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The momentum of a particle is interpreted as a gauge potential that defines the phase factors and gauge field in terms of the properties of the space–time manifold. Equating the field and the phase factors with the corresponding electromagnetic quantities results in the Lorentz and the Klein–Gordon equation respectively.
It is clarified that the conclusions reached in ͓Phys. Rev. B. 66, 033110 ͑2002͔͒ result from erroneous application of the Hellmann-Feynman theorem, and a result deduced there for twofold degenerate eigenvalues is generalized.The Hellmann-Feynman theorem has been widely used for a variety of calculations for its computational convenience, e.g., to calculate the forces acting on the nuclei in molecules. 1 The theorem states essentially that the derivative of the eigenvalue of a parameter dependent self-adjoint operator is equal to the expectation value of the derivative of the operator with respect to the eigenvector of the operator. According to the calculations of Zhang and George, 2 this equality does not hold for degenerate states of C 60 , where the parameter is the location of the nucleus. It is pointed out here that this conclusion is based on the application of an incorrect relation deduced from the Hellmann-Feynman theorem for the nondegenerate case without due consideration of the legitimacy of the operations. The equality is shown to be valid with properly selected eigenstates, which can be obtained within the framework of the degenerate perturbation theory. Also, a minor result deduced there for twofold degeneracy is extended to the general case.Consider the eigenvalue equation,
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Access and use of this website and the material on it are subject to the Terms and Conditions set forth at Effect of plasma on ultra short pulse laser material processing Li, Chengde; Vatsya, S.R.; Nikumb, S.K.
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