Andromache (PBK) by Euripides, P. T. Stevens
By Euripides, P. T. Stevens
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These approaches proposed for ground band energies a series expansion in terms of J (J + 1) term. The weak point of these expansions is that they do not converge for high angular momenta. The first attempt to avoid this difficulty was due to Holmberg and Lipas [HoLi68] who proposed a square root of a linear expression of J (J + 1). This expression proves to work better than a quadratic expression in J (J + 1). Here we raise the question whether this formula can be improved so that it can be extended to the region of states with high angular momenta.
32) (panel b) as well as their product (c) are plotted as function of d for angular momentum projected states 2 Classical Versus Quantal Features in a Projected Coherent State (a) (b) (c) common value is reached for d larger than the maximum value shown in Fig. 4. 33) as a function of d. One notices that the departure from the classical limit is an increasing function of angular momentum. Also, this is increasing with the nuclear deformation. It is worth noticing that for large deformation, the J = 2 values become indistinguishable from each other.
Instead of finding the classical trajectories and then quantizing them, here we first quantize the energy by replacing 2 r• A → −i ∂ . 26) Thus, one arrives at the Schrödinger equation: − 2 A L2 A A ∂2 + + r 2 u(r ) = u(r ). 28) one obtains the following differential equation: x ∂2 1 ∂ + 2s + − x + 2 ∂x 2 ∂x 1 2s 2 − s − 2L 2 − −s + √ 2x 4 2 AA f (x) = 0. 29) This should be compared with the differential equation for the Laguerre polynomials: x ∂2 ∂ + n Lm + (m + 1 − x) n (x) = 0. 30) Indeed, the two equations are identical provided the following equations hold: 1 1 1 + m = 2s + , n = √ − − s, 2s 2 − s − 2L 2 = 0.