Abstract
This chapter provides the elements of quantum scattering theory which shall be used throughout this book. The general scattering problem is reviewed in the context of scattering and time evolution operators and recognises that the observables are the pre- and post-scattering asymptotic states. Quantum perturbation theory is developed with the two linked goals of deriving an expression for the transition probability rate between two quantum states (Fermi’s Golden Rule) and the scattering amplitude and differential cross section (first Born approximation). The Born approximation is derived through the Lippmann–Schwinger equation. The chapter concludes with a phase shift analysis of the scattering problem.
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Notes
- 1.
Collision and scatter have been differentiated by some authors so that the former correspond to multichannel final states, and the latter was linked to a single channel final state. Such a distinction will not be made here.
- 2.
Bremsstrahlung is sometimes referred to as being inelastic. However, it is an elastic process by our definition as the pre- and post-scatter energies of the electron and of the electron/photon/recoiling nucleus set are the same.
- 3.
Although this will not always be the case, particularly for high-Z atomic targets. The effect of a non-negligible atomic electron velocity is to reduce the rate at which energy is lost. The correction factor for this effect is derived in Chap. 12.
- 4.
When we do consider the multi-body problem in which the target is made up of multiple charges (e.g. the scatter of a projectile due to the combined Coulomb interactions with the nucleus and electrons of an atom), the multi-body problem will be simplified and reduced to the sum of independent two-body scattering calculations in which it is assumed that the interactions between individual targets are uncorrelated.
- 5.
In the case of a ‘dressed ion’ projectile (i.e. one carrying atomic electrons), this cannot be the case as the electrons themselves can participate in the scattering.
- 6.
In the case of α-decay, as a result of its two-body nature, the energy spectrum of the α-particle is discrete with levels corresponding to the energy channels generated by the excited states of the daughter nucleus. In the case of β-decay, as a result of its three-body final state, the energy spectrum of the electron/positron is continuous.
- 7.
With the exception of photonuclear reactions which have a photon energy threshold of about 10 MeV, well above the energies of photons emitted by radionuclides in medical use.
- 8.
However, in the case of internal radiation dosimetry, the charged particle is rarely detected, but its transport can be calculated from the known physics of its interactions with matter, such as that described in this book.
Bibliography and Further Reading
Abramowitz M, Stegun IA (eds). Handbook of mathematical functions. New York: Dover Publications; 1972.
Belkic D. Principles of quantum scattering theory. Bristol: Institute of Physics Publishing; 2004.
Churchill RV, Brown JW, Verhey RF. Complex variables and applications. 3rd ed. New York: McGraw-Hill Inc; 1974.
Dyson F. Advanced quantum mechanics. Singapore: World Scientific Publishing; 2007.
Goldberger ML, Watson KM. Collision theory. Mineola: Dover Publications; 2004.
Gribov VN, Nyiri J. Quantum electrodynamics. Cambridge: Cambridge University Press; 2001.
Jackson JD. Classical electrodynamics. 3rd ed. Hoboken: Wiley; 1999.
McParland BJ. Nuclear medicine dosimetry: advanced theoretical principles. Berlin: Springer; 2010.
Messiah A. Quantum mechanics. New York: Wiley.
Sigmund P. Particle penetration and radiation effects. Berlin: Springer; 2008.
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McParland, B.J. (2014). Elements of Quantum Scattering Theory. In: Medical Radiation Dosimetry. Springer, London. https://doi.org/10.1007/978-1-4471-5403-7_2
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DOI: https://doi.org/10.1007/978-1-4471-5403-7_2
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