Dong S.-H. Wave Equations in Higher Dimensions

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Part IV

Applications in Relativistic Quantum

Mechanics

Chapter 13

Dirac Equation with the Coulomb Potential

1 Introduction

The exact solutions of quantum system with a 1/r type potential are of impor-

tance in quantum mechanics [1, 2, 444]. Due to the recent interest of the higher-

dimensional ﬁeld theory, many problems related to the Schrödinger equation and

Klein-Gordon equation in (D + 1) dimensions have been discussed. To ﬁll in the

gap between them, we have carried out the Dirac equation with this potential in

(D +1) dimensions [91].

The purposes of this Chapter are two-fold. First, we exhibit the exact solutions

of the hydrogen atom by the conﬂuent hypergeometric equation approach. The sec-

ond is to investigate the variations of energy difference E(n,l,D)and the energy

levels E(n,l,D) on the dimension D [87] and also to study the variations of energy

levels E(n,l,ξ) on the potential strength ξ =Zα for a given D.

This Chapter is organized as follows. The exact solutions of the radial equations

will be displayed via the conﬂuent hypergeometric equation approach in Sect. 2.

The variations of energy difference E(n,l,D) and energy levels E(n,l,D) on

the dimension D as well as the variations of energy levels E(n,l,ξ) on the po-

tential strength ξ = Zα shall be elucidated in Sect. 3. The Dirac equation with a

Coulomb potential plus a scalar potential will be discussed in Sect. 4. Some con-

cluding remarks are given in Sect. 5.

2 Exact Solutions of Hydrogen-like Atoms

As demonstrated in Chap. 4,theD-dimensional radial Dirac equations are written

(r) +

(r) =[E −V(r)−M]F

(r),

−

(r) +

(r) =[E −V(r)+M]G

(r),

(13.1)

S.-H. Dong, Wave Equations in Higher Dimensions,

DOI 10.1007/978-94-007-1917-0_13, © Springer Science+Business Media B.V. 2011

157

158 13 Dirac Equation with the Coulomb Potential

with

K =±(2l +D −1)/2. (13.2)

Let us study the solutions of radial equations (13.1) by the conﬂuent hyperge-

ometric equation approach. This is different from power series expansion method

used in Chap. 4.

Consider the Coulomb-like potential, i.e.,

V(r)=−

,ξ=Zα. (13.3)

Introduce a new variable ρ for bound states |E|<M,

ρ = 2r



−E

. (13.4)

Substitution of this, together with Eq. (13.3), into Eq. (13.1) leads to

dρ

(ρ) +

(ρ) =



−



M −E

M +E



(ρ),

dρ

(ρ) −

(ρ) =



−



M +E

M −E

−



(ρ).

(13.5)

Deﬁne the wavefunction 

(ρ) of the forms

(ρ) =

√

M −E[

(ρ) +

−

(ρ)],

(ρ) =

√

M +E[

(ρ) −

−

(ρ)].

(13.6)

Substitutions of them into Eq. (13.5) allow us to write down



dρ



(ρ) +

dρ



−

(ρ)



[

(ρ) +

−

(ρ)]



−



M +E

M −E



[

(ρ) −

−

(ρ)],



dρ



(ρ) −

dρ



−

(ρ)



−

[

(ρ) −

−

(ρ)]



−



M −E

M +E



[

(ρ) +

−

(ρ)].

(13.7)

Their addition and subtraction yield

dρ



(ρ) ∓



ξE

√

−E

−





(ρ)

=−



ξM

√

−E





∓

(ρ). (13.8)

Deﬁne the following notations

τ =

ξE

√

−E

,τ



ξM

√

−E

. (13.9)

2 Exact Solutions of Hydrogen-like Atoms 159

Equation (13.8) is simpliﬁed to

dρ



(ρ) ∓



−





(ρ) =−

K ±τ





∓

(ρ), (13.10)

from which we can obtain an important second-order differential equation



dρ



−

τ ±1/2

−





(ρ) =0, (13.11)

where

−ξ

. (13.12)

For a weak potential, we have

η =



−ξ

> 0. (13.13)

It indicates that Eq. (13.11) is a special case of the Tricomi equation [20]



a +





α +



y = 0. (13.14)

From the behaviors of the wavefunction at the origin and at inﬁnity, we deﬁne



(ρ) =ρ

−ρ/2

(ρ). (13.15)

Substitution of this into (13.11) leads to

dρ

(ρ) +



−1 +

1 +2η



dρ

(ρ)

τ −η −1/2 ±1/2

(ρ) =0, (13.16)

whose solutions are the conﬂuent hypergeometric functions

(ρ) =a

(η −τ ;2η +1;ρ),

−

(ρ) =b

(1 +η −τ ;2η +1;ρ).

