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Configuration mixing calculations in soluble models
Nuclear Physics A403 (1983) 381-395 @ North-Holland Publishing Company
CONFIGURATION MIXING CALCULATIONS IN SOLUBLE CODED M. C. CAMBIAGGIO +, A. PLASTINO t and L. SZYBISZ’
Repartamentode Fisica, Facultadde Ciencias Exactas, UniversidadNaciolzalde La Plata, CC67,1900 La Plata, Argentina
H. G. MILLER NRIMS,
CSIR, Pretoria OQOi,South Africa
Received 4 October 1982 Abstract: Configuration mixing calculations have been performed in two quasi-spin models using basis states which are solutions of a particular set of Hartree-Fock equations. Each of these solutions, even those which do not correspond to the global minimum, is found to contain interesting physical information. Relatively good agreement with the exact lowest-lying St&f3 has been obtained. In particular, one obtains a better approximation to the ground state than that provided by Hartree-Fock.
1. Introduction When studying the structure of a nucleus one is generally only interested in the lowest-lying bound states. Consequently, if one does not need the corresponding complete set of eigenstates, one tries to avoid the large diagonalizations which occur in a shell-model calculation. This is the motivation for shape mixing calculations in deformed nuclei. Here one mixes solutions of the Hartree-Fock or shapes. (HI?) equations, which generally possess different deformations Configuration mixing calculations have also been done with configurations constructed from one-particle-one-hole (fp-lh) excitations of the HF ground state and/or lp-lh excitations of the additional HF solutions. Recently Miller el al. ’ -3) have proposed a systematic approach for generating a set of basis states via variational methods. These states are then used in configuration mixing calculations providing a good description of the lowest-lying states of a many-body system. The basis states may be orthogonal lS2) or nonorthogonal 3, but we shall be concerned at present with the latter case only. In this + Member of the Scientific Research Career of the Consejo Naaonal y Tkcnicas, Argentina. 381
de Investigacion~
Cientificas
382
M. C. Cambiagglo
approach
the basis
equations
(for example,
states
et al. 1 Configuration
are all the. solutions
HF equations).
As usual,
mixmg
of the ground the variational
state
variational
method
breaks
some symmetry, which is then restored by projection. Finally, the hamiltonian is again diagonalized in the non-orthogonal, undercomplete, set of projected states. Reasonably rapid convergence at the lower end of the exact eigenvalue spectrum -is expected if the variational energies of the basis states form a monotonically increasing sequence. Miller and Schroder 3, have tested their method -by calculating the low-lying T = 0 positive-parity states in 20Ne and have achieved very good agreement with relatively few basis states. The results obtained in ref. 3, are quite interesting in that they show the power of the new approach. We feel that further tests may be of interest, in particular within the context of some simple models which allow for “exact” HF calculations and that, moreover, make it possible to employ projection techniques without much difficulty. We think that, in this way, a better understanding of the advantages and disadvantages of the proposal of Miller et al. ’ -“) may be obtained.
2. Review of the models We shall consider two models in this work, each with N identical particles distributed in two N-fold degenerate single-particle (s.p.) levels. Two quantum numbers characterize a given s.p. state. One of them adopts the values G = - 1 (lower level) and CJ= + 1 (upper level). The other, which may be called p-spin, singles out a state within the N-fold degeneracy. Several two-level models of this type are reviewed in ref. “).
2.1. THE LIPKIN
MODEL
This simple model was suggested by Lipkin and coworkers ‘) in order to test new approaches for the solution of the many-body problem. The interaction is a monopole force scattering two particles from one level into the other. We let p run from 1 to N and introduce the quasi-spin operators 5,
(2.1) which fulfill the angular momentum operators the hamiltonian is written
commutation as 5,
rules [SU(2)].
In terms of these
(2.2)
383
M. C. Cambiaggm et al. 1 Configuration mixing
The energy is given in units of the s.p. energy E. As the hamiltonian
(2.2) commutes
with the quasi-spin I2 = 2 +@+J_ The exact eigenvalues,
obtained
diagonalizing
+J^-JI+),
(2.3)
the hamiltonian,
can be labelled
by
the quasi-spin J and depend on the number of particles N and the coupling constant v. The maximum quasi-spin value is J,,, = +N, as it can be seen from (2.1).
2.2. SU(2)*SU(2)
MODEL
This is a generalization of the SU(2) Lipkin model proposed by Cambiaggio and Plastino “)_ For convenience, we allow the p-spin to run from -iN to +N in steps of one, excluding zero, and define, further, the quasi-spin operators “) S
and, also
The S-operators commute among themselves in the same manner momentum operators. We introduce, additionally V+ = P’ q and
obtain
two
= $(J^+ +s^+),
= 5(.JZ+ O), quasi-spin
operator commutes The hamiltonian
@+ = @? = $3,
-sI+),
WZ = $(& - O),
sets : P,,
P_,
pz and
(2.6) m+,
I%,
qz. Any
given
P
with all &-operators [SU(2)*SU(2)]. (in units of E) may now be written as 6,
fi, = & +$,@2
_ pz _ @/“)
= ~+~z+~k(2~~z+~+~_+~_~+). The exact eigenstates
as the angular
(2.7)
are given by 6, E = J,++k[J(J+l)-W(W+l)-V(V+l)],
(2.8)
384
and, consequently,
M. C. Cambiaggio et al. 1 Configuration mixing
can also be labelled
by J. The maximum
value of I/ and
W is
+lv.
3. Calculation
of basis states
3.1. THE HF TRANSFORMATION
The first step in the application of the method developed by Miller and SchrGder 3, is the generation of a set of basis states which are all the solutions of the HF ground-state variational equations. Consequently, we need a general HF transformation that provides us with several such solutions, even if it violates the conservation of‘ the quasi-spin J, which is a good quantuin number for the exact solution in both models. This is not the case with the usual HF transformation ‘) applied in the Lipkin model which is a single rotation of all particles in the J-space [SU(2)]. It does not violate 2’ but yields at most an additional “deformed” HF solution when the interaction strength exceeds a critical value. As we are interested in finding other solutions we must look for a more general HF transformation. The most general transformation which.does not mix the different p-values is the Cayley-Klein transformation. This corresponds to an independent three dimensional rotation of each particle’s quasi-spin. In this case, the total HF energy depends on 3N parameters. In principle one should solve the HF equations derived from this transformation. But this is not possible analytically. However, by reducing the size of the variational space, i.e. by reducing the number of variational parameters from 3N to, for example, 6, one may still hope to obtain physically significant results without losing the advantage of having analytical solutions. Moreover, one can try mixing different p-values. With this objective in mind, we note that the J-operators may be expressed in using eq. (2.6). Hence, the Lipkin terms of the V and W spin operators, hamiltonian (2.2) may be written as (3.1) and we may use a simplified HF transformation which is a uniform rotation of all the quasi-spins in the V-space plus a similar uniform rotation in W-space this same transformation is also used in the [su(2)*su(2)]. Of course SU(2)*SU(2) model, i.e., with the hamiltonian (2.7). In both cases, we rotate all quasi-spins (eq. (2.6)) with respect to the three Euler angles which we call CI,/?, y in the V-space and cp, 9, $ in the W one. It should be noted that this transformation mixes different p-values and does not conserve ?‘, but it preserves the symmetry of interchange of V and W.
