Lectures on Quantum Mechanics by Steven Weinberg

Author:Steven Weinberg
Language: eng
Format: epub, pdf
Publisher: Cambridge University Press
Published: 2022-08-07T00:00:00+00:00

(5.4.12)

This only applies if the zeroth-order degeneracy is removed in first order. If any of the first-order perturbations of energies for which are equal to then Eq. (5.4.10) tells us nothing about , but instead implies that if and but then , where

(5.4.13)

We noted in Section 5.1 that when there are states with the same values of the zeroth- and first-order energies we can take the unperturbed state vectors to be any orthonormal linear combination of these states. Since is an Hermitian matrix, by the same reasoning as we applied in Section 5.1 to , we can choose these linear combinations to diagonalize this matrix, so that if and but then in the new basis . This completely determines the unperturbed states unless some of the second-order energies are equal. In this case we must look to higher orders of perturbation theory to remove the degeneracy and fix the unperturbed states.

It is generally not easy to do the sums over states in Eqs. (5.4.5) or (5.4.9). In some cases the sum can diverge; there are ultraviolet divergences that occur when the matrix elements do not fall off rapidly enough for high-energy states to make the sum converge, and there are infrared divergences that occur when there is a continuum of states with energies extending down to . The treatment of these infinities has been a major preoccupation of theoretical physicists since the 1930s.

There are two cases that allow to be more easily calculated. In the first case, the energies of all the states with for which is appreciable for a given state are clustered at a value , with . The completeness of the orthonormal state vectors allows us to write

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