(13.17)

It is shown from Eqs. (13.6), (13.15) and (13.17) that G

(ρ) and F

(ρ) can be

directly obtained by the combinations of the conﬂuent hypergeometric functions.

We now study the relation between the coefﬁcients a

and b

. Before proceeding,

we recall the following recurrence relations between the conﬂuent hypergeometric

functions [192]

(α;γ ;z) = α

(α +1;γ +1;z)

αγ

[γ

(α +1;γ ;z) −γ

(α;γ ;z)],

(α +1;γ +1;z) =(α −γ)

(α;γ +1;z) +γ

(α;γ ;z),

(α +1;γ ;z) = (z +2α −γ)

(α;γ ;z)

+(γ −α)

(α −1;γ ;z).

(13.18)

160 13 Dirac Equation with the Coulomb Potential

It is shown from Eqs. (13.11), (13.17) and (13.18) that



η −τ





(1 +η −τ ;2η +1;ρ) =0. (13.19)

Since both a

and b

cannot be vanishing, we obtain the following relation between

them

τ −η



. (13.20)

From Eq. (13.6) we thus have

(ρ) = N

√

M −Eρ

−ρ/2

×[(τ



+K)

(η −τ ;2η +1;ρ)

+(τ −η)

(1 +η −τ ;2η +1;ρ)],

(ρ) = N

√

M +Eρ

−ρ/2

×[(τ



+K)

(η −τ ;2η +1;ρ)

−(τ −η)

(1 +η −τ ;2η +1;ρ)],

(13.21)

where the normalization factor

(τ



+K)

−1





−E



−1/2

(13.22)

is to be determined.

We now study the eigenvalues of this quantum system. The quantum condition is

obtained from the ﬁniteness of the solutions at inﬁnity

τ −η =n



=0, 1, 2,.... (13.23)

When n



=0, η =τ , and

=τ

+ξ

=(τ



)

. (13.24)

Therefore, K has to be positive to avoid a trivial solution.

Introduce the principal quantum number n

n =|K|−(D −3)/2 +n



=|K|−(D −3)/2 +τ −η

=l +1 +n



=1, 2,.... (13.25)

The n may be equal to 1 only for K = (D − 1)/2 and is equal to other positive

integers for both signs of K. The energy E can be obtained from Eqs. (13.9), (13.11)

and (13.25)

E(n,l,D) =M

|ξ|



1 +

(



−ξ

+n −l −1)



−1/2

. (13.26)

For a large D,wehave

E(n,D) M

|ξ|

[1 −2ξ

−2

+4ξ

(2n −3)D

−3

−···], (13.27)

which implies that the energy is independent of l for a large D.

2 Exact Solutions of Hydrogen-like Atoms 161

Fig. 13.1 The energy difference E(1, 0,D) decreases with the dimension D ∈ (0, 0.9], while

increases with the dimension D ∈[1.1, 1.9] and then decreases again with the dimension D ≥2.1.

The E(1, 0,D) is symmetric with respect to the point (1.5, 0), so are those E(n, 0,D).The

parameter ξ =0.05 is taken here and also in Figs. 13.2–13.6

For a small ξ ,wehave

E(n,l,D) M



1 −

2[n +(D −3)/2]

−

2[n +(D −3)/2]



2n +D −3

2l +D −1

−



, (13.28)

where the ﬁrst term on the right hand side is the rest energy M (c

=1), the second

one is from the solutions of the Schrödinger equation, and the third one is the ﬁne

structure energy, which removes the degeneracy between the states with the same n.

It is found from Eq. (13.28) that the energy levels E(n,l,D) are almost independent

of the quantum number l for a small potential parameter ξ asshowninFig.13.1.

We now determine the normalization factor N

from the normalization condi-

tion



†



dV =1. (13.29)

Since n



=τ −η is a non-negative integer, we may express the conﬂuent hyperge-

ometric functions by the associated Laguerre polynomials (5.21). Through a direct

calculation, we obtain the normalization factor

−E

)

1/4

(2η +1)



(τ +η +1)

2Mτ



(K +τ



)(τ −η)!