385
M. C. Cambiaggio et al. 1 Configuration mixing 3.2. THE LIPKIN
MODEL
Applying the above-mentioned transformation one obtains the following expectation value: E = (HFlfi,IHF)
to the Lipkin hamiltonian
(3.1)
= +V(COS~+COS~)
-~~N[(N-2~(cos2ccsin2~+cos2~sin28)+2Nsin~sin~cos(~+~)]. Notice that it is independent then
(3.2)
of the Euler angles y and $. The HF equations are,
dE dE -_=-=:_
da
aE
aE
ag NapXi=
0,
(3.3)
which provide the solutions listed in table 1. It is to be noted that solutions (a), (b), (d) and (e) are the ones obtained with the usual HF transformation ‘). However, seven lzew solutions are found with this more general transformation. Four of them occur if u 2 v: = 2/(N - 2) and other two [(i) and (k)] occur only if ZI2- z$’= 1, i.e. for coupling constants larger than a certain critical value. A similar situation occurs for solutions (d) and (e) but in this case the critical coupling constant points out the existence of a ground-state phase transition, as is well-known. Consequently, we look for the possible physical meaning of these new critical
TABLE1 Solutions of the HF equations for the Lipkin model V-space
W-space
Solution
Range of v G(
B
9
8
0 “) oa)
0
:
$9=0! rp=a
0 7t
C
0 “)
;
cp=cc
d e f g h i
0 jz. 0 0 $ $c
j k
ti 0
arcos [l/u(N- l)] arcos [- I/v(N- l)] arcos [2/4N- 2)] arcos [Z/u(N-2)] arcos [ -2/v(N- 2)] arc& [ - Z/u(N- 2)] arcos (l/v) arcos (- l/u)
0 $7~ +7r +7r 0 0 & x
all v all u n all u 0 = B ” 2 l/(N-1) 0 = p 0 2 l/(N- 1) 0 u 2 2/(N-2) H u 2 2/(N-2) 0 v 2 2/N-2) u 2 2/(N-2) LIZ1 0:s U21 e=p
Energy
-)N $N 0 -$N[u(N- 1)+ I/~(N- I)] +wu(N- l)+ l/v(N- l)] -$N&(N-2)+ 1+ I/U(N-2)] -~~~u(N-2)lt l/o(N-2)] $N[$u(N-2)1+ l/u(N-2)] $V&(N-2)+ l+ l/u(N-2)] -+N(u+ l/v) $N(v+ l/u)
“) Actually, for the solutions (a), (b) and (c) the angles a and cp are equal but undetermined, adopted c( = cp = 0 for the calculations.
we
386
LIPKIN-MODEL
20
200
15
150
N=12 6-..-..Ei-
6---
&zz.z 6---
6,s -
6-
6,5,4,3 -
44’
+-..-.-
5
&----53+ 4/---:-
S,5,4,3,2 -
0
6,5,4,3,2,7,0 -
100
10
4546 3& 54q.------
6,5,4 -
s,5,4,3,2,1
6-
5'
5--5s3---34642--‘Y----32‘1 ,5*4,3_ 2,1,0
50
0
l---
s,5,4,3,2,t
--
I& 4-
-
:j== kf-----
6,5,4,3 ~
c------
s,5,4,3,2
fC-----
-5
$===
32 ‘542‘S 433’ S’ 55--...--
-50
6,5,4 -5
gII 46---
6,5 -
-10
+===
.lOO
6---6-
S-
-75
-If
+===
-150
-20 9-----
v=o
kO.6
Fig. 1. Exact solutions for the Lipkin model for N = 12 and some values of u.
M. C. Cambiaggio et al. 7 Configuration mixing coupling
constants
and find that they indicate
crossings
381
of excited levels which also
are a sort of phase transition “). With respect to solutions (f), (g), (h) and (i), the appearance of these new excited HF levels occurs at a coupling constant that is very near the one in which two exact excited levels cross each other. Namely, the 1 becomes lower than the third level with J = JmaX. Of second level with J = J,,,course, since the exact levels are symmetric about zero, the crossing also occurs for the corresponding positive energy eigenvalues. This, is illustrated in fig. 1 for N = 12. These crossings can also be clearly seen in fig. 2b of ref. “) (notice that the coupling constant scale is different from the one used in the present paper). As N increases, more crossings appear due to the fact that two succesive levels with the same J approach each other when v increases (see fig. 1). However, the critical coupling constant v: always seems to indicate the above-mentioned crossing which orders the four lowest (and the four highest) levels in two pairs with J = Jr,,,, and J = J,,, - 1. The difference between the exact and the approximate U: decreases when N increases. On the other hand, in order to try to understand the physical meaning of the second new ‘critical coupling constant (vy = 1) we note that it is independent of N. We have therefore looked for a crossing of excited levels which always occurs at the same v and find that this is what happens with the crossing of the second J = 2 level with the third J = 3 level, which takes place at v -.0.6. Here, we conclude that v: points out this particular crossing which orders the first two odd-J levels below (and above) zero. This feature is also displayed in fig. 1. Moreover, the new solutions [(c), (f), (g), (h), (i), (j) and (k)] do not correspond to wave functions with the quasi-spin J as a good quantum number. The usual HF solutions, instead, have good J = J,,,. This can be seen studying the transformation applied and noting that it does not violate 3’ if tl = 40 and /I = 8.
