16 August 1999

Ž .Physics Letters A 259 1999 215–219www.elsevier.nlrlocaterphysleta

Negative-energy levels of the Dirac equation in N dimensions

Mehmet Simsek 1

˙ ˙Department of Physics, Faculty of Arts and Sciences, Gazi UniÕersity, 06500 Ankara, Turkey

Received 30 March 1999; received in revised form 18 June 1999; accepted 28 June 1999Communicated by P.R. Holland

Abstract

In this study, bounded real spectra of the Dirac equation have been found for a spherically symmetric completelyimaginary linear potential in N dimensions. Negative-energy states lie on the regions m)E )ym, and E -ym, whenn n

the potential parameter A, k and the mass m satisfy certain constraints. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 03.65.-w; 03.65.Ge; 11.10.QrKeywords: Dirac equation; Imaginary linear potential

Ž .The problem of the Dirac equation DE has beenthe subject of much discussion for decades, becauseof its importance for the prediction of antiparticles,and also for the description of the particles spin andmagnetic moments. Including relativity, the DE isone of the extensions of the Schrodinger equation¨Ž .SE in quantum mechanics. However, at the non-relativistic limit, the systems can be examined bysolving the SE without spin. The fact that nonrela-tivistic quantum mechanics with local interactionsloses its general properties.

As is well known, the existence of negative en-ergy solutions is a problem since the positive-energystates should make rapid, radiative transitions to

w xthem 1 . Now, we want to ask the following ques-tion: do the bound states lie only just below m? Toanswer this question, let us consider non-HermitianDirac Hamiltonians with an imaginary potential.

1 E-mail: msimsek@quark.fef.gazi.edu.tr

It should be noted that there are many solutions offor the standard SE and Lagrange equations forcomplex potentials in classical and quantum mechan-

w xics 2–4 . However, very recently, it has been foundthat the energy spectra are real when the quantum

w xtheories respect PT invariance 2,4 . Although timereversal invariance does not lead to any conservationlaw, parity conservation implies some selection rulesand it is meaningful to discuss the conservation ofangular momentum which is a result of rotationalinvariance. In general, what is needed to check paritynonconservation is to examine a physical observablethat allows us to distinguish between our world and a‘world reflected in a mirror’. The existence of ellip-tical orbits in planetary physics is not an evidenceagainst the invariance of the laws of gravitationonder rotations, unless some elliptical orbits are pref-

w xered to others 5 . In this paper we shall not pursuePT invariance, whereas we focus ourselves on thesolution of the DE for a particle in a sphericallysymmetric completely imaginary linear potential,

0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00431-4

( )M. SimsekrPhysics Letters A 259 1999 215–219˙ ˙

216

iA r, which is not PT symmetric, but invariant underall rotations.

Let us start whith two coupled first order radialw xequations 6,7 ,

d Ly F r s VyWym G r , 1aŽ . Ž . Ž . Ž .ž /dr r

and

d Lq G r s WyVym F r . 1bŽ . Ž . Ž . Ž .ž /dr r

Combining them, we obtain a second order equationw xin N dimensions 7 :

2d L Ly1Ž .y q2m EyV F rŽ . Ž .2 2dr r

1 dV d Ls y yž /2mqEyV dr dr r

2y EyV F r , 2Ž . Ž . Ž .

Ž .where Ls N y2 sr2, N sNq2 j, WsEqm,j j

and ss"1.To remove the first derivative, dFrdr, of the

Ž .right-hand side of Eq. 2 we substitute the transfor-Ey VŽ . Ž .mation, F r s 1q c r . Furthermore, we( 2m

can construct the Schrodinger-like equation as fol-¨Ž .lows: If 2mqEyVs0, we cannot use Eq. 2 , and

this transformation cannot be substituted. On theother hand, if 2mqEyV/0 in that region, andalso the rest energy is large, then we expand terms

Ž . Ž .w Ž .like 1r 2m q E y V ; 1r2m 1 y E y V r2mx 2q . . . . Neglecting 1rm and higher order tems,

w xthen one can get Schrodinger-like equation DE as 7¨Ž ."scs1 ,

2d ky1 ky3Ž . Ž .y q qU rŽ .2dr 4r

2q2 Eqm V r yV r csec . 3Ž . Ž . Ž . Ž .

