Abstract
The harmonic oscillator provides a unique opportunity to study with simple mathematics the properties acquired by a mechanical system in permanent contact with the zeropoint field, and to assess with a specific example the merit of the assumption that the classical particle becomes through this interaction a quantum object. Of course, due to its linearity the oscillator is far from being a fair representative of the general quantum system, but nevertheless the importance of getting a concrete expression of the relationship between quantum theory and SED can hardly be overestimated.
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References
See, e.g., Wang and Uhlenbeck (1954), Papoulis (1965) or van Kampen (1981). 2 An important instance is studied in chapter 11.
A more detailed discussion can be seen in de la Pena (1980) and Marc and McMillan (1983).
The oscillator model of the atom, frequently called the Lorentz model and sometimes the Drude-Lorentz model, can account for a wide range of optical and nonoptical effects. In quantum theory its use can be shown to be legitimate (with the proper oscillator strengths included), provided, e.g., that the probability for the atom to remain in its initial state is close to unity [Cray et al. 1982; see also Allen and Eberly 1975, Cohen-Tannoudji et al. 1977, vol. II, p. 1318].
Some pertinent references are: Zwanzig (1960, 1961), Stratonovich (1963, 1968), Lax (1966), Brissaud and Frisch (1974), Terwiel (1974), Haken (1975), van Kampen (1976, 1981), Santos (1978, 1985b), Claveric (1979, 1980b), San Miguel and Sancho (1980), Cetto et al. (1984), Cetto (1984), etc. The quoted works by Claverie contain detailed comparisons between the different approximations.
We will meet other examples of fluctuation-dissipation relations below. In general this kind of relationships —first discovered in the study of Brownian motion— connect coefficients that measure the dissipation in the system with coefficients related to the source of the fluctuations under equilibrium conditions. They show that fluctuations and dissipation are concomitant, so that it is not possible to have one without the other. For a general introduction to the fluctuation-dissipation theorem, see, e.g., Reichl (1980).
The first Fokker-Planck type equation in SED was constructed by Marshall (1963) in his pioneering studies of the harmonic oscillator, using different approximations and variables from those discussed here. A version of this equation in terms of integrals of motion is given in Marshall (1980b).
General procedures to separate a Fokker-Planck equation into its reversible and irreversible parts in the time-dependent case are given, e.g., in van Kampen (1981).
For a discussion of Welton’s model see, e.g., Milonni 1994, section 3.6.
Recall that the factor ħω in the integrand is the energy ε(ω); it is precisely this linear dependence of ε on ω that gives rise to results which differ qualitatively from the Brownian ones.
The first nonrelativistic quantum calculation of the Lamb shift for the atomic case is due to Bethe (1947). A detailed discussion of the Lamb shift in QED can be found in Milonni (1994), chapters 3, 4 and 16. Explicit evaluations of the nonrelativistic QED prediction for the Lamb shift of the harmonic oscillator are given in appendix B of Santos (1974a) and in Goedecke (1984).
A thorough exposition of the orthogonality principle and its applications can be found in Papoulis 1965, sections 7.4 and 7.5.
This is also the differential equation for the radial Schrödinger amplitude for the s states of the H atom; the generalization to other states can be achieved by considering the multidimensional case. This relationship between the oscillator and the Coulomb problem is well known [see, e.g., Hillery et al. 1984]. The functions W n appear for the first time in Groenewold (1946).
In França and Marshall (1988) it is shown that the functions W n form a basis for the representation of phase-space Wigner states. For a discussion on the use of W n in optical problems see section 13.3.
We are here merely applying the factorization procedure to determine the eigenvalues. The interesting point is, however, that the idea that quantization follows as the result of algebraic overdetermination of the eigenvalues of a pair of operators that leads to consistency requirements, can be developed for all textbook problems; see de la Pena and Montemayor (1980) and Fernandez and Castro (1984).
For an introductory discussion of the master equation see, e.g., van Kampen 1981, chapter 5.
The QED problem was clarified in Milonni et al. (1973) and Senitzky (1973); for detailed discussions see Cohen-Tannoudji (1986) and Milonni (1994). An introductory quantum discussion on the linewidth and the related Wigner-Weisskopf theory can be found in the last cited book, chapter 4. The first calculation of A nm for the SED harmonic oscillator was given in Marshall (1968).
For more details see de la Peña and Cetto (1979). Note that in this and related papers, D xp differs in sign with respect to the one used here.
Detailed accounts of this classical problem of QED can be seen, e.g., in Knight and Milonni (1980), Loudon (1973), chapter 8, or Walls and Milburn (1994), chapter 11. At higher incident intensities the stimulated emissions can contain nonlinear contributions, such as the Rabi oscillations of the state probabilities and other effects that have been entirely neglected in the simple calculation presented above.
An introduction to cavity quantum electrodynamics is given in Haroche and Kleppner (1989); see also Berman (1994).
Cetto and de la Perla (1988c); first results of this kind were derived within SED already by Marshall (1965a).
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© 1996 Springer Science+Business Media Dordrecht
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de la Peña, L., Cetto, A.M. (1996). The Harmonic Oscillator. In: The Quantum Dice. Fundamental Theories of Physics, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8723-5_7
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DOI: https://doi.org/10.1007/978-94-015-8723-5_7
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