Approach to the shifted 1/N expansion for the Klein - Gordon equation

PHYSICAL REVIEW A VOLUME 43, NUMBER 11 1JUNE 1991

Approach

to

the

shifted

1/N

expansion

for

the Klein-Gordon

equation

Omar Mustafa and Ramazan Sever

Department

of

Physics, Middle East Technical University, 06531Ankara, Turkey (Received 30 October 1990j

A different approach to the shifted 1/N expansion technique isdeveloped todeal with the Klein-Cxordon particle trapped in a spherically symmetric potential. Properly modifying the definition of

the perturbative expansion ofthe energy eigenvalue, and without making any approximation in the determination ofthe parameters involved, we obtain suSciently good results compared with the

ex-actones for the Coulomb problem. The calculations are carried out to the second-order correction ofthe energy series.

I.

INTRODUCTION

The 1/N expansion technique has proved itself in

solv-ing the Schrodsolv-inger equation for a large number

of

physi-cally interesting potentials yielding sufFiciently accurate results. '

It

has also been applied to solid-state phys-ics ' _{and quantum-field theory.}

This technique has recently been modified by Imbo and co-workers'

'

and called the shifted

1/N

expansion. A suitable shifting parameter was introduced, which has the meaning

of

an additional degree

of

freedom, that consid-erably improves the analytical structure

of

the perturba-tion series for the eigenvalues and surpasses most approx-imation methods in its domain

of

applicability and accu-racy as well. '

To

the best

of

our knowledge, only a few groups' have so far applied the

1/N

expansion technique (unshift-ed and shifted) to study the relativistic bound-state ener-gies

of

spin-0 and spin-—,' particles.

It

has been noted,

however, that the rate

of

convergence

of

the unshifted ex-pansion is very slow for the relativistic part

of

the energy eigenvalue as compared to that for the nonrelativistic part. Panja and Dutt have extended this technique and introduced a shifting parameter to deal with relativistic particles (with and without spin).

For

the Coulomb case, exact analytical expressions and highly convergent

ex-pansion were restored forthe relativistic correction

of

or-der

1/c2.

In this paper, an alternative approach to the shifted

1/N

expansion technique is introduced to work out the energy eigenvalues

of

a Klein-Gordon

(KG)

particle. We have defined the energy eigenvalue series as

E

=

Eo

+E

&

/k

+

E2

/k

+

' _{and determined the shift} parameter requiring that E~

=0.

This approach provides remarkably very accurate re-sults for the energy eigenvalues for the Coulomb poten-tial. The calculations

of

the energy eigenvalues were car-ried out tothe second-order correction.

In Sec.

II

we develop the formalism

of

this technique for the Klein-Gordon particle.

Exact

numerical results for the KG-Coulomb potential ' _{are presented in}

_Sec.

III,

together with our results. Section

IV

contains con-clusions.

II.

THEMETHOD

The radial part

of

the

KG

equation (in units

fi=c

=1)

for a scalar particle

of

mass m moving in a spherically symmetric potential

V(r)

is given by'

'

d

₊

(k

—

1)(k

—

3)

_—

I

[E

—

V(r)]

—

m I

U„(r)=0,

2 2

dr 4r

where k

=N+2l

and

U„(r)

is the reduced radial wave function. In terms

of

the shifting parameter a,

i.e.

, k

=k

—

a,

r

Eq.

(1)becomes

d

,

+,

+

I2EV(r)

—

V(r)']

U„(r)=(&'

—

m')

_U„(r)

.

dr 4r

It

isconvenient toshift the origin by defining

x

=k'"(r

r,

)/r,

,

—

and touse the following expansions:

(3)

5788 OMAR MUSTAFA AND RAMAZAN SEVER 43

V(r)=(k IQ)[V(ro)+

V'(ro)rox/k'

+

V"(ro)rox

12k+

.

_),

E=E

+E

/k+E

/k

+-.

.

where _{Q is}a scale, whose magnitude isto be determined later. Equations (4},when substituted in Eq. (2),yield

(4a) (4b) d

U„(x)

dx2 k (2

—

a)

(1

—

a)(3

—

a)

+4

2+

_4k 1

—,

2x

+

3x

—

.

U

(x)

2,

'k

+

Eo+

+

+

Q k

k'

V(r

)+

V'(r

),

+

V"(r

)

+

' ' ' _U,

_{x)

'2

V(ro)+

V'(ro)

+

V"(ro)

+

. .

U„(x)=e„U„(x),

k1/2 2k

where

e„=

(ro2/Q)[ k {

E

₀

—

m ₎

+

2EOE,

+

(E,

+

2EOE~₎

lk

+

_] . (6)

Equation {5)isa Schrodinger-like equation for the one-dimensional anharmonic oscillator problem which has been dis-cussed in detail by Imbo, Pagnamenta, and Sukhatme. ' Therefore, following their formalism, we obtain

s„=

k [—

„'+2roEo

V(ro)/Q

—

roV(ro)

IQ]+

[(1+2n„)w/2

—

(2

—

a)/2]

+(1/k

)

[(1

—

a)(3

—

a)/4+(1+2n„)7~+3(1+2n„+2n„)e4

(1/w—

)[Y&+6(1+2n„)s+3+(11+30n„+30n„)s3]

j,

(7)

where

n„

isthe radial quantum number and and

c

=c.

