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15 - Quantum Dynamics

Published online by Cambridge University Press:  11 May 2023

Uri Peskin
Uri Peskin
Affiliation:
Technion - Israel Institute of Technology, Haifa
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Summary

Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

Keywords

time evolution operatorHeisenberg pictureinteraction picturequantum-classical correspondenceGaussian wave packetcoherent stateunitary transformationstime-dependent perturbation theoryHeisenberg equationtransition probabilities
Type
Chapter
Information
Quantum Mechanics in Nanoscience and Engineering , pp. 286 - 319
Publisher: Cambridge University Press
Print publication year: 2023

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References

Meyer, H.-D., Manthe, U., and Cederbaum, L. S., “The multi-configurational time-dependent Hartree approach,” Chemical Physics Letters 165, 73 (1990).Google Scholar
Wang, H. and Thoss, M., “Multilayer formulation of the multiconfiguration time-dependent Hartree theory,” The Journal of Chemical Physics 119, 1289 (2003).CrossRefGoogle Scholar
Alon, O. E., Streltsov, A. I., and Cederbaum, L. S., “Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems,” Physical Review A 77, 033613 (2008).Google Scholar
Car, R. and Parrinello, M., “Unified approach for molecular dynamics and density-functional theory,” Physical Review Letters 55, 2471 (1985).Google Scholar
Sakurai, J. J. and Napolitano, J., “Modern Quantum Mechanics,” 2nd ed. (Addison Wesley, 2011).Google Scholar
Peskin, U. and Moiseyev, Nimrod, “The solution of the time-dependent Schrödinger equation by the method: Theory, computational algorithm and applications,” The Journal of Chemical Physics 99, 4590 (1993).Google Scholar
Klauder, J. R. and Skagerstam, B.-S., “Coherent states: applications in physics and mathematical physics” (World Scientific, 1985).Google Scholar
Mukamel, S., “Principles of nonlinear optical spectroscopy” (Oxford University Press, 1999).Google Scholar
Tannor, D. J., “Introduction to quantum mechanics: A time-dependent perspective” (University Science Books, 2007).Google Scholar

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  • Quantum Dynamics
  • Uri Peskin, Technion - Israel Institute of Technology, Haifa
  • Book: Quantum Mechanics in Nanoscience and Engineering
  • Online publication: 11 May 2023
  • Chapter DOI: https://doi.org/10.1017/9781108877787.016
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  • Quantum Dynamics
  • Uri Peskin, Technion - Israel Institute of Technology, Haifa
  • Book: Quantum Mechanics in Nanoscience and Engineering
  • Online publication: 11 May 2023
  • Chapter DOI: https://doi.org/10.1017/9781108877787.016
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Quantum Dynamics
  • Uri Peskin, Technion - Israel Institute of Technology, Haifa
  • Book: Quantum Mechanics in Nanoscience and Engineering
  • Online publication: 11 May 2023
  • Chapter DOI: https://doi.org/10.1017/9781108877787.016
Available formats
×