3.3. SU(2)*SU(2) MODEL
In this model the expectation state gives
value of the hamiltonian
E = (HFIE?,JHF) +&kN2[sin
(2.7) with respect to the HF
= -+N(cos/?+cos~)
/Isin &OS (LX - q) +cos j3cos 01,
(3.4)
which is independent of y and I/. The HF equations (3.3) provide the solutions shown in table 2. These results coincide with the ones obtained in ref “) and no new solutions are found. As in the previous model, some of these solutions [(b) and (c)] have good J = Jmax while the others [(a) and (d)] do not correspond to a fixed quasi-spin value.
388
Ad. C. Cambiaggio
et al. / Configuration
mixing
TABLE 2 Solutions
of the HF equations
for the SU(2)*SU(2)
V-space
model
W-space
Solution
Range c(
B
0 “1 0 “1 0 “1 P “) For the solutions
of k
cp
0
0
0 “1
?I
all k
0
0 “)
0
all k
0 “1 0
arcos :2/kN)
(a), (b) and (c) the angles
Energy
-$kN’ -fN+&kN=
all k k 2 2/N
CI and cp are undetermined;
;N-?+kN= -+k-&kN=
we adopted
a = cp = 0
for the calculations.
4. Configuration
mixing calculations
4.1. FORMALISM
As it was mentioned in the introduction one must restore the symmetry violated by the HF transformation before proceeding to do the configuration mixing. Some of the HF wave functions obtained in the preceding section do not possess good quasi-spin J and, consequently, we project out this quantum number. We diagonalize f2 within our relevant multiplet, this procedure yielding the set of wave functions IV> w,J,
c
n>=
CJ,,K, WIV,
w, vz,w.
(4.1)
v,w, Here n distinguishes among states .of the same J. The projection given by b, = c IV, W, J, n)
operator
P, is
(4.2)
For obtaining the projected wave function
I@J>= $
3
(4.3)
IHF),
where NJ is the norm, and the projected energy E
(PHF)
=
J
(HFIf@,IHF) (HFJ&HF)
’
(4.4)
we need an explicit expression for the HF wave function IHF) =
c 6%
WV,, KV’,
W,
v,,K>.
(4.5)
M. C. Cambiaggio et al. 1 Conjiguration mixing
389
As in refs 7,9), th e coefficients D(V,, W,) can be easily obtained in terms of the Euler angles
where the d-functions are defined by Edmonds lo). Here we have taken y = $ = 0 because the HF solutions of tables 1 and 2 are independent of these particular angles. It is to be noted that eqs. (4.1)-(4.6) allow us to do an exact projection. Once the projected wave functions (4.3) have been calculated, the hamiltonian is diagonalized in this non-orthogonal set, solving the following equation: i
(Hji - EJNj,)a, =
0.
(4.7)
i=l
Here m is the number of projected states considered, Hj, and Nji are the energy matrix and the overlap matrix, respectively, evaluated with the projected states, EJ are the energy eigenvalues obtained after the configuration mixing (c.m.) and ai are the mixing coefficients. In order to solve the problem (4.7) we follow the prescription described in ref. ll). 4.2. RESULTS
The results obtained in the Lipkin model for N = 8 and v = 1.1 are shown in fig. 2. The quantum number which characterizes the levels is the quasi-spin 9. Notice that we have eleven HF solutions for J = J,,, = $N, but we only have seven solutions for the other values of J because, as it was already mentioned, solutions (a), (b), (d) and (e) in table 1 have good J = J,,,,,. Studying the spectrum obtained after performing c.m. with the projected states which arise from all the HF states (case J), it can be seen that an excellent agreement with the exact values is obtained, except for J = 3. As can be seen in fig. 2 (A -J) increasing the number of projected HF basis states from 5 to 11 does not always significantly improve the quality of the spectrum of the J = 3 states. This is not too surprising considering our selection of the variational subspace in which HF basis states have been obtained. It is obviously too restrictive in this case to obtain a good description of all the states in the spectrum but adequate enough to ensure good agreement with a large number of states in the spectrum. It is interesting to note that the degeneracy present in the exact solutions due to the symmetry of interchange of V and W (i.e. two states for each level) is not always reproduced in the c.m. calculations. For example, the degeneracy of the J = 1 levels is not restored until at least six projected HF basis states have been used in the c.m. calculations ,(see fig. 2, B-F). We have studied the quality of the agreement obtained for different values of v
EXACT
44
HF
PHF
\
:F
"F
",yfjy_"o
33---4---'&----
34---
22---4----
l,OL--
43.2 1-----
4-
I+---
,1:0-
412 3-
j-
$= 2---2-
3--444-
4-
34434-
4-
z24----
+====
4,3‘2,3p-
3.me1
4----34422----
3-
;w==
2---Z$ZZ
&----4-
(&.-.---4-
3--z----2---4---
1-----
43,2,1,0-
1-
4Z-----2---3-
4---
3---3-
i-
i$--_
4z
Ii-----
fZ==
z-
2---$z=
?-----
310z2 4T== 1.0 --
CM
45
l+-.---
4-
2---3--2-
4,3.117
v
4-
"yg-404x
43----
2----
5-----
4..--.._ 1D-
2---
$======
3---
4----
g===== +===
24-
4.3 -
4----
k--
'7
3lo'2-
+====
4----
2----
44.3-
4----
LIPKIN-MODEL
3-
2---.-. g===
p=
5 ~
L-----
-
43t-----
:---3-
4.5 $====
"z
4-
V=l.l
N=8
/
Fig. 2. Results for the Lipkkin model for N = 8 and u = 1.1. ~on~~ratioll mixing results m cases A, B, , . . , J were calculated using the two, three, . . . . eleven lowest HF solutions, respectively.
-20
-15
-10
-5
20 LLI
5
10
15
20
M. C. Cambiaggio
et al. / Configuration
391
mixing
and N. For a given N it does not change significally as a function of the coupling constant when v is larger than one. For v 5 1 the agreement with the exact levels is dramatically worse due to the reduced number of HF solutions available. Both features are illustrated in fig. 3. With respect to the variation with the number of particles the overall agreement I
F-
LIPKIN-MODEL
15-
N-8 80-
44-
44-
4-
44-
60 10 3---_ 3-
40 34-
4-
5-
33’
33-
4-
4-
4-
4ZZ-
Z-
422-
3l-
li_ 2L 1-
4422’
20 -
3l-
o - “~$--
?L 4qF-
‘1 -
l-
it__
3-
;F
3224-
2-
-2o-
2-
4’z 4-
4-
322444-
443-
33-
-40-
-60 +-
33e
44-
4-
EXACT
CM
-80
v-1.0
Fig. 3. Results
for the Lipkin
-
t model
z-
4, 4-
CM
EXACT v= 5.0
for N = 8 and u = 1 and 5.
