By F. Iachello, R. D. Levine
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Additional info for Algebraic Theory of Molecules (Topics in Physical Chemistry Series)
The simple examples discussed in this section illustrate the main properties of the algebraic method. By introducing the unitary algebra in 2 = (1+1) dimensions, one can simultaneously describe harmonic and anharmonic oscillators. Within the general description, two limiting cases can be solved exactly: (1) the purely harmonic oscillator V(x) = V0x2 corresponding to the subalgebras U(l) of U(2) and (2) the Morse oscillator V(x) = V0(l - e'^'^)2, corresponding to the subalgebra O(2) of U(2). For those two cases one can obtain explicit Summary of Elements of Algebraic Theory 35 expressions for the energy eigenvalues.
Therefore we limit our discussion to Lie algebras. It has become customary to denote both algebras and groups by the same capital letters. We shall follow this notation in this book as well. 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.
As a result, they are diagonal in the basis [^, A , 2 , . . , X,v], Summary of Elements of Algebraic Theory 25 The eigenvalues / have been evaluated for any Casimir operator of any Lie algebra, and a summary of the results is given in Appendix A. Using the expressions of the appendix, we find, for example, that the eigenvalues of the Casimir operator of SO(3), J2, in the representation I/ > is a familiar result. 6 Algebraic realization of quantum mechanics In Chapter 1 we have discussed the familiar realization of quantum mechanics in terms of differential operators acting on the space of functions (the Schrodinger wave function formulation, also called "wave mechanics").
Algebraic Theory of Molecules (Topics in Physical Chemistry Series) by F. Iachello, R. D. Levine