By F. Iachello, R. D. Levine
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Extra info for Algebraic Theory of Molecules (Topics in Physical Chemistry Series)
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. 1 Elements of the algebra of U(4) Explicit form No. of components Notation -VSlVxTC]^ 1 n •N/^xn]^ J [T^Xflf 3 5 [7I t XCT + O t XJl]^ ) 3 i[7l t x6-a t X7l]^ ) 0 3 1 [a'-xa]' ' Q D D' "a The operators /, Q, D, D' have the physical meaning of the angular momentum, quadrupole, coordinate, and momentum operators, respectively. The Hamiltonian operator for molecular rotational-vibrational spectra, which in the uncoupled form is written as in 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. 6/JV. The scaling of ot with N is because of the harmonic limit where F+ + £_ tends to f>N(r - re) [Eq. 87)]. 79). The limit of the operator F+ + F_ is If one realizes the operators a* and a in terms of differential operators one obtains We shall return to the geometrical content of the algebraic approach in several places, and in particular in Chapter 7.
21)]. This form of the algebra, called the uncoupled form, is not well suited for the analysis of the problem since one wants states of good angular momentum. The corresponding operators must then have definite transformation properties under rotations. This is equivalent to saying that Cartesian coordinates are not particularly useful to solve the Schrodinger equation with a spherically symmetric potential and that one prefers to use spherical coordinates. To this end, one divides the four boson operators, b^a, into a scalar, c^, and a vector nj (p, = 0, ±1).