392
M. C. Cambiaggio
3Q-
et al. / Configuratzon
LIPKIN-MODEL
mixing
N=lO 5-
i- 5,+,3,2,l,o l!F 4_ Y-27
Fig. 4. Results
becomes
worse
when
for the Lipkm
N increases.
model
for N = 10 and v = 1.1.
This is to be expected
because
the number
of
exact levels increases with N while the number of HF solutions remains the same. In particular, the worst results are obtained when J,,, is odd in which case the about zero. As an example we give the c.m. levels for J,,, are not symmetric results for N = 10 and u = 1.1 in fig. 4. Finally, the results obtained in the SU(2)*SU(2) model for N = 8 and k = 0.28 are shown in fig. 5. Here we have four projected states with J = J,,, but only two for the other values of J. This is due to the fact that solutions (b) and (c) of table 2 have good J = J,,,. The agreement with the exact values obtained for the lowest states for each J is very good. However, again as the coupling constant k increases the situation gets worse. The only difference between the projected spectrum and
M. C. ~Cambiaggio et al. / Con~g~ra~~onrni~~~g
6
N=8
~U(2)~~U~2)-~ODEL
5
4-
4
L-
3
4.3x
393
@--4-
4-
w2 z; 1 0 -1 -2 -3 -4
‘o-
t$z
4,
3f_ 1’ $=
@;_
3’
@
EXACT
o-
/
l4 -4.Eas 2\ y---
HF
- PHF kz0.28
l4-B 2, 3/-
CM
Fig. 5. Results for the SU(2)*SU(Z) model for N = 8 and k = 0.28
the c.m. one is the position of the second level with .I = J,,, (J = 4 in fig. 5). In the projected spectrum it practically coincides with the lowest level of the same J whereas in the cm. spectrum it lies near the third (and not the second) level with the same J. This can be understood by considering the quantum number J,, which has not been used in these calculations but, nevertheless, is a good quantum number for the exact levels in the SU(2)*SU(2) model. If we investigate which J, values are present in the projected wave functions we find that those coming from solution (a) in table 2 have J, = 0 while those coming from solution (d) correspond to a mixture of J, values but all with the same parity (odd or even J&J. Moreover, solution (b) corresponds to J, = -Jmax and solution (c) to J, = Jmx. Consequently, the c.m. separates the levels according to the J, values and this is the reason why we do not find an approximation to the second level with J = J,,,, as it has the opposite parity. Similar results are found for other values of the number of particles. N.
394
M. C. Cumbiaggio et al. 1 Configuration mixing
5. Conclusions
In the present work we have shown that one may still obtain physically significant results even if one reduces the number of variational parameters considered in the HF transformation. But one should not be surprised if one does not obtain a good description of all the lowest-lying eigenstates and that the convergence rate is not as rapid as the one expected when using a more general variational ansatz. This of course is analogous in some respects to performing variational calculations in a Nilsson basis and varying only one parameter per s.p. orbital (i.e. the deformation). Here the variational problem will be easier to solve (smaller variational space) but the convergence rate in the c.m. calculations may not be as good as in the case where HF basis states have been used. Another important conclusion arrived at in this work is that all the solutions of the HF ground-state variational equations found in the present work, which do not correspond to the global minimum and that are usually not taken into account, contain interesting physical information. This is clearly illustrated by the new critical coupling constants found in the Lipkin model which point out crossings in the exact excited levels, as has been discussed in subsect. 3.2. Similarly, in the SU(Z)*SU(Z) model, solution (a) provides an approximation for all the levels with J, = 0 (see fig. 5 and the corresponding discussion). This physical meaning of all the HF solutions justifies their use as basis states for a cm. calculation, as proposed by Miller and Schroder 3, and performed in the present work. However, care must be taken to be sure that one considers the lowest HF solutions. If such is not the case, differences as great as the ones observed between figs. 2 and 3 may occur, depending on which HF solutions are missing. An additional point to be careful about is the possible presence of linearly dependent projected basis states. They appear as states with norm zero on diagonalizing the overlap matrix Nji in eq. (4.7) and must be eliminated before going on. This problem is always present when one works with an overcomplete basis, for example in trying to obtain an approximation to the three levels (six states) with J = 1 in the Lipkin model using seven projected states. It is interesting to mention that with the present method one always obtains a better approximation to the ground state than that provided by HF alone 12). This is true in the Lipkin model for example if one only mixes the HF ground state with the trivial solution (a) in table 1 [for u 2 l/(N - l)]. Nevertheless, the present results are not as good as the ones obtained in ref. “) by means of a more sophisticated multiconfiguration ‘Hartree-Fock calculation (notice, however, that the results of ref. “) are restricted to v < 1). Finally, it should be noted in the present work that the agreement with the exact lowest-lying states obtained in the Lipkin model is worse (except for N = 8) than that in the SU(2)*SU(2) model. Two of the authors (M.C.C. and L.S.) thank the staff at the Departamento
de
M. C. Cambiaggio
et al. / Configuration
mxing
395
Fisica of the Comisibn National ‘de Energia Atbmica for the kind hospitality afforded to them during the performance of this work.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
H. G. Miller, R. M. Dreizler and G. Do Dang, Phys. Lett. 77B (1978) 119 H. G. Miller and R. M. Dreizler, Nucl. Phys. A316 (1979) 32 H. G. Miller and H. P. Schriider, Z. Phys. A304 (1982) 273 M. C. Cambiaggio, A. Plastino, L. Szybisz and M. de Llano, Rev Mex. Fis. 27 (1981) 223 A. Ghck, H. J. Lipkm and N. Meshkov, Nucl. Phys. 62 (1965) 118 M. C. Cambiaggio and A. Plastino, Z. Phys. A291 (1979) 277 D. Agassi, H. J. Lipkin and N. Meshkov, Nucl. Phys. 86 (1966) 321 M. C. Cambiaggio, A. War, F. J. Margetan, A. Plastino and J. P. Vary, to be published A. Faessler and A. Plastino, Z. Phys. 220 (1969) 88 A. R. Edmonds, Angular momentum in quantum mechanics (University Press, Princeton, 1960) A. Faessler, F. Griimmer and A. Plastino, 2. Phys. 260 (1973) 305 G. Fichera, Annah di Mathematics pura ed Aplicata (IV) CVIII (1976) 367
CONFIGURATION MIXING CALCULATIONS IN SOLUBLE CODED M. C. CAMBIAGGIO +, A. PLASTINO t and L. SZYBISZ’
Repartamentode Fisica, Facultadde Ciencias Exactas, UniversidadNaciolzalde La Plata, CC67,1900 La Plata, Argentina
H. G. MILLER NRIMS,
CSIR, Pretoria OQOi,South Africa
Received 4 October 1982 Abstract: Configuration mixing calculations have been performed in two quasi-spin models using basis states which are solutions of a particular set of Hartree-Fock equations. Each of these solutions, even those which do not correspond to the global minimum, is found to contain interesting physical information. Relatively good agreement with the exact lowest-lying St&f3 has been obtained. In particular, one obtains a better approximation to the ground state than that provided by Hartree-Fock.