Where

1 1 d2V L dVU r s q , 4Ž . Ž .2ž /2m 2 r drdr

and the abbreviations

ksNq2 jq1 when js lly1r2,

or ksNq2 jy1 when js llq1r2;

Ls ky1 r2 or y ky3 r2Ž . Ž .according to js lly1r2 or llq1r2 and

esE Eq2m . 5Ž . Ž .With these considerations, we now have producedthe radial DE as in the radial SE for a spherically-symmetric potential. To solve the Schrodinger-like¨

w xDE equation, we will extend our previous works 8to the relativistic case, i.e., we establish a factorizedform of radial DE for a radially symmetric com-pletely imaginary linear potential in N dimensions.

Thus, to find exact andror quasi-exact solutionsof the Schrodinger-like DE equation in N dimen-¨

Ž .sions, we will choose f r in the following factor-ized form:

f r s f r exp g r , 6Ž . Ž . Ž . Ž .which is applied to the inverse-power and also to

w xreducible potentials 8 . In practice, however, choos-Ž . Ž .ing the form f r s f r exp g r fot the waveŽ .w xfunction is usual 9,11 . If we substitute this form of

Ž . Ž .f r in Eq. 3 , then the second derivative can bewritten as

X 2XX XXf r s g r qg rŽ . Ž . Ž .

XX X Xf r q2 f r g rŽ . Ž . Ž .q f r . 7Ž . Ž .

f rŽ .Ž . Ž .So, comparing Eq. 3 and Eq. 7 , it is easy to see

that

2XX XV r yesg r qg rŽ . Ž . Ž .eff

f XX r q2 f X r gX rŽ . Ž . Ž .q , 8Ž .

f rŽ .where

2V r sU r q2 Eqm V r yV rŽ . Ž . Ž . Ž . Ž .eff

ky1 ky3Ž . Ž .q . 9Ž .24r

Note that, if one considers obtaining well-behavedsolutions, first of all one should ensure, while choos-

( )M. SimsekrPhysics Letters A 259 1999 215–219˙ ˙

217

Ž . Ž . Ž .ing g r and f r , the normalisability of f rŽ Ž . Ž . .f r™` ™0 and f r™0 ™0 , which has re-

w xcently been discussed in some detail by us 8 andalso in the references therein.

Now, we limit ourselves to find the fraction of theŽ .right hand side of the Eq. 8 corresponding to the

function of the potential and the energy given. To dothis, with the radially symmetric imaginary linearpotential,

V r s iAr , 10Ž . Ž .one can write

2d f iALr2m2 2s A r q2 iA mqE rqŽ .2 rdr

ky1 ky3 r4Ž . Ž .q yE 2mqE f .Ž .2r

11Ž .

Ž .Eq. 8 in N-dimensional space for the reducedŽ .radial wave function f r has the structure of the

one-dimensional Schrodinger like equation for a¨Ž .spherically symmetric complex potential V r ,eff

w 2 2V r s A r q2 iA mqE rŽ . Ž .eff

iALr2m ky1 ky3 r4Ž . Ž .q q .2r r

12Ž .

Thus, to solve the transformed Schrodinger like¨Ž . Ž . Ž .Eq. 11 for V r we choose f r in the nodaleff

form,n

Ž .nf r s rya , ns1,2, . . . , f r s1,Ž . Ž .Łn i 0is1

13Ž .

Ž .and for g r , we make following ansatz for theground state and the other states:

g r sa r 2 qb rqcln r , 14Ž . Ž .where a Žn. values correspond to the nodes of wavei

Ž . Ž .function f r that are determined by roots of f r .We can only speak generally, this approach is

w xsimilar to the usual power series methods 9 , thew xSUSY factorization method 10 , and the modified

w xRiccati approach 11 . Because the right hand side of

Ž .Eq. 7 equal to the energy plus effective potentialŽ .for any given state ns0,1,2 . . . , the solutions are

exact or quasi-exact when certain relations holdamong the parameters.

