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Correlated hyperspherical harmonics.

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Ann. Physik 1 (1992) 110-116
Annalen
der Physik
0 Johann Ambrosius Barth 1992
Correlated hyperspherical harmonics
W. Zickendraht’ and J. S. Levinger’
‘Fachbereich Physik, Philipps-Universitat, W-3550 Marburg, Germany
’Department of Physics, Rensselaer Polytechnic Institute, Troy, New York 12180, U. S. A.
Received 29 November 1991, accepted 6 January 1992
Abstract. The expansion of the wavefunction for a bound three particle state in the five-dimensional
hyperspace of hyperspherical harmonics in some cases suffers bad convergence, especially for weakly
bound states. For this reason correlated hyperspherical harmonics are proposed, of which the ordinary
hyperspherical harmonics are one special choice. The “best” suited correlated hyperspherical harmonics
are chosen from an infinite set of complete orthogonal systems by a Ritz variational calculation.
Keywords: Few-body systems.
1 Introduction
Hyperspherical harmonics were introduced almost thirty years ago [ 1,2,3] and have been
used widely especially for three-body problems (see the summary article [4] with many
references). They constitute a complete orthogonal system of functions - in the case of
three particles in a five-dimensional angular space - and the wave function of the system
is expanded in these functions. The number of functions in this expansion for sufficient
accuracy depends on the strength of correlations between the particles. Three identical
particles interacting by equal oscillator potentials for example would be uncorrelated corresponding to a shell model description, the particles moving in a central potential and the whole expansion of the wave function would shrink to a single term.
Recently Valk and Levinger have applied hyperspherical harmonic expansions to an
exactly solvable three-body problem of Crandall et al. [ 5 , 6 ] demonstrating their effectiveness and limitations.
It is the purpose of this note to extend the application of hyperspherical harmonics by
introducing what will be called correlated hyperspherical harmonics. From an infinite set
of complete orthogonal functions in a first step the “best” system for the problem
considered is chosen. In the second step the wave function is expanded in this chosen
system in the same way as is done for ordinary hyperspherical harmonics.
To illustrate the method it is applied again to the Crandall model.
2 Model and coordinates
For completeness the Crandall model is sketched in this section. The potentials in this
model are chosen in such a way, that the Schrodinger equation can be solved analytically
[5, 61. Including these potentials the Schrodinger equation is written in the form:
W. Zickendraht, J. S. Levinger, Correlated hyperspherical harmonics
[A,,
+
1
+ As] + -21K q 2 + --K
2
E] Y
-
2v2
111
(r, q)
=
1'"
[r,
0
with
=
[Z(M+2 m )
+ r2 - 2r3] .
The first and second particles have mass m; the third has mass M. ri are the space vectors
of three particles. K and Q are the parameters characterizing the potential energy of the
Crandall model:
(r,
2
-
r,)2
1
+K (r2 - r3)2 + Q/(rl
2
-
1.
r2)2
3 Correlated hyperspherical harmonics
To proceed to the definition and illustration of correlated hyperspherical harmonics we
restrict the problem to total orbital angular momentum zero. So after introducing Euler
angles and internal coordinates, Eq. (1) will contain only the internal coordinates in this
case. For the internal coordinates a suitable choice in the case of the Crandall model are
p, the hyperradius, 6, the angle between 4 and q and E ; which gives the ratio q/r.
q
= p
cos
&
< = p sin E
(4)
The Schrodinger equation is then
ty @, E )
p
=
1
+
1
j T K p 2 (1
+ 2m/M.
+ p) +
1
-
4
K p 2 (1 - p ) cos 2~
Ann. Physik 1 (1992)
112
The eigenvalue E and eigenfunction w@, E ) are given in Crandall [6] and Valk [ 5 ] .
The ground state alone is considered. As the potential does not depend on 6 , the
wavefunction depends on p and E only. The hyperspherical harmonics are functions of
E alone and so are the correlated hyperspherical harmonics.
The systems from which the final system will be chosen are indexed with A , which can
have any non-negative values. The complete orthogonal set (for fixed 1)is obtained as
the solution of:
The eigenfunctions are
x;’) ( E )
=
N, cos2’
EF
-n, 2 1
+ n + 2; 2A +
2
with eigenvalue
y,
=
(2n
+ 2A) (2n + 2 1 + 4 ) .
(9)
N,, is the normalization factor, and n is a non-negative integer. Any set
course be used to expand v / @ ,
xi’)
(E)
can of
E):
The purpose of this note is to show that there is probably a “best” system. The value of
A, for which one may hope to get most rapid convergence, is to be determined. It is simply
done by taking as an Ansatz of zeroth order the lowest function of every system, to
calculate the energy:
(1 I ) is substituted into (9,which is multiplied by cos2‘ E . The integration over
performed giving an ordinary differential equation in the hyperradius
+ -K
2
41+3
4 1 + 6 p2
3
+
1
T P K2 1 + 3
P2
- Ex
1fo@)
The solution is straightforward. One finds for the energy:
+ 3 @ + 1)/4 ”*
2(21 + 3 )
+ 41 + 1 + (m/h2) Q (A +
4A + I
1
3A2
1
1)
1‘7
.
