ON A THEOREM FOR ONE-BODY WIGNER DISTRIBUTION FUNCTIONS

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ON A THEOREM FOR ONE-BODY WIGNER DISTRIBUTION FUNCTIONS

E. de Guerra, J. Martorell

To cite this version:

E. de Guerra, J. Martorell. ON A THEOREM FOR ONE-BODY WIGNER DISTRIBUTION FUNC- TIONS. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-95-C6-101. �10.1051/jphyscol:1984611�.

�jpa-00224212�

ON A THEOREM FOR ONE~BODY WIGNER DISTRIBUTION FUNCTIONS

E. Moya de Guerra and J . M a r t o r e l l

Dpto. de F-Csica Atomiaa y Nuclear, Facultad de Cienaias, Badajos, Spain

*Dpto. de F-Csica Atomiaa y Nuclear, Universidad de Palma de Mallorca, Palma de Mallorca, Spain

Résumé - Nous présentons un théorème pour la transformée de Wigner de la Matrice densité à un corps dans l'approximation du champ moyen.

Le cas particulier de l'Oscillateur Harmonique est étudié en détail.

Abstract - Within the context of the mean field approximation a theorem is given for the Wigner transforms of determinantal one-body density matrices.

Application to Harmonic Oscillator wave functions is dicussed in detail.

The one-body Wigner distribution function plays a crucial role in the semiclassical methods that are currently used to study the static and dynamical properties of the many body nuclear system /l/. It is then desirable to have, at least at the static level, simple analytical expressions of the quantum-mechanical one-body Wigner distribution function that serve as reference for comparison to semiclassical approximations.

At present the best description of the static one body density matrix is obtained, in the mean field approximation, using the Density Dependent Hartree-Fock (DDHF) approximation with realistic effective interactions. Its corresponding Wigner transform should then be used for the purpose of comparison mentioned above.

Unfortunately this requires lengthy computational work for every particular nucleus and the simplicity of direct analytical comparison is lost. This problem can however be overcome by approximating the radial single particle wave functions by Harmonic Oscillator (H.O.) wave functions. In a previous work /2/ we found that the Wigner transform obtained by this approximation is indeed very similar in all respects to that obtained from DDHF calculations, and can therefore be used as reference with the advantage of having a simple analytical expression.

It is well known /3/ that for a closed shell corresponding to an isotropic, three-dimensional, harmonic oscillator potential, the Wigner transform is a simple functional of the Wigner transform of the single particle H.O. hamiltonian. Most nuclei do not however correspond to shell closure, but rather to open shells in either isotropic or anisotropic Harmonic Oscillators.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984611

C6-96 JOURNAL DE PHYSIQUE

The purpose of this work is to show that still in those cases the one body Wigner distribution function admits simple analytical expressions and how those can be obtained.

The starting point for this is the following theorem valid in the mean field approximation:

"Let p and h be the static one body density matrix and single particle hamiltonian of a system of A independent fermions, and let

pi)

Then the Wigner transform of p , f(a, p), depends on R and p only through its dependence on the Wigner transforms of the operators belonging to the set S".

This theorem is very useful to guide the election of the independent variables on which f(z, 5) depends in any particular case, its proof is given in reference /2/.

Here we shall only apply it to the case that h is a spherically symmetric harmonic oscillator hamiltonian and the density matrix corresponds to the ground state of an even-even, spin saturated system. In this case, due to reflexion and time RV&

invariance the Wigner transform,

can in principle depend on R 2 , p2 and ( ~ . p ) ~ , but the above theorem restricts in this case the number of independent variables to two*:

and

- ₁ ² _{= (R X 6)' ( 3 )}

Indeed in this case the operators belonging to S to be considered are h, 1 and any 2 product or power of those.

But the Wigner transforms of the latter can, in turn, be expressed in terms of h and

2 4

9 '

. Consider for instance the Wigner transforms of h , h12 and 1 . Using the relation /4/

* Here and in what follows we denote with a tilde, 5, the Wigner transform of any single particle operator, 8, defined in analogy to eq. (l).

f * C +

The notation A means /4/ A = V . V - ^V .V R P P R '

The vyiables R and p are adimensional (in units of a and l/ao, respectively, with a =l/Muo, c=&l throughout).

- ² - 2 3 h = $ ( ( R ~ + ~ ~ ) * - ? ) 4 = ( h ) -- 4

- 3 " 3 11-

h = ( h ) - 4 h , e t c . ...

- ^-2

and h , 1 a r e t h e o n l y independent v a r i a b l e s l e f t .

