Abstract
Resonance phenomena constitute some of the most interesting and striking features of scattering experiments. This chapter discusses in detail the connection between quasistationary states and resonance phenomena, and culminates in the derivation of the Breit-Wigner formula. In Section XVIII.2 the concept of “time delay” is introduced and its relation to the phase shift derived. Various formulations of causality are given in Section XVIII.3. In Section XVIII.4 the causality condition is used to derive certain analyticity properties of the S-matrix. These properties are discussed further in Section XVIII.5. In Section XVIII.6, the central section of this chapter, the connection between quasistationary states, defined by a large time delay, and resonances, defined by characteristic structures in the cross section, is derived. Section XVIII.7 describes the observable effects of virtual states. Section XVIII.8 discusses the effect resonances have on the Argand diagram. The actual appearance of resonances in experimental data when the effects of the resonant phase shift, the nonresonant background, and the limited resolution of the apparatus are taken into account is discussed in Section XVIII.9.
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References
Time delay has been introduced previously in a similar way by F. T. Smith, Phys. Rev. 118, 349 (1960) based on
E. P. Wigner, Phys. Rev. 98, 145 (1955).
Gel’fand and Shilov (1964), Vol. 1.
Cf. the discussion in Section II.9.
One obtains dispersion relations when these analytic properties are expressed in terms of integral relations between different matrix elements for real values of the variables.
Taylor (1972), Chapter 12.
N. G. van Kampen, Phys. Rev. 91, 1267 (1953), Section II.
For more on this subject, see Nussenzveig (1972).
N. G. van Kampen, Physica (Utrecht), 20, 115 (1954).
This derivation is based on Goldberger and Watson (1964), Section 8.5.
Cf. Appendix XVII.A.
In addition to their other quantum numbers like l,
We did not give a complete proof of the statements in Section XVIII.5, but argued that causality, the finiteness of the binding energies for possible bound states, and the finite range of the interaction are sufficient, though certainly much less is necessary.
L. Fonda, Fortschr. Phys. 20, 135 (1972).
G. Bialkowski has helped me with the writing of this section.
The original parameters introduced by U. Fano [Phys. Rev. 124, 1866 (1961)] were the negative of the parameters defined by (9.2).
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© 1986 Springer Science+Business Media New York
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Bohm, A. (1986). Resonance Phenomena. In: Quantum Mechanics: Foundations and Applications. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-01168-3_18
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DOI: https://doi.org/10.1007/978-3-662-01168-3_18
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