=
0
f
E
is
113
W. Zickendraht, J. S . Levinger, Correlated hyperspherical harmonics
1 is now treated as a parameter. So a Ritz variational calculation is performed, that is
the value of A is determined, for which (13) yields a minimum:
d Ex =
dA
0.
(14) determines A as a function of p and Q.
With A fixed the calculation is improved by including terms with n > 0 in the Ansatz
(10). The coupled system of equations for the functions f,@) is found to be:
+ (1
-
P)
(21
+
21(2A + 1 )
2n
1)(2A + 2n
+
+ 3)
]
P2
- EO]
f n
with
Cnn= - (2n
4A
+
+ 2A)(2n + 2 1 + 4) +
t
n,! T ( n ,
n,! r ( n ,
=
n' for n' > n
n,
=
n for n' < n
+ 4A(2A +
l)]
+ 2n + 2)
(n
21
A2
+ 3 / 2 ) r ( 2 1 + n , + 3/2)(2A + n , +
+ 3/2)T(2A + n , + 3/2)(2A + n, +
n,
d, = -
4
(21
[ - 2mQ
+ 2n + 3
+
l)(n
l)!
I)!
1
(1 7)
+ 3/2)(21 + n + 2)(2A + n + 3/2)
(A + n + t)(A + n + 2)
Results
Tab. 1 shows our results. All energies are given in units of h (K/m)'12,following Valk
[ 5 ] , Tab. 3. The other parameters of the Crandall model are the strength Q of the inverse
square potential, and the mass ratio m/M. We use Valk's 25 values of these two
parameters, to explore the (Q, m / M )plane. The third, fourth and last columns are copied
Ann. Physik 1 (1992)
114
Table 1 Q and m / M are parameters in the Crandall model, Eq. (1). Energies are given in units of h
(K/m)”2; (C
and g2 and the last column E are copied from Table 3, Ref. 4. The parameter rl is
chosen to minimize E,, Eq. (13). The energy E f ) is the eigenvalue of Eq. (15) for 2 coupled differential
equations.
FF)
2Qm/h2
m/M
0.1
I
4
16
60
0.1
0
0
0
0
0
1
4
16
60
0. I
1
4
16
60
1
1
0.1
I
4
16
60
0.1
I
4
16
60
1
I
I
4
4
4
4
4
25
25
25
25
25
1800
1 800
I 800
1 800
1 800
(C F)
62
1
Ex
EL2)
3.05
3.37
4.00
5.37
8.00
4.29
4.63
5.30
6.76
9.55
6.77
7.20
8.05
9.89
13.42
15.42
16.27
17.94
21 .58
28.53
128.3
135.1
148.5
177.7
233.5
3.05
3.35
4.03
5.38
8.28
4.15
4.50
5.15
6.73
10.31
6.22
6.59
7.44
9.66
14.77
13.50
14.26
15.97
20.65
31.57
111.2
117.1
131.0
1 69.1
258.4
0.0
0.183
0.500
1.19
2.50
0.184
0.454
0.994
3.08
4.46
0.618
1.15
2.16
4.26
8.22
4.7
5.6
7.6
12.2
21.3
127.0
130.0
136.0
160.0
222.0
3.05
3.37
4.00
5.37
8.00
4.25
4.56
5.18
6.60
9.14
6.5 I
6.79
7.35
8.63
11.17
13.74
13.89
14.29
15.38
17.74
96.67
96.72
96.89
97.5 1
99.26
3.05
3.37
4.00
5.37
8.00
4.17
4.49
5.12
6.52
9.10
6.22
6.52
7.1 1
8.44
1 1.02
12.97
13.15
13.62
14.77
17.32
94.01
94.08
94.3 1
95.1 1
97.20
E
3.05
3.37
4.00
5.37
8.00
4.15
4.46
5.10
6.47
9.10
6.05
6.32
7.00
8.37
11
.oo
12.26
12.58
13.21
14.58
17.21
91.56
91.88
92.5 1
93.88
96.51
from Valk. (6‘FF> uses the Haftel-Mandelzweig [7] Ansatz, discussed below. (Also see
Valk, section V.) This energy is the expectation value of the Hamiltonian, using Haftel’s
trial function. C2 gives the numerical solution of two coupled equations found using
Simonov harmonics. The last column is Crandall’s exact eigenvalue, E.
As discussed above, the value of the parameter A is chosen to minimize the energy El,
Eq. (13). We see that A becomes large as we go to large values of Q, or large values of
m / M , or both. The next column El gives the minimum energy for n = 0, using a single
hyperspherical harmonic. The eigenvalue Eo in Eq. (15) is found, for the tabulated value
of parameter A, by numerical solution of ( 1 5 ) using the renormalized Numerov method.
We evaluate for two coupled ordinary differential equations, and denote the eigenvalue
as Ei2’. (The superscript reminds us that we have two coupled equations, by choosing n
as zero or one.)
