# Nuclear Ground-State Properties from Mean-Field Calculations

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Additional info for Nuclear Ground-State Properties from Mean-Field Calculations

Example text

It is obvious that the solution, nonsingular for r -»• 0 (t —>• — x), corresponds to a pendulum which at t -*• —oo is in the position of unstable equilibrium a = 0 (or = rnr). When t —> — oo, we have a(t) -+ 7r —»• jel. (33) When t -> +oo, we have a(t) -> ± | , and, owing to the presence of the damping, for any t we get \a(t)\ < it. , it decreases as 1/r. 7(r), and consequently for B (r). is characterized by four parameters: three parameters T^o* which define the reference point, and a parameter 7, which has the meaning of an inverse radius of the region beyond which B\ decreases.

For the sake of simplicity we assume that *i,2n=ai,2*M+0i,2«n. (60) Let us now return to the integration which was performed when we derived (46). Multiplying (46) by x\[ J a ^r8ix\-2a'{xyz')-\ and interchanging the integrations over a' and Q\ we get 1 x\ f a'do' fd/2' , . n . » ^Ti~7rra ^ ~ 1 TTT^ -T—C^ZvSmFSip^-zz), 2-rr Xi(pi — p2) a J (ai —a +5) J 4-n- ] = £= ffi-g' «i« n r/3i. S = a/32- "2-Qi Pi-Pi At the choice of e according to (60) we have sin F(x, z') = sin F(z, —. z'). N. vz'v=*2a piz^, (ai==a2~a), • - lit "02 f a'da'p!

E.. ae ~ >/e. Then a ~ e2 also decreases with large r. As / further increases, e (the binding energy) increases, and the expansion in powers of e deteriorates. However, the solution still exists. When / increases to rhe point where a second level with a small binding energy e appears, then, by repeating the above operations, we shall show that a second solution appears. We therefore get a degree of ambiguity equal to the number of eigenvalues of the operator • . Actually, owing to the fact that a(oc) can equal not only zero but mr as well (the pendulum can sail through the equilibrium position several times from different directions), the degree of ambiguity of the fields B{ is even larger.