Analytical solutions of the klein - gordon equation with the woods-saxon potential for arbitrary l - state

ANALYTICAL SOLUTIONS OF THE

KLEYN-GORDON EQUATION WITH THE

WOODS-SAXON POTENTIAL FOR

ARBITRARY - STATE

l

V.H.Badalov, H.I.Ahmadov, S.V.Badalov

.

r

n

In the paper the analytical solution of the Klein

Gordon equation for the Woods - Saxon potential is presented. In our calculations we have applied the Nikiforov - Uvarov method by using the

Pekeris approximation to the centrifugal potential for arbitrary states. The exact bound state

energy eigenvalues and the corresponding

eigenfunctions are obtained for a particle bound on the various values of the quantum numbers and .

l

0

V

• The Woods-Saxon potential [9] is defined by • (1)

• where is the potential depth, is the width of the potential or the nuclear radius and the

parameter is the thickness of the superficial layer inside of which the potential falls from

value outside of a nucleus up to value inside a nucleus. ) ( 1 ) ( 0 ₀ 0 a R e V r V a R r     _ 0 R a 0  V V  V0





) 0 ( , 0 ) ( ) 1 ( ) ( 2 ) ( 2 2 2 4 2 0 2 2 2             _      R r r r l l c c m V E dr r dR r dr r R d 

where is the angular momentum quantum number. After introducing the new function , Eq.(2) takes the form

) ( ) (r rR r u  l





0 ) ( ) 1 ( ) ( 2 2 2 4 2 0 2 2 2          _     u r r l l c c m V E dr r u d 

The radial part of the Klein-Gordon equation [11] with Woods-Saxon potential is

(2)

2 2 2 1 2 0 48 48 8 12 4 1











      , C , C C





            1 2 ₂ 0 1 1 x x l e C e C C ) r ( V~   

According to the Pekeris approximation, we shall replace potential 2 0 2 2 ) 1 ( ) ( r m l l r V_l    _{with expression}

In order to define the constants and ,we also expand this potential in the Taylor series around the point

1 0 ,C C C₂ ) ( 0 r R₀ x  

• we obtain



(1 )



( ) 0 , (0 1) ) ( ) 1 ( 2 1 ) ( ₂ 2 2 2 2              u z z z z z z z u z z z z u    1 0 1             a R r e z where





2 0 2 2 2 4 2 0 2 2 ( 1)   l l C c a c m E       2 1 2 2 2 0 2 2 ( 1)   l l C c a EV _    2 2 2 2 2 2 0 2 ( 1)   l l C c a V      (4)

2

1

4

)

1

(

192

1

0

2 2 2 2 0 4 0 4













c

a

V

R

l

l

a

n

_r



2 0 0

1

3

4

0

R

)

l

(

l

ca

V









For the bound states 2 4 _{, we get}

0 2 _{m c} E 

(5)

• The exact energy eigenvalues of the Klein-Gordon equation with the Woods-Saxon potential are

derived as                                                                                        2 2 2 2 0 2 2 2 2 2 0 4 0 4 2 2 2 2 0 4 0 4 0 2 2 2 2 0 2 2 2 2 2 0 4 0 4 3 0 3 0 4 1 2 4 ) 1 ( 192 1 1 2 4 ) 1 ( 192 1 2 1 4 1 2 4 ) 1 ( 192 1 ) 1 ( 32 1 2 c a V n c a V R l l a n c a V R l l a V c c a V n c a V R l l a R l l a V E r r r l n_r      

. 16 ) 1 ( 32 4 1 2 4 ) 1 ( 192 1 12 4 1 ) 1 ( 2 1 2 2 2 3 0 3 2 2 2 2 0 2 2 2 2 2 0 4 0 4 2 0 2 0 2 0 2 2 2 0                                                   a R l l a c a V n c a V R l l a R a R a R l l c m _r     0  l _{nr  0}

If the conditions (5) and (6) are satisfied

simultaneously, the bound states exist. From Eq.(5) is seen that if , then one gets . Hence, the

Klein-Gordon equation for the standard Woods-Saxon potential with zero angular momentum has not bound states.

0

R

According to Eq.(7) the energy eigenvalues depend on the depth of the potential the width of the potential , and the surface thickness . Any energy eigenvalue must be less than . If constraints imposed on , and are satisfied, the bound states appear. From Eq.(6) is seen that the potential depth increases when the parameter increases, but the parameter

is decreasing and vice versa. Therefore, one can say that the bound states exist within this potential. Thus, the energy spectrum Eq.(7) are limited, i.e. we have only the finite number of energy eigenvalues.

0 V 0 V

a

r n V₀ l n_r E a R₀

The corresponding radial wave functions are given by expressions

) (z u_nrl

)

z

(

P

)

z

(

z

C

)

z

(

u

_{n l n l n}( , ) r r r

1

1

2

2 2 2 2 2 2 _{2 2}







       

where is the Jacobi polynomials

and are the normalization constants determined using constraint. ) 2 1 ( ) 2 , 2 ( 2 2 2 z P r n      



( )



1 0 2 _



 dr r u_nrl l n_r C



 



    _{ }  _  n _ n n n ) , ( n z ( z ) dz d ) z ( z ! n ) z ( P 1 2 1 1 1

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