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Extra info for General relativity and gravitation Vol. 5
216). 223) i=1 j=1 If they are normalized to unity we have (1j1(k)11j1(1)) = Dk,l , (1j1(k)lifI1j1(1») = ckDk,l, k, 1= 1, ... N . 224) The method of diagonalizing in a subspace is particularly useful if we are looking for approximations to describe not only the ground state of a system. 215) sets an upper bound for the ground state energy and hence we know, the lower the value of E[ 1j1], the closer it is to the exact ground state energy El. 215), and it is not always a good thing to approximate it by a model state with as Iowan energy as possible.
178). This additional phase shift characterizes the resonance. 179) for tan 8. e. 182) 36 1. Review of Quantum Mechanics 2 Fig. 8. 183». :1. Around the resonance energy ER the phase 8 rises more or less suddenly by 7r. 181); at E = ER - r and E = ER + r the phase has risen 1/4 and 3/4 of 7r respectively. 183) is illustrated for constant values of the parameters ER and r in Fig. 8. 183) is called a Breit-Wigner resonance. 183). 185). 4 Resonances and Channels 37 of the resonance. For a broader resonance, however, the unique definition of its position and width can become a difficult problem (see also Sect.
1. Around the resonance energy ER the phase 8 rises more or less suddenly by 7r. 181); at E = ER - r and E = ER + r the phase has risen 1/4 and 3/4 of 7r respectively. 183) is illustrated for constant values of the parameters ER and r in Fig. 8. 183) is called a Breit-Wigner resonance. 183). 185). 4 Resonances and Channels 37 of the resonance. For a broader resonance, however, the unique definition of its position and width can become a difficult problem (see also Sect. 3). The derivative of the phase shift with respect to energy is also a measure for the strength of the closed channel component in the solution of the coupled channel equations.