/uj,

j

=1,

2,3,4 (8)

s,

=2

—

a,

e2=

—

3(2

—

a)/2,

e3

=

—

1+

(ro/3Q)

[Eo V"'(ro

)

—

V(ro)

V"'(ro

)

—

3V'(ro)

V"

(ro)

],

e4=

—,

'+(ro/12Q)[EO V""(ro)

—

V(ro)

V""(ro)

—

4V'(ro) V'"(ro)

—

3

V"(ro)

V"(ro)]

.

Comparing the terms

of Eq.

(7)with those

of Eq.

(6)and equating terms

of

same order in k implies

(ro/Q)(Eo

—

m )

=

_[—,

'+2roEo

V(ro)/Q

—

roV(ro)

/Q],

(ro/Q)(2EOE,

)

=

[(1+2n„)w/2

—

(2

—

a)/2],

(ro

/Q)(E

&

+2EoE2)

=

I

(1

—

a)(3

—

a)/4+(1+2n„)F2+3(1+2n„+2n„)ez

—

_(1/w)[e,

_{+6(1+2n„)e,}

_{73+(11+30n„+30n„)s3]}

j . (10) (12)

From Eq. (10)we obtain

Eo=

V(ro)+m (1+Q/4m ro)'

where ro ischosen tominimize

Eo.

That is,

(13)

where mis given by

w

=

[3+

r,

V"

(r,

)/V'(r,

)

—

4r',

V'(r,

)

V'(r, )/Q]

'

and Qsatisfies

Eq.

(15),which can be written as

(17) dEo d Eo

=0,

_,

&0,

Io dpo (14) a

=2

—

{1+2n„)w,

(16) therefore, ro satisfies the equation

roV'(ro)(1+Q/4m

ro)'

=Q/4m

. (15)

To

solve for the shifting parameter a, the next contribu-tion to the energy eigenvalue is chosen to vanish, '

i.e.

,

E, =0,

which implies that

Q

=[roV'(ro)]

(2+2y),

(18)

where

y=

[1+[2m

lroV'(ro)]

j'

(19)

Equations (16) and (18) along with Eqs. (17) and (19),

with Q

=

k,

read

1+2l

+(1+2n„}w

=roV'(ro)(2+2y)',

(20) which is an explicit equation in ro. Once ro is

deter-mined,

Eq.

(13)gives Eo and

Eq.

(12) gives

Ez.

Finally,

Eq.

(4)gives

43 APPROACH TOTHE SHIFTED 1/N EXPANSION FOR

THE.

.

.

5789

TABLE

I.

The energy levels for spin-0 particle inthe Coulomb potential in units of10 me

.

States 1s 2s 2p 3$ 3p 3d 99 997 335.93 99 999 334.00 99999 334.00 99 999 704.00 99 999 704.00 99 999 704.00 ED+Ex/k 99 997335.89 99999333.99 99999 334.00 99999 704.00 99999 704.00 99999 704.00 Exact (Ref. 25) 99 997 355.86 99999333.98 99999 334.00 99 999703.99 99 999 704. 00 99 999 704.00

E

=Eo+(1/2E&ro

)_[(1

—

a)(3

—

a)/4+

(1+2n„)E2+3(1+2n„+2n„)E4

—

_{( /tc)[82~+}

₆₍₁₊

_2n„)Z,

Z3+

(11+30n„+

30n„)E3]

I

.

(21)

III.

APPLICATION TOTHECOULOMB POTENTIAL

For

the Coulomb potential,

V(r)

=

/3/r,

P=e-Eqs. (17), (19),and (20) yield

w

=[(y —1)/(y+1)]'i

y =

[1+(2mro//3)

]'~~,

and (22) (23) (24)

ones given by

Ref. 25.

The agreement

of

our results with the exact ones is

better than that

of

Panja and Dutt, especially for small l states. Moreover, the convergence

of

the results listed in

Table

I

seems to be fast in a sense that the second-order contribution

to

the energy series,

E2/k,

isvery small (of the order

of

10 —10

")

compared with the contribu-tion

of

the leading term,

Eo. It

should be pointed out, however, that the accuracy as well as the convergence

of

our results increases as the principal quantum number n

of

the state increases.

P(2y+2)'~

=

1+2l

+(1+2n„)[(y—

1)/(y+1)]'~

(25) We have numerically solved

Eq.

(25) for ro in terms

of

mc and found the energy eigenvalues.

In Table

I,

the numerical results for the energy eigen-values calculated by the leading term Eo

of

the energy series and by

Eo+E2/k

are compared with the exact

IV. CONCLUSION

In this paper we have developed a formalism

of

the shifted

1/X

expansion technique for the Klein-Gordon equation with radially symmetric potentials.

For

the Coulomb potential the method looks quite attractive as it

yields highly accurate results. We have also seen that the convergence increases as the principal quantum number increases.

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