notes in Physics, 348. Springer Verlag, Berlin 1989, https://doi.org/10.1007/3-540-51916-5.
  • Civitarese, O., Gadella, M., Physical and Mathematical Aspects of Gamow States, Phys. Rep., 396 (2004), 41-113, https://doi.org/10.1016/j.physrep.2004.03.00.
  • Berggren, T., On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes, Nucl. Phys. A, 109 (2) (1968), 265-287, https://doi.org/10.1016/0375-9474(68)90593-9.
  • Berggren, T., Expectation value of an operator in a resonant state, Phys. Lett. B, 373 (1-3) (1996), 1-4, https://doi.org/10.1016/0370-2693(96)00132-3
  • Civitarese, O., Gadella, M., Id Betan, R., On the mean value of the energy for resonance states, Nucl. Phys. A, 660 (1999), 255-266, https://doi.org/10.1016/S0375-9474(99)00405-4.
  • Civitarese, O., Gadella, M., On the concept of entropy for quantum decaying systems, Found. Phys., 43 (2013), 1275-1294, https://doi.org/10.1007/s10701-013-9746-0.
  • Civitarese, O., Gadella, M., The definition of entropy for quantum unstable systems: A view-point based on the properties of Gamow states, Entropy, 20 (4) (2018), 231, https://doi.org/10.1007/s10701-014-9860-7.
  • Amrein, W. O., Jauch, J. M., Sinha, K. B., Scattering Theory in Quantum Mechanics. Physical Principles and Mathematical Methods, 917 Bejamin, Reading, Massachusetts, USA, 1977.
  • Reed, M., Simon, B., Scattering Theory, Academic, New York, 1979.
  • Gelfand, I. M., Vilenkin, N. Ya., Generalized Functions. Applications of Harmonic Analysis, Academic, New York, 1970.
  • Maurin, K., Generalized eigenfunction expansions and unitary representations of topological groups, Polish Scientific Publishers, Warszawa, 1968.
  • Castagnino, M., Gadella, M., Id Betan, R., Laura, R., Gamow functionals on operator algebras, J. Phys. A: Math. Gen. 34 (2001), 10067-10083
  • Gadella, M., et al., To appear in Entropy.
  • Segal, I. E., Postulates for General Quantum Mechanics, Annal. Math., 48 (1947), 930-948, https://doi.org/10.2307/1969387.
  • Brateli, O., Robinson, B., Operator Algebras and Quantum Statistical Mechanics, Vol I and II, Springer, New York, 1979.
  • Reed, M., Simon, B., Functional Analysis, Academic Press, New York, 1972.
  • Antoniou, I. E., Laura, R., Suchanecki, Z., Tasaki, S., Intrinsic irreversibility of quantum systems with diagonal singularity, Phys. A, 241 (1997), 737-772, https://doi.org/10.1016/S0378-4371(97)00167-2.
  • van Hove, L., The approach to equilibrium in quantum statistics, Physica, 23 (1957), 441-480, https://doi.org/10.1016/S0031-8914(57)92891-4.
  • van Hove, L., The ergodic behaviour of quantum many-body systems, Physica, 25 (1959), 268-276, https://doi.org/10.1016/S0031-8914(59)93062-9.
  • Castagnino, M., Gadella, M., Id Betan, R., Laura, R., The Gamow functional, Phys. Lett. A, 282 (2001), 245-250, https://doi.org/10.48550/arXiv.quant-ph/0209146.
  • Bohm, A., Bryant, P. W., Uncu, H., Wickramasekara, S., Schleich, W. P., The beginning of time observed in quantum jumps, Fort. Phys., 65 (2017), 1700015, http://dx.doi.org/10.1002/prop.201700015.
There are 51 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Review Articles
Authors

Manuel Gadella 0000-0001-8860-990X

Carlos San Mıllan This is me 0000-0001-7506-5552

Project Number PID2020-113406GB-I00
Publication Date June 30, 2022
Submission Date April 7, 2022
Acceptance Date May 16, 2022
Published in Issue Year 2022

Cite

APA Gadella, M., & San Mıllan, C. (2022). Averages of observables on Gamow states. Communications Faculty of Sciences University of Ankara Series A2-A3 Physical Sciences and Engineering, 64(1), 51-65. https://doi.org/10.33769/aupse.1100013
AMA Gadella M, San Mıllan C. Averages of observables on Gamow states. Commun.Fac.Sci.Univ.Ank.Series A2-A3: Phys.Sci. and Eng. June 2022;64(1):51-65. doi:10.33769/aupse.1100013
Chicago Gadella, Manuel, and Carlos San Mıllan. “Averages of Observables on Gamow States”. Communications Faculty of Sciences University of Ankara Series A2-A3 Physical Sciences and Engineering 64, no. 1 (June 2022): 51-65. https://doi.org/10.33769/aupse.1100013.
EndNote Gadella M, San Mıllan C (June 1, 2022) Averages of observables on Gamow states. Communications Faculty of Sciences University of Ankara Series A2-A3 Physical Sciences and Engineering 64 1 51–65.
IEEE M. Gadella and C. San Mıllan, “Averages of observables on Gamow states”, Commun.Fac.Sci.Univ.Ank.Series A2-A3: Phys.Sci. and Eng., vol. 64, no. 1, pp. 51–65, 2022, doi: 10.33769/aupse.1100013.
ISNAD Gadella, Manuel - San Mıllan, Carlos. “Averages of Observables on Gamow States”. Communications Faculty of Sciences University of Ankara Series A2-A3 Physical Sciences and Engineering 64/1 (June 2022), 51-65. https://doi.org/10.33769/aupse.1100013.
JAMA Gadella M, San Mıllan C. Averages of observables on Gamow states. Commun.Fac.Sci.Univ.Ank.Series A2-A3: Phys.Sci. and Eng. 2022;64:51–65.
MLA Gadella, Manuel and Carlos San Mıllan. “Averages of Observables on Gamow States”. Communications Faculty of Sciences University of Ankara Series A2-A3 Physical Sciences and Engineering, vol. 64, no. 1, 2022, pp. 51-65, doi:10.33769/aupse.1100013.
Vancouver Gadella M, San Mıllan C. Averages of observables on Gamow states. Commun.Fac.Sci.Univ.Ank.Series A2-A3: Phys.Sci. and Eng. 2022;64(1):51-65.

Communications Faculty of Sciences University of Ankara Series A2-A3 Physical Sciences and Engineering

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