Aggregates of Particles by D.R. BATES (Eds.) PDF

By D.R. BATES (Eds.)

ISBN-10: 1483227413

ISBN-13: 9781483227412

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1. COMPLEX ATOMS 39 For field strengths intermediate between the limiting cases of the Zeeman effect and the Paschen-Back effect it is necessary to choose one or other of the two representations and to calculate the entire energy matrix including all off-diagonal elements. The energies are then obtained on solving the secular equation (compare (46)). **, L = 27 L i i One may show that the exact electrostatic Hamiltonian (44) commutes with L 2 ,L„S 2 and Sz. We therefore introduce functions W(ySLMsML) where SMS are total spin quantum numbers and LML are quantum numbers for the total orbital angular momentum.

If we choose Γ to stand for the quantum numbers {n^xmkmSxtn^mKms%t... nNlNmiNmSN) (50) this condition will not, in general, be satisfied. We may, however, choose states which are linear combinations of the states with representation (50) and we may do this in such a way that (47) will be satisfied. In order to develop a systematic procedure we require two theorems on commuting operators. 2 Commuting Operators Let A be an operator with eigenvalues a, B an operator with eigenvalues b and let A and B commute: AB — BA = 0.

Thus for the 2p2 configuration the allowed values* of (SL) are 15,1Z) and 3 P . For a configuration such as 2p3p containing two inequivalent p electrons the allowed values of (SL) would be ^S^P, 1 /), 3 5, 3 P and 3Z). * For an alternative derivation 8«β one could list the values of fn^m^m^m^ allowed by the exclusion principle and deduce the allowed values of SL by an obvious generaliza­ tion of the argument leading to (63). 57 1. 4 The Electrostatic Energies: Theory For two inequivalent electrons the wave function may be writtenf ^ ( * l W 2 S i | * l ' X 2 ) = Ψ(5\σ1>σ2) y= ' (119) {φ{η111η2121\τντ2) + ( - Vj^n^n^L^^)}.

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Aggregates of Particles by D.R. BATES (Eds.)

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