Ž . Ž . Ž .Rewriting Eqs. 7 and 11 by substituting g rŽ .and f r for the ground and excited states as fol-n

lows, so, one can find the relations between thecorresponding potential parameters and a,b,c and

Žn. Ž .a as for three setsi

n n2Žn. Žn.4a a q2b a qn 2cqny1 s0,Ž .Ý Ýi i

is1 is1

n nŽn. Žn. Žn. Žn. Žn. Žn.4a a a a qa q4b a aŽ .Ý Ýi j i j i j

i-j i-J

nŽn.q ny1 2cqny2 a s0,Ž . Ž . Ý i

is1

nŽn. Žn. Žn. Žn. Žn. Žn.4a a a a a qa qaŽ .Ý i j k i j k

i-j-k

nŽn. Žn. Žn.q6b a a a q ny2Ž .Ý i j k

i-J-k

nŽn. Žn.= 2cqny3 a a s0. 15Ž . Ž .Ý i k

is1

The corresponding relations between the potentialŽ . Ž .parameter A,k,m mass and E energy , and wave

function parameters a,b,c and a Žn. arei

niLAŽn.[2b cqn q4a a ,Ž . Ý i2m is1

ns0,1,2, . . . a Žn.s0 for ns0 , 16Ž .Ž .i

A2 [4a2 , 2 iA Eqm [4ab ,Ž .

ky1[2c and E Eq2mŽ .2[y b q2 a 2cq2nq1 . 17Ž . Ž .

Ž . Ž . Ž .From Eq. 15 and Eqs. 16 and 17 it is possible toobtain the coefficients a,b,c and a Žn. as a functioni

Ž .of A,k, and m mass parameters. There are notzeros on the real line, i.e., a Žn. always imaginary.i

The index n denotes the number of nodes of thewave function in 0-r-`.

( )M. SimsekrPhysics Letters A 259 1999 215–219˙ ˙

218

Furthermore, it was found that the restriction onŽ .the coefficient, A, of the potential from Eqs. 16

Ž .and 17 is

m2

"As . 18Ž .kq2n

It is the limitation for parameter A, given the massand the parameters of bound states. For ns0, thereal ground state energy of the imaginary potential,

Ž .Eq. 10 , is

LE sym 1y , for ns0, 19Ž .0 2k ky1Ž .

and we follow the same steps for ns1. The firstexited state is

LE sy m 1y1 22 kq2 k y1Ž .Ž .

=

28 ky1 kq2 k y1Ž .Ž .Ž .k" q1 ,( 2ž /Lž /

for ns1. 20Ž .Hence, other exact negative energies and correspond-ing wave functions in N-dimensional space can be

Ž . Ž . Ž . Ž .determined from Eqs. 15 – 17 and Eqs. 6 , 13Ž .and 14 , respectively. Next, we obtain the ground

state wave functions, with corresponding parameters,as

Ey iAr "AŽky1.r2 2F r s 1q r exp y rŽ . ( ž /ž2m 2

y "i Eqm r , for ns0, 21Ž . Ž .Ž . /and the above statements are valid for nG1,

2nEy iArŽ2 n. Žky1.r2F r s 1q rya rŽ . Ž .( Ł i2m is1

="A

2exp y r y "i Eqm r .Ž .Ž .ž /ž /222Ž .

It is to be noted that, several features can beverified for different conditions: For simple illustra-

Ž .tions we will use the abbreviations in 5 . For exam-

ple, lower lying energies for ks3, i.e., in threedimensions:

. Ž .i for Ls ky1 r2; E sy0.91667m, E s0 1

y0.6460m, or y1.27897m;. Ž .ii for L s y k y 3 r2; E s 1.0m, E s0 1

y0.36755m, or y1.63246m, respectively.It is interesting that, although there is no real

eigenvalue of the one-dimensional Schrodinger equa-¨w xtion for an imaginary potential 2 , yix, at the

relativistic case the radial DE has, as seen from Eqs.Ž . Ž .19 and 20 , for a radially symmetric completelyimaginary linear potential in N dimensions. Conse-

Žquently, the first exited state lies lower when Ls k. Ž .y1 r2 than when Lsy ky3 r2, in the region

m)E )ym. However, in the region E -ym,1 1

the contrary is true. Note that the negative energysolutions are the interesting aspects of this study.Moreover, there are the pecularities attending the

w xnegative energy solutions, as illustrated in Ref. 12for one dimensional potentials and similar peculiari-ties are present also for strongly attractive potentials.

Summing up, in this paper an algorithm has beenperformed for the generation of a solution of SE tothe Schrodinger like DE for an imaginary linear¨potential in N-dimensions. Negative-energy levelslying in the regions m)E )ym, and E -ymn n

have been pointed out. There are restrictions betweenk, the mass m and the potential parameter, A, as

Ž .given in Eq. 18 .

Acknowledgements

The author wishes to thank Prof. N. Aktekin for astimulating discussion.

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