Our values for Eh and Ei2)are self consistent in each case, obeying the rule that must
hold for all trial functions: Eh 2 El2) 2 E. Of course Valk’s approximate eigenvalues
(flFF) and C; must also each be greater than the exact eigenvalue E - though there is no
necessary inequality between (8FF) and 8’.
The Table shows that our Ansatz works very well for the Crandall model, as compared
to the approximate results quoted from Valk. For m/Mof zero, either the Haftel Ansatz
or our Ansatz give eigenvalues accurate to the three significant figures quoted, while for
W. Zickendraht, J. S. Levinger, Correlated hyperspherical harmonics
115
large Q, the two-term Simonov energyE2 is 3 1 / 2 % above the exact eigenvalue. (See Valk,
Table 3 for convergence of the expansion in the optimal subset of H.H.)
For m/M 2 1, our one-term analytic expression El2)is always more accurate than
the expectation value (6FF). Our two-term expansion result El2) is below the value t2
found with Simonov harmonics. This is anticipated, since our use of the parameter A
allows more flexibility in the trial function than in the use of Simonov H. H. (The one
exception for 2 Qm/fi2 = 0.1 and m / M = 1 is likely due to round-off error in numerical
work here, or in Ref. [4]. But either result is within 1Yo of the exact eigenvalue for this
case.)
Our Ansatz is particularly successful for cases with large m/M, and large Q. For the
last row in the Table, either (8fF) or (s7, are more than twice the exact eigenvalue. Even
use of 5 members of the optimal subset [6] gives an energy 50% above the exact value.
But our correlated H. H. represent an order of magnitude improvement in accuracy; one
term is within 3% and two terms are within 1 % of the exact energy.
5
Relation to other methods
Here we give a short discussion of this paper’s method in relation to the techniques
discussed in Ref. [ 5 ] .
a) Relation to the Simonov approach
The coordinates A and I of Ref. [ 5 ] were replaced by E and 6. Because there is no 6dependence of the potential in the Crandall-model, the ground state has no 6dependence which makes the problem simpler. From the complete system of E , 6
dependent hyperspherical harmonics only those without &dependence contribute.
The first step of the method consists in determining the “best” system of hyperspherical harmonics from an infinite set of systems. This is done by a Ritz variational
calculation and determines the parameter A characterizing a special system of
(correlated) hyperspherical harmonics. A = 0 would exactly correspond to the
Simonov approach (the only difference being the use of the coordinates E , 6 instead
of A and A). The special system of hyperspherical harmonics is now used in the second
step, which is exactly like the Simonov approach, leading to E2.
Relation to the “Correlation Function Approach” of Haftel
The first step discussed under a) determines a correlation function. So there is a
similarity to the “Correlation Function Method”. But the second step makes the
difference: In the “Correlation Function Approach’’ the wavefunction is a product of
the correlation function and a series of HHS. The product (correlation function) x
(HH) d o not form a complete system.
Relation to the Correlated Hyperspherical-Harmonic Expansion of Rosati et al. [8]
In our Eq. (8) we introduce a prefactor with a specified functional dependence, and
one variational parameter. Rosati et al. also use a prefactor, found from their
Eqs. (2.9) and (2.10) as the solution of the two-body problem, with a modified twobody potential. The modification introduces a parameter, which is adjusted by the
variation principle. Like Haftel et al. they use the standard set of hyperspherical
harmonics, combined with this prefactor. They apply their technique to the trinucleon
with the Malfliet-Tjon (V) potential.
In our method the expansion of the wavefunction is in a complete system consisting
of products (correlation function) x (hypergeometric function), where the hypergeometric functions depend on the correlation parameter A.
116
Ann. Physik 1 ( 1 992)
So one could say that the method discussed here is in between the Simonov approach
on the one hand and the methods of Haftel et al. and Rosati et al., which are similar to
each other, on the other hand.
6 Summary
In short the summary is this: The use of correlated hyperspherical harmonics improves
the convergence of three particle calculations considerably.
One of the authors (W.Z.) would like to express his gratitude for the hospitality experienced at the
Department of Physics a t the Rensselaer Polytechnic Institute during the spring and summer of 1989.
The work was supported by Deutsche Forschungsgemeinschaft.
References
[ I ] W. Zickendraht, Proc. Natl. Acad. Sci. (US) 52 (1964) 1565
[2] W. Zickendraht, Ann. Phys. (N. Y.) 35 ( 1 965) 18
[3] Yu. Simonov, Sov. J. Nucl. Phys. 3 (1966) 461
[4] V. B. Mandelzweig, Nucl. Phys. A 508 (1 990) 63
[5] H. S. Valk, J. S. Levinger, Ann. Phys. (N.Y.) 199 (1990) 141
[6] R. Crandall, R. Bettega, R. Whitnell, J. Chem. Phys. 83 (1985) 698
[7] M. I. Haftel, V. B. Mandelzweig, Ann. Phys. (N.Y.) 189 (1989) 29
[8] S . Rosati, M. Viviani, A. Kievsky, Few Body Systems 9 (1990) 1
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