1. Introduction When studying the structure of a nucleus one is generally only interested in the lowest-lying bound states. Consequently, if one does not need the corresponding complete set of eigenstates, one tries to avoid the large diagonalizations which occur in a shell-model calculation. This is the motivation for shape mixing calculations in deformed nuclei. Here one mixes solutions of the Hartree-Fock or shapes. (HI?) equations, which generally possess different deformations Configuration mixing calculations have also been done with configurations constructed from one-particle-one-hole (fp-lh) excitations of the HF ground state and/or lp-lh excitations of the additional HF solutions. Recently Miller el al. ’ -3) have proposed a systematic approach for generating a set of basis states via variational methods. These states are then used in configuration mixing calculations providing a good description of the lowest-lying states of a many-body system. The basis states may be orthogonal lS2) or nonorthogonal 3, but we shall be concerned at present with the latter case only. In this + Member of the Scientific Research Career of the Consejo Naaonal y Tkcnicas, Argentina. 381
de Investigacion~
Cientificas
382
M. C. Cambiagglo
approach
the basis
equations
(for example,
states
et al. 1 Configuration
are all the. solutions
HF equations).
As usual,
mixmg
of the ground the variational
state
variational
method
breaks
some symmetry, which is then restored by projection. Finally, the hamiltonian is again diagonalized in the non-orthogonal, undercomplete, set of projected states. Reasonably rapid convergence at the lower end of the exact eigenvalue spectrum -is expected if the variational energies of the basis states form a monotonically increasing sequence. Miller and Schroder 3, have tested their method -by calculating the low-lying T = 0 positive-parity states in 20Ne and have achieved very good agreement with relatively few basis states. The results obtained in ref. 3, are quite interesting in that they show the power of the new approach. We feel that further tests may be of interest, in particular within the context of some simple models which allow for “exact” HF calculations and that, moreover, make it possible to employ projection techniques without much difficulty. We think that, in this way, a better understanding of the advantages and disadvantages of the proposal of Miller et al. ’ -“) may be obtained.
2. Review of the models We shall consider two models in this work, each with N identical particles distributed in two N-fold degenerate single-particle (s.p.) levels. Two quantum numbers characterize a given s.p. state. One of them adopts the values G = - 1 (lower level) and CJ= + 1 (upper level). The other, which may be called p-spin, singles out a state within the N-fold degeneracy. Several two-level models of this type are reviewed in ref. “).
2.1. THE LIPKIN
MODEL
This simple model was suggested by Lipkin and coworkers ‘) in order to test new approaches for the solution of the many-body problem. The interaction is a monopole force scattering two particles from one level into the other. We let p run from 1 to N and introduce the quasi-spin operators 5,
(2.1) which fulfill the angular momentum operators the hamiltonian is written
commutation as 5,
rules [SU(2)].
In terms of these
(2.2)
383
M. C. Cambiaggm et al. 1 Configuration mixing
The energy is given in units of the s.p. energy E. As the hamiltonian
(2.2) commutes
with the quasi-spin I2 = 2 +@+J_ The exact eigenvalues,
obtained
diagonalizing
+J^-JI+),
(2.3)
the hamiltonian,
can be labelled
by
the quasi-spin J and depend on the number of particles N and the coupling constant v. The maximum quasi-spin value is J,,, = +N, as it can be seen from (2.1).
2.2. SU(2)*SU(2)
MODEL
This is a generalization of the SU(2) Lipkin model proposed by Cambiaggio and Plastino “)_ For convenience, we allow the p-spin to run from -iN to +N in steps of one, excluding zero, and define, further, the quasi-spin operators “) S
and, also
The S-operators commute among themselves in the same manner momentum operators. We introduce, additionally V+ = P’ q and
obtain
two
= $(J^+ +s^+),
= 5(.JZ+ O), quasi-spin
operator commutes The hamiltonian
@+ = @? = $3,
-sI+),
WZ = $(& - O),
sets : P,,
P_,
pz and
(2.6) m+,
I%,
qz. Any
given
P
with all &-operators [SU(2)*SU(2)]. (in units of E) may now be written as 6,
fi, = & +$,@2
_ pz _ @/“)
= ~+~z+~k(2~~z+~+~_+~_~+). The exact eigenstates
as the angular
(2.7)
are given by 6, E = J,++k[J(J+l)-W(W+l)-V(V+l)],
(2.8)
384
and, consequently,
M. C. Cambiaggio et al. 1 Configuration mixing
can also be labelled
by J. The maximum
value of I/ and
W is
+lv.
3. Calculation
of basis states
3.1. THE HF TRANSFORMATION
The first step in the application of the method developed by Miller and SchrGder 3, is the generation of a set of basis states which are all the solutions of the HF ground-state variational equations. Consequently, we need a general HF transformation that provides us with several such solutions, even if it violates the conservation of‘ the quasi-spin J, which is a good quantuin number for the exact solution in both models. This is not the case with the usual HF transformation ‘) applied in the Lipkin model which is a single rotation of all particles in the J-space [SU(2)]. It does not violate 2’ but yields at most an additional “deformed” HF solution when the interaction strength exceeds a critical value. As we are interested in finding other solutions we must look for a more general HF transformation. The most general transformation which.does not mix the different p-values is the Cayley-Klein transformation. This corresponds to an independent three dimensional rotation of each particle’s quasi-spin. In this case, the total HF energy depends on 3N parameters. In principle one should solve the HF equations derived from this transformation. But this is not possible analytically. However, by reducing the size of the variational space, i.e. by reducing the number of variational parameters from 3N to, for example, 6, one may still hope to obtain physically significant results without losing the advantage of having analytical solutions. Moreover, one can try mixing different p-values. With this objective in mind, we note that the J-operators may be expressed in using eq. (2.6). Hence, the Lipkin terms of the V and W spin operators, hamiltonian (2.2) may be written as (3.1) and we may use a simplified HF transformation which is a uniform rotation of all the quasi-spins in the V-space plus a similar uniform rotation in W-space this same transformation is also used in the [su(2)*su(2)]. Of course SU(2)*SU(2) model, i.e., with the hamiltonian (2.7). In both cases, we rotate all quasi-spins (eq. (2.6)) with respect to the three Euler angles which we call CI,/?, y in the V-space and cp, 9, $ in the W one. It should be noted that this transformation mixes different p-values and does not conserve ?‘, but it preserves the symmetry of interchange of V and W.
