One Dimensional Scattering Theory based on

2025-05-02 0 0 541.58KB 16 页 10玖币

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1

Electrostatic multipole contributions

to the binding energy of electrons

A. D. Alhaidari(a) and H. Bahlouli(b)

(a) Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438, Saudi Arabia

(b) Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

Abstract: The interaction of an electron with a local static charge distribution (e.g., an atom or

molecule) is dominated at large distances by the radial

Coulomb potential. The second order

effect comes from the non-central electric dipole contribution

cos r



. Moreover, the third order

effect is due to the electric quadrupole potential,

(3cos 1) 2r





. We use the tridiagonal

representation approach to give a reasonably accurate account for the combined effects of all these

contributions to the binding energy of the electron but with an effective quadrupole interaction. As

an application, we obtain the bound states of a valence electron in an atom with both electric dipole

and quadrupole moments.

Keywords: electric dipole, electric quadrupole, electron binding, tridiagonal representations,

orthogonal polynomials, Bessel polynomial

1. Introduction

In the atomic units,

1Me  

, the time-independent three-dimensional Schrödinger

equation for an electron (mass M and charge

e

) in the field of a static charge distribution

described by an electrostatic potential function

()Vr

reads as follows

2( ) ( ) 0V r E r





    



, (1)

where



is the three-dimensional Laplacian, E is the electron energy and

()r



is the

associated wavefunction. In spherical coordinates, the potential function

()Vr

associated with

the local charge distribution at distances much larger than the size of the distribution can well

be represented by a multipole expansion, which when written up to and including the linear

electric quadrupole, reads as follows (see, for example, Ref. [1])

2(3cos 1)

cos

() Q

V r d q

r r r





   

, (2)

where Q is the net effective positive nuclear charge felt by the valence electron, d is the electric

dipole moment along the positive z-axis, and q is the linear electric quadrupole moment of the

charge distribution. We took the Bohr radius,

0 0 0

44a Me

 



, as the unit of length.

Now, the quadrupole term in the potential (2) destroys separability of the wave equation (1)

making its solution a highly non-trivial task. Therefore, we consider an effective electric

quadrupole interaction where the angular factor

 

23cos 1





is replaced by a dimensionless

2

angular parameter



such that



   

since

0 cos 1





. This could be considered as

being some average over the angular dependence and results in an effective quadrupole

potential

, where the effective electric quadrupole moment is





. Consequently, the

radial part of the wave equation (1) becomes

2 2 3

1 ( 1) ( ) 0

d Q p Er

dr r r r

 





     





, (3)

where



is a quantum number that depends on the electric dipole moment d and the azimuthal

angular momentum quantum number

0, 1, 2,...m  

[2]. If

0d

then



becomes the orbital

angular momentum quantum number

0,1,2,...

. On the other hand, for a non-zero dipole

moment, Eq. (3.6) and Eq. (3.8) in Ref. [2] give

 





as one of the eigenvalues of an infinite

symmetric tridiagonal matrix whose elements are given by

 

, , , 1 , 1

( 2 ) ( 1)( 2 1)

2( ) 1 4 ( 1) 1 4

i j i j i j i j

i i m i i m

i m i m

T i m d d

  



   

    

    

. (4)

For pure dipolar interaction (

0p

), the condition

 





eliminates quantum anomalies

associated with the orbital inverse square potential [3]. However, for

0p

this condition is not

required for a physical solution.

In section 2, we start by formulating the problem using the “tridiagonal representation approach

(TRA)”. In section 3, we obtain the TRA solution for non-zero positive net charge Q. In section

4, we present numerical results where we show the effect of the pure quadrupole (

0d

)

contribution to the electron binding energy after filtering out the Coulomb (electric monopole)

contribution. In section 5, a realistic model is presented where we obtain the bound states of a

valence electron in an atom with electric dipole and quadrupole moments. Then, we conclude

in section 6 with some relevant remarks.

2. TRA formulation of the problem

The combined

r

, and

r

power-law potential in Eq. (3) for arbitrary physical

parameters

 

,,Qp



has no known exact solution in the published literature. Nonetheless, in

this work we give a highly accurate analytic approximation in a properly chosen finite basis

set. To avoid quantum anomalies due to the most singular term in the potential, the effective

electric quadrupole interaction term

pr

, we require that the angular parameter



be chosen

such that





is positive. We employ the tools of the TRA in the formulation and solution

of this problem. For details on the TRA and how it is used in solving the wave equation, one

may consult [4,5] and references cited therein. We start by expanding the wavefunction in a

pointwise convergent series as

( ) ( )

r f x





, where

 

()



is a set of square integrable

functions that produce a tridiagonal matrix representation for the wave operator (3). We choose

the dimensionless variable x as

1xr





, where



is an arbitrary real positive scale parameter

with inverse length dimension. In terms of this variable, the wave equation (3) becomes

