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A unified theory of nuclear reactions. II
ANNALS
OF PHYSICS:
Abstracts
18, 480-481
(1962)
of Papers
to Appear
in Future
Issues
A Unified Theory of Nuclear Reactions. II. HERMAN FESHBACH. Massachusetts Institute of Technology, Cambridge, Massachusetts The effective Hamiltonian method for nuclear reactions described in an earlier paper with the same title, part I, is generalized so as to include all possible reaction types, as well as the effects arising from the identity of particles. The principal device employed, as in part I, is the projection operator which selects the open channel components of the wave function. It is found that the formal structure of part I providing a unified description for direct and compound nuclear reactions including the coupled equation description for direct reactions remains valid in this wider context. A Hapur-Peierls expansion may also be readily obtained. The concept of channel radii is not needed nor is any decomposition of the wave function for the system into angular momentum eigenstates required, so that the expressions for transition amplitudes and widths are invariant with respect to the angular momentum for coupling scheme. Since the open channels can only be defined in an asymptotic sense, the corresponding projection operators are not unique. As a consequence the projection operator method has a flexibility which in the first place is consonant with the wide range of phenomena which can occur in nuclear reactions and in the second place can effectively exploit an intuitive understanding of the phenomena. Example of projection operators are obtained including one which leads to the Wigner-Eisenbud formalism, another which is appropriate for the stripping reaction and finally one which takes the Pauli exclusion principle into account. Note that explicit representations of the projection operators are not required for the development of general formal results but are necessary if, eventually, quantitative calculations are made.
Radiation Damping in Classical Electrodynamics. TED CLAY BRADBURY. Los Angeles State College, Los Angeles, California. The problem of radiation from a charged particle in uniform acceleration is considered. It is shown that, by transforming from an inertial frame to the accelerated frame in which the particle is at rest, the magnetic field, and hence the radiation, is transformed away. Exact solutions of Maxwell’s equations in the accelerated frame are obtained. Dirac’s classical equations of motion of a charged-particle are rederived for the special case of one-dimensional motion. By doing the calculation in the permanent (noninertial) rest frame of the charged particle, it is shown that it is not necessary to use advanced fields as was done by Dirac. The calculation requires no modification of the energy-momentum tensor. The theory still contains a divergent term (as does Dirac’s) but in a modified form. A simple solution for one-dimensional motion is considered. By consideration of a case where an electron enters from a region of no field, passes through the region of the field and t.hen again into a region of no field, it is shown that the conventional power radiation formula gives the same answer for the total power radiated as does Dirac’s equation. Hyperbolic motion is considered as a limiting case of motion through a finite region of space where there is a uniform electric field. The region where the field is defined is then allowed to become large compared to the electron radius. 480
OF PHYSICS:
Abstracts
18, 480-481
(1962)
of Papers
to Appear
in Future
Issues
A Unified Theory of Nuclear Reactions. II. HERMAN FESHBACH. Massachusetts Institute of Technology, Cambridge, Massachusetts The effective Hamiltonian method for nuclear reactions described in an earlier paper with the same title, part I, is generalized so as to include all possible reaction types, as well as the effects arising from the identity of particles. The principal device employed, as in part I, is the projection operator which selects the open channel components of the wave function. It is found that the formal structure of part I providing a unified description for direct and compound nuclear reactions including the coupled equation description for direct reactions remains valid in this wider context. A Hapur-Peierls expansion may also be readily obtained. The concept of channel radii is not needed nor is any decomposition of the wave function for the system into angular momentum eigenstates required, so that the expressions for transition amplitudes and widths are invariant with respect to the angular momentum for coupling scheme. Since the open channels can only be defined in an asymptotic sense, the corresponding projection operators are not unique. As a consequence the projection operator method has a flexibility which in the first place is consonant with the wide range of phenomena which can occur in nuclear reactions and in the second place can effectively exploit an intuitive understanding of the phenomena. Example of projection operators are obtained including one which leads to the Wigner-Eisenbud formalism, another which is appropriate for the stripping reaction and finally one which takes the Pauli exclusion principle into account. Note that explicit representations of the projection operators are not required for the development of general formal results but are necessary if, eventually, quantitative calculations are made.
Radiation Damping in Classical Electrodynamics. TED CLAY BRADBURY. Los Angeles State College, Los Angeles, California. The problem of radiation from a charged particle in uniform acceleration is considered. It is shown that, by transforming from an inertial frame to the accelerated frame in which the particle is at rest, the magnetic field, and hence the radiation, is transformed away. Exact solutions of Maxwell’s equations in the accelerated frame are obtained. Dirac’s classical equations of motion of a charged-particle are rederived for the special case of one-dimensional motion. By doing the calculation in the permanent (noninertial) rest frame of the charged particle, it is shown that it is not necessary to use advanced fields as was done by Dirac. The calculation requires no modification of the energy-momentum tensor. The theory still contains a divergent term (as does Dirac’s) but in a modified form. A simple solution for one-dimensional motion is considered. By consideration of a case where an electron enters from a region of no field, passes through the region of the field and t.hen again into a region of no field, it is shown that the conventional power radiation formula gives the same answer for the total power radiated as does Dirac’s equation. Hyperbolic motion is considered as a limiting case of motion through a finite region of space where there is a uniform electric field. The region where the field is defined is then allowed to become large compared to the electron radius. 480