385
M. C. Cambiaggio et al. 1 Configuration mixing 3.2. THE LIPKIN
MODEL
Applying the above-mentioned transformation one obtains the following expectation value: E = (HFlfi,IHF)
to the Lipkin hamiltonian
(3.1)
= +V(COS~+COS~)
-~~N[(N-2~(cos2ccsin2~+cos2~sin28)+2Nsin~sin~cos(~+~)]. Notice that it is independent then
(3.2)
of the Euler angles y and $. The HF equations are,
dE dE -_=-=:_
da
aE
aE
ag NapXi=
0,
(3.3)
which provide the solutions listed in table 1. It is to be noted that solutions (a), (b), (d) and (e) are the ones obtained with the usual HF transformation ‘). However, seven lzew solutions are found with this more general transformation. Four of them occur if u 2 v: = 2/(N - 2) and other two [(i) and (k)] occur only if ZI2- z$’= 1, i.e. for coupling constants larger than a certain critical value. A similar situation occurs for solutions (d) and (e) but in this case the critical coupling constant points out the existence of a ground-state phase transition, as is well-known. Consequently, we look for the possible physical meaning of these new critical
TABLE1 Solutions of the HF equations for the Lipkin model V-space
W-space
Solution
Range of v G(
B
9
8
0 “) oa)
0
:
$9=0! rp=a
0 7t
C
0 “)
;
cp=cc
d e f g h i
0 jz. 0 0 $ $c
j k
ti 0
arcos [l/u(N- l)] arcos [- I/v(N- l)] arcos [2/4N- 2)] arcos [Z/u(N-2)] arcos [ -2/v(N- 2)] arc& [ - Z/u(N- 2)] arcos (l/v) arcos (- l/u)
0 $7~ +7r +7r 0 0 & x
all v all u n all u 0 = B ” 2 l/(N-1) 0 = p 0 2 l/(N- 1) 0 u 2 2/(N-2) H u 2 2/(N-2) 0 v 2 2/N-2) u 2 2/(N-2) LIZ1 0:s U21 e=p
Energy
-)N $N 0 -$N[u(N- 1)+ I/~(N- I)] +wu(N- l)+ l/v(N- l)] -$N&(N-2)+ 1+ I/U(N-2)] -~~~u(N-2)lt l/o(N-2)] $N[$u(N-2)1+ l/u(N-2)] $V&(N-2)+ l+ l/u(N-2)] -+N(u+ l/v) $N(v+ l/u)
“) Actually, for the solutions (a), (b) and (c) the angles a and cp are equal but undetermined, adopted c( = cp = 0 for the calculations.
we
386
LIPKIN-MODEL
20
200
15
150
N=12 6-..-..Ei-
6---
&zz.z 6---
6,s -
6-
6,5,4,3 -
44’
+-..-.-
5
&----53+ 4/---:-
S,5,4,3,2 -
0
6,5,4,3,2,7,0 -
100
10
4546 3& 54q.------
6,5,4 -
s,5,4,3,2,1
6-
5'
5--5s3---34642--‘Y----32‘1 ,5*4,3_ 2,1,0
50
0
l---
s,5,4,3,2,t
--
I& 4-
-
:j== kf-----
6,5,4,3 ~
c------
s,5,4,3,2
fC-----
-5
$===
32 ‘542‘S 433’ S’ 55--...--
-50
6,5,4 -5
gII 46---
6,5 -
-10
+===
.lOO
6---6-
S-
-75
-If
+===
-150
-20 9-----
v=o
kO.6
Fig. 1. Exact solutions for the Lipkin model for N = 12 and some values of u.
M. C. Cambiaggio et al. 7 Configuration mixing coupling
constants
and find that they indicate
crossings
381
of excited levels which also
are a sort of phase transition “). With respect to solutions (f), (g), (h) and (i), the appearance of these new excited HF levels occurs at a coupling constant that is very near the one in which two exact excited levels cross each other. Namely, the 1 becomes lower than the third level with J = JmaX. Of second level with J = J,,,course, since the exact levels are symmetric about zero, the crossing also occurs for the corresponding positive energy eigenvalues. This, is illustrated in fig. 1 for N = 12. These crossings can also be clearly seen in fig. 2b of ref. “) (notice that the coupling constant scale is different from the one used in the present paper). As N increases, more crossings appear due to the fact that two succesive levels with the same J approach each other when v increases (see fig. 1). However, the critical coupling constant v: always seems to indicate the above-mentioned crossing which orders the four lowest (and the four highest) levels in two pairs with J = Jr,,,, and J = J,,, - 1. The difference between the exact and the approximate U: decreases when N increases. On the other hand, in order to try to understand the physical meaning of the second new ‘critical coupling constant (vy = 1) we note that it is independent of N. We have therefore looked for a crossing of excited levels which always occurs at the same v and find that this is what happens with the crossing of the second J = 2 level with the third J = 3 level, which takes place at v -.0.6. Here, we conclude that v: points out this particular crossing which orders the first two odd-J levels below (and above) zero. This feature is also displayed in fig. 1. Moreover, the new solutions [(c), (f), (g), (h), (i), (j) and (k)] do not correspond to wave functions with the quasi-spin J as a good quantum number. The usual HF solutions, instead, have good J = J,,,. This can be seen studying the transformation applied and noting that it does not violate 3’ if tl = 40 and /I = 8.