3

   

2 2 2

222

( ) 2 1 2 ( ) 0

x d d Q

r x x p x r

dx dx x x

  

    



        





. (5)

where





. A proper choice of square-integrable basis that could support a tridiagonal

matrix representation for the wave operator has the following elements

( ) ( )

x x e Y x









, (6)

where

()



is the Bessel polynomial on the positive real line whose relevant properties are

shown in the Appendix and





. The degree of the polynomial is limited by the negative

parameter



0,1,2,..,nN

and N being the largest integer less than





. Therefore, the

basis set

 

()N





is finite. Consequently, this basis can produce a faithful physical

representation for the bound states of a system whose discrete spectrum is also finite and with

a maximum size of

1N

. Nonetheless, it can also support a quasi-exact solution (finite portion

of the spectrum) for a system with an infinite discrete spectrum and with an accuracy that

increases with the size of the basis. We will shortly discover that for a fixed positive net charge

Q, the size of the basis for the system under consideration increases with the bound state energy

level. This is consistent with the fact that higher excited states have larger number of nodes

(oscillation) requiring a larger degree of the polynomial

()



for accurate representation,

which means larger basis size.

Using the differential equation of the Bessel polynomial (A4) in the Appendix and choosing





, we can evaluate the action of the wave operator on the basis elements giving

   

 

222

2 1 2 2

2 1 4

( ) 2 ( )

x x e n p x Y x



   

   

 



        





(7)

A tridiagonal representation of the wave operator is obtained if and only if the right-hand

side of Eq. (7) becomes a sum of terms proportional to

()



and

1()





with constant factors

(modulo an overall multiplicative function that does not depend on n). That is, [4,5]

 

1 1 1

( ) ( ) ( ) ( ) ( )

n n n n n n n

x x a x b x c x

    

  

  

, (8)

where

 

n n n

a b c

are x-independent parameters and

()x



is a node-less entire function.

Consequently, the three-term recursion relation (A2) dictates that the expression inside the

square brackets in (7) must be linear in x. Therefore, the terms proportional to

x

and

x

must vanish and thus we must choose the basis parameters



and



as follows

22E





, (9a)

2QE



  

. (9b)

Hence, our TRA solution will be restricted to negative energies (i.e., bound states). Moreover,

since



is negative then the net charge of the electrostatic distribution, Q, seen by the electron

must be positive. Additionally, since N is the largest integer less than





, then relation (9b)

implies that the size of the basis increases as

becomes smaller (i.e., as the energy level gets

higher) as noted above.

4

In the following section, we present the TRA solution of the problem which is written in terms

of a new orthogonal polynomial defined in [6] by its three-term recursion relation and initial

values.

3. TRA solution of the problem

With the basis parameters



and



given by (9), the action of the wave operator (7) on the basis

elements reduces to the following

   

 

2 1 2 11

( ) 4 2 ( )

x E x e n p x Y x



   

 

     





. (10)

Substituting this action in the wave equation

( ) ( ) 0

r f x







and using the recursion

relation of the Bessel polynomial (A2), we obtain the following three-term recursion relation

for the expansion coefficients

   

( )( 1)

( 1)(2 2 3) ( )(2 2 1)

F n F

p F F

n n n n









   





    



  











      



(11)

where we have written

0nn

f f F

making

01F

. If we define

 

  

( 1) ! 2 1

2 2 1 2 1







  

where

 

()

( 1)( 2)...( 1)

a a a a a n 



     

is the Pochhammer symbol (a.k.a. shifted

factorial), then this recursion becomes

   

( )( 1)

2( ) ( 1)

P n P

p n n

n n n n









   





    



  











      





(12)

Comparing this recursion relation to that of the polynomial

( ; )



defined in [6] by its three-

term recursion relation and initial values, which is shown here in the Appendix as (A10), we

conclude that

( ; )

P B z





with

12pE



  

 

22z p E



   

. (13)

Moreover, the k-th bound state wavefunction for the electron reads

 

( ) ( ) ( ; ) (1 )

k k n n n

r f E r e G B z Y r

  

   







, (14)

where

    

2 2 1 2 1 ( 1) ! 2 1

G n n

  

     

. For a given set of physical parameters and

bound state energy

, the basis parameters



and



are given by (9) whereas z and



are given

by (13). Therefore, to have a full representation of the wavefunction (14), we only need an

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1ElectrostaticmultipolecontributionstothebindingenergyofelectronsA.D.Alhaidari(a)andH.Bahlouli(b)(a)SaudiCenterforTheoreticalPhysics,P.O.Box32741,Jeddah21438,SaudiArabia(b)PhysicsDepartment,KingFahdUniversityofPetroleum&Minerals,Dhahran31261,SaudiArabiaAbstract:Theinteractionofanelectronwithalocal...

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