The hamiltonian as a conceptual tool in quantum physics: fundamentals, interpretation, and applications
DOI:
https://doi.org/10.71112/ffwncd38Keywords:
Hamiltonian, quantum mechanics, open systems, Lindblad equation, physical interpretation.Abstract
The Hamiltonian is one of the fundamental concepts in quantum mechanics. It is traditionally introduced as the operator associated with the total energy of a system; however, its role extends far beyond this formal interpretation, as it organizes the dynamics, defines the relevant observables, and structures the physical description of quantum systems. This work presents a theoretical analysis of the Hamiltonian as a conceptual tool, addressing its role in closed systems and its extension to open systems described by the Lindblad equation. Its implications for the interpretation of quantum dynamics are discussed, as well as its relevance in contemporary applications such as quantum optics and dissipative systems. The study is developed through an analytical review of the literature, highlighting the importance of the Hamiltonian not only as a mathematical operator, but as a central element in the construction of physical knowledge.
Downloads
References
Bocanegra-Garay, I. A., Hernández-Sánchez, L., Ramos-Prieto, I., Soto-Eguibar, F., & Moya-Cessa, H. M. (2024). Invariant approach to the driven Jaynes-Cummings model. SciPost Physics, 16(1), 007. https://doi.org/10.21468/SCIPOSTPHYS.16.1.007
Breuer, H.-P., & Petruccione, F. (2002). The theory of open quantum systems. Oxford University Press.
Carmichael, H. J. (1993). An open systems approach to quantum optics. Springer.
Carmichael, H. J. (2008). Statistical methods in quantum optics 2: Non-classical fields. Springer.
Datta, S. (1996). Electronic transport in mesoscopic systems. Cambridge University Press.
Dirac, P. A. M. (1976). Principles of quantum mechanics (4th ed.). Oxford University Press.
Feynman, R. P. (1967). The character of physical law. The MIT Press.
Gardiner, C. W., & Zoller, P. (2004). Quantum noise (3rd ed.). Springer.
Glauber, R. J. (1963). The quantum theory of optical coherence. Physical Review, 130(6), 2529–2539. https://doi.org/10.1103/PhysRev.130.2529
Gorini, V., Kossakowski, A., & Sudarshan, E. C. G. (1976). Completely positive dynamical semigroups of N-level systems. Journal of Mathematical Physics, 17(5), 821–825. https://doi.org/10.1063/1.522979
Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to quantum mechanics (3rd ed.). Cambridge University Press.
Haroche, S., & Raimond, J.-M. (2006). Exploring the quantum world: Atoms, cavities, and photons. Oxford University Press.
Heisenberg, W. (1958). Physics and philosophy: The revolution in modern science. Harper & Brothers.
Hernández-Sánchez, L., Bocanegra-Garay, I. A., Flores-Rosas, A., Ramos-Prieto, I., Soto-Eguibar, F., & Moya-Cessa, H. M. (2026). Numerical solution of the Lindblad master equation using the Runge-Kutta method implemented in Python. Revista Mexicana de Física E, 23(1), Article 010214. https://doi.org/10.31349/REVMEXFISE.23.010214
Hernández-Sánchez, L., Bocanegra-Garay, I. A., Ramos-Prieto, I., Soto-Eguibar, F., & Moya-Cessa, H. M. (2024). Effects of classical drivings on the power broadening of atomic lineshapes. Journal of the Optical Society of America B, 41(8), C68–C73. https://doi.org/10.1364/JOSAB.522587
Hernández-Sánchez, L., Bocanegra-Garay, I. A., Ramos-Prieto, I., Soto-Eguibar, F., & Moya-Cessa, H. M. (2025). Effective non-Hermitian formulation of the Lindblad equation for interacting quantized fields at finite temperature. Physical Review A, 112(2), Article L021702. https://doi.org/10.1103/gwzr-cjp6
Hernández-Sánchez, L., Ramos-Prieto, I., Soto-Eguibar, F., & Moya-Cessa, H. M. (2023). Exact solution for the interaction of two decaying quantized fields. Optics Letters, 48(20), 5435–5438. https://doi.org/10.1364/OL.503837
Hernández-Sánchez, L., Ramos-Prieto, I., Soto-Eguibar, F., & Moya-Cessa, H. M. (2024a). Discriminando superposiciones de estados coherentes mediante formas de línea. Revista Mexicana de Física, 70(1), Article 011302. https://doi.org/10.31349/REVMEXFIS.70.011302
Hernández-Sánchez, L., Ramos-Prieto, I., Soto-Eguibar, F., & Moya-Cessa, H. M. (2024b). Effects of squeezing on the power broadening and shifts of micromaser lineshapes. Photonics, 11(4), Article 371. https://doi.org/10.3390/PHOTONICS11040371
Hernández-Sánchez, L., Ramos-Prieto, I., Soto-Eguibar, F., & Moya-Cessa, H. M. (2025). Dynamics of the interaction between two coherent states in a cavity with finite temperature decay. Dynamics, 5(1), Article 4. https://doi.org/10.3390/DYNAMICS5010004
Jaynes, E. T., & Cummings, F. W. (1963). Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proceedings of the IEEE, 51(1), 89–109. https://doi.org/10.1109/PROC.1963.1664
Kittel, C. (2005). Introduction to solid state physics (8th ed.). Wiley.
Landau, L. D., & Lifshitz, E. M. (1981). Quantum mechanics: Non-relativistic theory (3rd ed., Vol. 3). Pergamon Press.
Leslie, E. B. (1998). Quantum mechanics: A modern development. World Scientific.
Lindblad, G. (1976). On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48(2), 119–130. https://doi.org/10.1007/BF01608499
Manzano, D. (2020). A short introduction to the Lindblad master equation. AIP Advances, 10(2), Article 025106. https://doi.org/10.1063/1.5115323
Nielsen, M. A., & Chuang, I. L. (2010). Quantum computation and quantum information (10th anniversary ed.). Cambridge University Press.
Prosen, T. (2008). Third quantization: A general method to solve master equations for quadratic open Fermi systems. New Journal of Physics, 10(4), Article 043026. https://doi.org/10.1088/1367-2630/10/4/043026
Rivas, A., & Huelga, S. F. (2011). Open quantum systems: An introduction. Springer.
Sakurai, J. J., & Napolitano, J. (2017). Modern quantum mechanics (2nd ed.). Cambridge University Press.
Schlosshauer, M. (2007). Decoherence and the quantum-to-classical transition. Springer.
Scully, M. O., & Zubairy, M. S. (1997). Quantum optics. Cambridge University Press.
von Neumann, J. (1955). Mathematical foundations of quantum mechanics. Princeton University Press.
Walls, D. F., & Milburn, G. J. (2008). Quantum optics (2nd ed.). Springer.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Dr. Leonardi Hernández Sánchez, Dr. Gabriel Efraín Villatoro Pérez, Dra. Noribeth Fuentes González (Autor/a)
This work is licensed under a Creative Commons Attribution 4.0 International License.