3.3. SU(2)*SU(2) MODEL
In this model the expectation state gives
value of the hamiltonian
E = (HFIE?,JHF) +&kN2[sin
(2.7) with respect to the HF
= -+N(cos/?+cos~)
/Isin &OS (LX - q) +cos j3cos 01,
(3.4)
which is independent of y and I/. The HF equations (3.3) provide the solutions shown in table 2. These results coincide with the ones obtained in ref “) and no new solutions are found. As in the previous model, some of these solutions [(b) and (c)] have good J = Jmax while the others [(a) and (d)] do not correspond to a fixed quasi-spin value.
388
Ad. C. Cambiaggio
et al. / Configuration
mixing
TABLE 2 Solutions
of the HF equations
for the SU(2)*SU(2)
V-space
model
W-space
Solution
Range c(
B
0 “1 0 “1 0 “1 P “) For the solutions
of k
cp
0
0
0 “1
?I
all k
0
0 “)
0
all k
0 “1 0
arcos :2/kN)
(a), (b) and (c) the angles
Energy
-$kN’ -fN+&kN=
all k k 2 2/N
CI and cp are undetermined;
;N-?+kN= -+k-&kN=
we adopted
a = cp = 0
for the calculations.
4. Configuration
mixing calculations
4.1. FORMALISM
As it was mentioned in the introduction one must restore the symmetry violated by the HF transformation before proceeding to do the configuration mixing. Some of the HF wave functions obtained in the preceding section do not possess good quasi-spin J and, consequently, we project out this quantum number. We diagonalize f2 within our relevant multiplet, this procedure yielding the set of wave functions IV> w,J,
c
n>=
CJ,,K, WIV,
w, vz,w.
(4.1)
v,w, Here n distinguishes among states .of the same J. The projection given by b, = c IV, W, J, n)
operator
P, is
(4.2)
For obtaining the projected wave function
I@J>= $
3
(4.3)
IHF),
where NJ is the norm, and the projected energy E
(PHF)
=
J
(HFIf@,IHF) (HFJ&HF)
’
(4.4)
we need an explicit expression for the HF wave function IHF) =
c 6%
WV,, KV’,
W,
v,,K>.
(4.5)
M. C. Cambiaggio et al. 1 Conjiguration mixing
389
As in refs 7,9), th e coefficients D(V,, W,) can be easily obtained in terms of the Euler angles
where the d-functions are defined by Edmonds lo). Here we have taken y = $ = 0 because the HF solutions of tables 1 and 2 are independent of these particular angles. It is to be noted that eqs. (4.1)-(4.6) allow us to do an exact projection. Once the projected wave functions (4.3) have been calculated, the hamiltonian is diagonalized in this non-orthogonal set, solving the following equation: i
(Hji - EJNj,)a, =
0.
(4.7)
i=l
Here m is the number of projected states considered, Hj, and Nji are the energy matrix and the overlap matrix, respectively, evaluated with the projected states, EJ are the energy eigenvalues obtained after the configuration mixing (c.m.) and ai are the mixing coefficients. In order to solve the problem (4.7) we follow the prescription described in ref. ll). 4.2. RESULTS
The results obtained in the Lipkin model for N = 8 and v = 1.1 are shown in fig. 2. The quantum number which characterizes the levels is the quasi-spin 9. Notice that we have eleven HF solutions for J = J,,, = $N, but we only have seven solutions for the other values of J because, as it was already mentioned, solutions (a), (b), (d) and (e) in table 1 have good J = J,,,,,. Studying the spectrum obtained after performing c.m. with the projected states which arise from all the HF states (case J), it can be seen that an excellent agreement with the exact values is obtained, except for J = 3. As can be seen in fig. 2 (A -J) increasing the number of projected HF basis states from 5 to 11 does not always significantly improve the quality of the spectrum of the J = 3 states. This is not too surprising considering our selection of the variational subspace in which HF basis states have been obtained. It is obviously too restrictive in this case to obtain a good description of all the states in the spectrum but adequate enough to ensure good agreement with a large number of states in the spectrum. It is interesting to note that the degeneracy present in the exact solutions due to the symmetry of interchange of V and W (i.e. two states for each level) is not always reproduced in the c.m. calculations. For example, the degeneracy of the J = 1 levels is not restored until at least six projected HF basis states have been used in the c.m. calculations ,(see fig. 2, B-F). We have studied the quality of the agreement obtained for different values of v
EXACT
44
HF
PHF
\
:F
"F
",yfjy_"o
33---4---'&----
34---
22---4----
l,OL--
43.2 1-----
4-
I+---
,1:0-
412 3-
j-
$= 2---2-
3--444-
4-
34434-
4-
z24----
+====
4,3‘2,3p-
3.me1
4----34422----
3-
;w==
2---Z$ZZ
&----4-
(&.-.---4-
3--z----2---4---
1-----
43,2,1,0-
1-
4Z-----2---3-
4---
3---3-
i-
i$--_
4z
Ii-----
fZ==
z-
2---$z=
?-----
310z2 4T== 1.0 --
CM
45
l+-.---
4-
2---3--2-
4,3.117
v
4-
"yg-404x
43----
2----
5-----
4..--.._ 1D-
2---
$======
3---
4----
g===== +===
24-
4.3 -
4----
k--
'7
3lo'2-
+====
4----
2----
44.3-
4----
LIPKIN-MODEL
3-
2---.-. g===
p=
5 ~
L-----
-
43t-----
:---3-
4.5 $====
"z
4-
V=l.l
N=8
/
Fig. 2. Results for the Lipkkin model for N = 8 and u = 1.1. ~on~~ratioll mixing results m cases A, B, , . . , J were calculated using the two, three, . . . . eleven lowest HF solutions, respectively.
-20
-15
-10
-5
20 LLI
5
10
15
20
M. C. Cambiaggio
et al. / Configuration
391
mixing
and N. For a given N it does not change significally as a function of the coupling constant when v is larger than one. For v 5 1 the agreement with the exact levels is dramatically worse due to the reduced number of HF solutions available. Both features are illustrated in fig. 3. With respect to the variation with the number of particles the overall agreement I
F-
LIPKIN-MODEL
15-
N-8 80-
44-
44-
4-
44-
60 10 3---_ 3-
40 34-
4-
5-
33’
33-
4-
4-
4-
4ZZ-
Z-
422-
3l-
li_ 2L 1-
4422’
20 -
3l-
o - “~$--
?L 4qF-
‘1 -
l-
it__
3-
;F
3224-
2-
-2o-
2-
4’z 4-
4-
322444-
443-
33-
-40-
-60 +-
33e
44-
4-
EXACT
CM
-80
v-1.0
Fig. 3. Results
for the Lipkin
-
t model
z-
4, 4-
CM
EXACT v= 5.0
for N = 8 and u = 1 and 5.