On t h e o t h e r hand t h e p r o o f o f t h e theorem i n t h i s r e s t r i c t e d c a s e is v e r y s i m p l e . S i n c e p s a t i s f i e s t h e quantum mechanical commutation r e l a t i o n

[ h , P ] = 0 ( 5 )

we f i n d , t a k i n g t h e Wigner t r a n s f o r m , t h a t f ( z , 5) must s a t i s f y

-

h s i n 4 ₂ f ( R , 5) ⁼ o

2 -2 and t h e c o r r e s p o n d i n g e q u a t i o n s f o r 1 , 1 .

But f o r t h e harmonic o s c i l l a t o r h a r n i l t o n i a n ( s e e e q . ( 2 ) ) c o n d i t i o n ( 6 ) i m p l i e s t h a t t h e P o i s s o n b r a c k e t o f h and f be z e r o :

o r , c a l l i n g zi=zi(E, E ) t h e i n d e p e n d e n t v a r i a b l e s on which f depends we f i n d t h a t t h e s e v a r i a b l e s must s a t i s f y

h , Zi = O f o r i = l , ..., N

I- 1

2 2

where N<3 i n t h i s c a s e . And from t h e P o i s s o n b r a c k e t s o f h w i t h R , p , (%.p)*:

(h, ^{R') = -2} 6.R ⁼

-v,

we f i n d t h a t t h e o n l y i n d e p e n d e n t v a r i a b l e s Zi s a t i s f y i n g eq. ( 7 ' ) a r e

C6-98 JOURNAL DE PHYSIQUE

This then proves that for spin saturated, time reversal invariant, spherically - _-2

symmetric states the one-body Wigner distribution function depends only on h and 1 when the single particle radial wave functions are those of a harmonic oscillator potential. Besides, if the configuration corresponds to closed harmonic oscillator shells the Wigner distribution function must be isotropic in - E (or 6) space and will therefore depend on h only.

The latter property is well-known from the explicit expression of the Wigner transform for a closed harmonic oscillator shell K /3/:

where spin degeneracy is taken into account.

The first property (not so well known) is very useful in constructing the explicit expression of the Wigner transform corresponding to closed harmonic oscillator j-subshells. This can be done in the following steps:

i) Take the one-body density matrix corresponding to a closed harmonic oscillator j-subshell (n l j ) , pn (I'1,F2), and use the Moshinsky-Brody transformation

1 1 1 brackets to write it as

where

ii) Take the Wigner transform /2/, /3/ of A NI, nl:

to get

( R , 6) ^{d cos(8} _R,P

This restricts the sum over 1 in eq. (13) to 1=0 and we end up with a function of p 2 and R 2 .

iv) Use the fact that Tn is symmetrical under the interchange of R2 and p2 to 1 1j1

write it as a function of R2+p2 and R2p2. Then using the explicit expression /2/, /5/ of the Moshinsky-Brody coefficients <no, NO, 0 ( nlll, nlll, 0 > we find that

with K =2nl+ll and the numerical coefficients ₁ u (m,s) given by recurrence K1 "'l

relations (see ref. / 2 / ) . For n =O (the most interesting case in practice) the 1

recurrence relation is as follows

K1 K1-m-s

)

(k,, n l = ~ (m. = -

gl(m+p ⁾

-

v) Finally, use the fact that -- f ( E , 6) depends only on h and 12, and that the angle average of the s power of l' is given by

to write f_ , ; ( R , 6) as a function of h and 1 -2

with

The result for a closed K-shell given in eq. (10) can be recoverd from eq. (17) by taking the sum

[ . , l

f 6 = E ^fn , ( 8 , ⁶⁾

K1 n = O j = 1 + 2 1 1 1

1 1 1

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Results for the n =O orbitals p

1 3/2' d5/2' f7/2' g9/2' h11/2' i13/2 obtained from eq. (17) can be found in ref. /2/. Here we quote the results for the one body distribution functions (in the H.O. approximation) of neutrons ( fn) and protons (fP) of several doubly closed nuclei

fn(160) = fP(160) = 16 exp. (-e/2) (e-2)

2 3 fP('02 ) = 16 exp. (-e/2) (-6+7e-2e +e /6)

where d and e are defined as

d = 41 -2 = 4~~~~ sir? 8

RP

Acknowledgements

This work has beensqported by the CAICYT (Spain). One of us (E.M.G.) is also thankful to the Dept. Investigaci6n Bgsica of the Junta de Energia Nuclear (Madrid, Spain) for its hospitality.

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