By F. Iachello, R. D. Levine
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Extra info for Algebraic Theory of Molecules (Topics in Physical Chemistry Series)
Thus N + I represents the number of bound states. When N —> oo one recovers the full oscillator spectrum. 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.
40 Chapter 2 The spherical components of n^ are related to the Cartesian components by Sometimes the operators o1',^ are denoted by With these definitions the creation operators o f , TtJ transform as spherical tensors under rotation. The annihilation operators do not. However, it is easy to construct operators that do transform as spherical tensors [Eq. 23)]. 1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. 25). Note that each tensor operator of multipolarity A, has 2X, + 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form.
It is increasingly the case that one refers to the operators of the algebra as the generators even when the group is not the object of direct interest. In making the connection to the differential equations form of quantum mechanics we shall use a realization of the operators X as differential operators. 4. Many other realizations of the same SO(3) algebra are discussed in Miller (1968). All admissible Lie algebras were classified by Cartan in 1905. Cartan's classification is given in Appendix A, where many other properties and definitions are provided.
Algebraic Theory of Molecules (Topics in Physical Chemistry Series) by F. Iachello, R. D. Levine