1/2

. (13.30)

162 13 Dirac Equation with the Coulomb Potential

3 Analysis of the Eigenvalues

We now analyze the properties of eigenvalues. Recently, Nieto has made use of the

concept of the continuous dimension D to study the bound states of quantum system

with a special potential [87]. With this spirit, we attempt to discuss what happened

to a continuous dimension D asshowninRef.[100].

It is shown from Eq. (13.25) that n = 1, 2, 3,... and l = 0, 1,...,n− 1. If the

principal quantum number n and the angular momentum quantum number l are

ﬁxed, the energy difference E(n,l,D) between eigenvalues for dimensions D

and D −1 can be calculated by

E(n,l,D)=E(n,l,D) −E(n,l,D −1)

|ξ|





1 +

(σ +

√

−ξ

)

−



1 +

(σ +



−ξ

)



, (13.31)

where

σ =n −l −1,K

=(2l +D −2)/2. (13.32)

It is found from Eq. (13.31) that the relation between E(n,l,D)and dimension D

is more complicated than that of the Schrödinger equation case [87]. The problem

arises from the factor



−ξ

(or



−ξ

). Therefore, for a weak potential it

is shown from Eqs. (13.2), (13.13) and (13.31) that

D =2(1/2 −l +K)



≥1 +2|ξ |, when l =0 and K>|ξ|,

≤1 −2|ξ |, when l =0 and K<−|ξ|,

(13.33)

and

D =2(1 −l +K)



≥2 +2|ξ |, when l =0 and K

> |ξ |,

≤2 −2|ξ |, when l =0 and K

< −|ξ |.

(13.34)

We present the variations of energy difference E(n,l,D) on the dimension D

in Figs. 13.1, 13.2, 13.3. We take the parameters M =1 and ξ =0.05 for deﬁnite-

ness. It is shown from Fig. 13.1 that there exist two singular points around D ∼ 1

and D ∼ 2forE(1, 0,D). As the dimension D increases, the energy difference

E(1, 0,D) ﬁrst decreases with the dimension D in the region (0, 0.9] and then

increases with the dimension D in the region [1.1, 1.9], and ﬁnally decreases again

with the dimension D ≥ 2.1. In particular, it is found that the energy difference

E(1, 0,D) is symmetric with respect to the point (1.5, 0). The singular points can

be easily explained from Eqs. (13.31), (13.33) and (13.34). That is, notice that there

exist singular points for K

=ξ

and K

=ξ

. Thus, it is shown from Eqs. (13.33)

and (13.34) that D =1 ±2ξ and D = 2 ±2ξ for l = 0, respectively. It is shown in

Fig. 13.2, where n =2 and l =1, 0, that there are no bound states for D<0.1. This

can also be explained by Eqs. (13.33) and (13.34). As the dimension D increases, the

energy difference E(2, 1,D) monotonically decreases with dimension D ≥ 0.1.

Finally, we show the E(3,l,D) as a function of dimension D in Fig. 13.3, where

3 Analysis of the Eigenvalues 163

Fig. 13.2 The plot of energy differences E(2,l,D) (l = 0, 1) as a function of dimension D.

Note that E(2, 1,D) decreases with the increasing dimension D ≥0.1 and the variation of the

E(2, 0,D) on the dimension D is very similar to E(1, 0,D)

Fig. 13.3 The plot of energy difference E(3,l,D) as a function of dimension D.Thered

dashed, green dotted and blue solid lines correspond to l =2, 1, 0, respectively

the red dashed, green dotted and blue solid lines correspond to l = 2, 1, 0, respec-

tively. Note that the energy difference E(3, 2,D) decreases monotonically with

the dimension D. The variations of the E(n, 2,D), E(n, 1,D)and E(n, 0,D)

areverysimilartotheE(3, 2,D), E(2, 1,D) and E(1, 0,D), respectively.

Second, we display the variations of energy E(n,l,D) on the dimension D in

Figs. 13.4, 13.5, 13.6. It is shown from Fig. 13.4 that the energy E(1, 0,D) de-

creases with the increasing dimension D ∈ (0, 0.9], but increases with the dimen-

sion D ≥ 1.1. Likewise, this singular point can be explained by Eqs. (13.26)or