By F. Iachello, R. D. Levine

ISBN-10: 0195080912

ISBN-13: 9780195080919

ISBN-10: 0195359739

ISBN-13: 9780195359732

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**Example text**

In the earlier days of the algebraic approach the harmonic limit (Levine, 1982) served as a useful guide to the connection with the geometrical picture. Since the harmonic limit is so well understood, taking it still provides a good intuitive link. The harmonic limit can be taken in a mathematically consistent way by the process of contraction (Gilmore, 1974). Returning to Eq. 26), one replaces the operators a and of by numbers, V#, and lets N -> oo. 4 The transition probabilities I < vlf Iv + Av > I 2 , logarithmic scale, versus v for Av = l , 2 , and 3.

The U(3) representations are characterized in general by three quantum numbers. However, in the reduction of totally symmetric states of U(4), only totally symmetric states of U(3) appear, characterized by a single quantum number, nK, which can take the values In the further reduction from U(3) to O(3), the values of the angular momentum, /, contained in a given representation nn of U(3) are Finally, in the further reduction from O(3) to O(2), one has the usual rule, discussed in Chapter 1, Chain (II).

92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as The coefficients e, K, and K' are linear combinations of the coefficients ec, en, u0, u2, v0, V[, M'O of Eq.

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