By F. Iachello, R. D. Levine
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28), that f can be thought of as a rotation operator so that its matrix elements can be computed (Levine and Wulfman, 1979) using the known results for the rotation matrices. When a
Since «T is an invariant, so is n\. One can thus write down the most general bilinear algebraic Hamiltonian with dynamic symmetry U(l) as The eigenvalues are This represents a truncated anharmonic oscillator with anharmonicity controlled by K. 1 states. The truncated harmonic potential and its spectrum of four (N = 3) bound Summary of Elements of Algebraic Theory 31 Note how the finite number of bound states arises very naturally in the algebraic approach. This example also illustrates the role of the "extra" quantum number, N.
Dynamical symmetries for one-dimensional problems can be studied by considering all the possible subalgebras of U(2). There are two cases We shall omit from here on the letter S in the orthogonal algebras since there is no difference in the algebraic structure of SO(«) and O(«). However, we will keep the letter S, if appropriate, in the unitary groups, since there is a difference in the algebraic structure of SU(«) and U(n). One-dimensional problems present on one hand the simplest (and most studied) example of algebraic theory, and on the other hand involve some subtle problems that are worthwhile elucidating.