392
M. C. Cambiaggio
3Q-
et al. / Configuratzon
LIPKIN-MODEL
mixing
N=lO 5-
i- 5,+,3,2,l,o l!F 4_ Y-27
Fig. 4. Results
becomes
worse
when
for the Lipkm
N increases.
model
for N = 10 and v = 1.1.
This is to be expected
because
the number
of
exact levels increases with N while the number of HF solutions remains the same. In particular, the worst results are obtained when J,,, is odd in which case the about zero. As an example we give the c.m. levels for J,,, are not symmetric results for N = 10 and u = 1.1 in fig. 4. Finally, the results obtained in the SU(2)*SU(2) model for N = 8 and k = 0.28 are shown in fig. 5. Here we have four projected states with J = J,,, but only two for the other values of J. This is due to the fact that solutions (b) and (c) of table 2 have good J = J,,,. The agreement with the exact values obtained for the lowest states for each J is very good. However, again as the coupling constant k increases the situation gets worse. The only difference between the projected spectrum and
M. C. ~Cambiaggio et al. / Con~g~ra~~onrni~~~g
6
N=8
~U(2)~~U~2)-~ODEL
5
4-
4
L-
3
4.3x
393
@--4-
4-
w2 z; 1 0 -1 -2 -3 -4
‘o-
t$z
4,
3f_ 1’ $=
@;_
3’
@
EXACT
o-
/
l4 -4.Eas 2\ y---
HF
- PHF kz0.28
l4-B 2, 3/-
CM
Fig. 5. Results for the SU(2)*SU(Z) model for N = 8 and k = 0.28
the c.m. one is the position of the second level with .I = J,,, (J = 4 in fig. 5). In the projected spectrum it practically coincides with the lowest level of the same J whereas in the cm. spectrum it lies near the third (and not the second) level with the same J. This can be understood by considering the quantum number J,, which has not been used in these calculations but, nevertheless, is a good quantum number for the exact levels in the SU(2)*SU(2) model. If we investigate which J, values are present in the projected wave functions we find that those coming from solution (a) in table 2 have J, = 0 while those coming from solution (d) correspond to a mixture of J, values but all with the same parity (odd or even J&J. Moreover, solution (b) corresponds to J, = -Jmax and solution (c) to J, = Jmx. Consequently, the c.m. separates the levels according to the J, values and this is the reason why we do not find an approximation to the second level with J = J,,,, as it has the opposite parity. Similar results are found for other values of the number of particles. N.
394
M. C. Cumbiaggio et al. 1 Configuration mixing
5. Conclusions
In the present work we have shown that one may still obtain physically significant results even if one reduces the number of variational parameters considered in the HF transformation. But one should not be surprised if one does not obtain a good description of all the lowest-lying eigenstates and that the convergence rate is not as rapid as the one expected when using a more general variational ansatz. This of course is analogous in some respects to performing variational calculations in a Nilsson basis and varying only one parameter per s.p. orbital (i.e. the deformation). Here the variational problem will be easier to solve (smaller variational space) but the convergence rate in the c.m. calculations may not be as good as in the case where HF basis states have been used. Another important conclusion arrived at in this work is that all the solutions of the HF ground-state variational equations found in the present work, which do not correspond to the global minimum and that are usually not taken into account, contain interesting physical information. This is clearly illustrated by the new critical coupling constants found in the Lipkin model which point out crossings in the exact excited levels, as has been discussed in subsect. 3.2. Similarly, in the SU(Z)*SU(Z) model, solution (a) provides an approximation for all the levels with J, = 0 (see fig. 5 and the corresponding discussion). This physical meaning of all the HF solutions justifies their use as basis states for a cm. calculation, as proposed by Miller and Schroder 3, and performed in the present work. However, care must be taken to be sure that one considers the lowest HF solutions. If such is not the case, differences as great as the ones observed between figs. 2 and 3 may occur, depending on which HF solutions are missing. An additional point to be careful about is the possible presence of linearly dependent projected basis states. They appear as states with norm zero on diagonalizing the overlap matrix Nji in eq. (4.7) and must be eliminated before going on. This problem is always present when one works with an overcomplete basis, for example in trying to obtain an approximation to the three levels (six states) with J = 1 in the Lipkin model using seven projected states. It is interesting to mention that with the present method one always obtains a better approximation to the ground state than that provided by HF alone 12). This is true in the Lipkin model for example if one only mixes the HF ground state with the trivial solution (a) in table 1 [for u 2 l/(N - l)]. Nevertheless, the present results are not as good as the ones obtained in ref. “) by means of a more sophisticated multiconfiguration ‘Hartree-Fock calculation (notice, however, that the results of ref. “) are restricted to v < 1). Finally, it should be noted in the present work that the agreement with the exact lowest-lying states obtained in the Lipkin model is worse (except for N = 8) than that in the SU(2)*SU(2) model. Two of the authors (M.C.C. and L.S.) thank the staff at the Departamento
de
M. C. Cambiaggio
et al. / Configuration
mxing
395
Fisica of the Comisibn National ‘de Energia Atbmica for the kind hospitality afforded to them during the performance of this work.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
H. G. Miller, R. M. Dreizler and G. Do Dang, Phys. Lett. 77B (1978) 119 H. G. Miller and R. M. Dreizler, Nucl. Phys. A316 (1979) 32 H. G. Miller and H. P. Schriider, Z. Phys. A304 (1982) 273 M. C. Cambiaggio, A. Plastino, L. Szybisz and M. de Llano, Rev Mex. Fis. 27 (1981) 223 A. Ghck, H. J. Lipkm and N. Meshkov, Nucl. Phys. 62 (1965) 118 M. C. Cambiaggio and A. Plastino, Z. Phys. A291 (1979) 277 D. Agassi, H. J. Lipkin and N. Meshkov, Nucl. Phys. 86 (1966) 321 M. C. Cambiaggio, A. War, F. J. Margetan, A. Plastino and J. P. Vary, to be published A. Faessler and A. Plastino, Z. Phys. 220 (1969) 88 A. R. Edmonds, Angular momentum in quantum mechanics (University Press, Princeton, 1960) A. Faessler, F. Griimmer and A. Plastino, 2. Phys. 260 (1973) 305 G. Fichera, Annah di Mathematics pura ed Aplicata (IV) CVIII (1976) 367