Dong S.-H. Wave Equations in Higher Dimensions

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xx List of Figures

Fig. 13.4 The energy E(1, 0,D) decreases with the increasing dimension

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

D ≥1.1. Note that it is symmetric with respect to axis D =1...164

Fig. 13.5 The variation of energy E(2, 0,D) (red dotted line)on

the dimension D is very similar to E(1, 0,D). The energy

E(2, 1,D) (blue dashed) increases with the increasing

dimension D. Specially, for D>1 the energy E(2, 1,D) almost

overlaps E(2, 0,D) .........................164

Fig. 13.6 The variations of energy levels E(3,l,D) (l = 2, 1, 0) on

dimension D are similar to E(2,l,D) (l = 1, 0). The energy

E(3, 2,D) almost overlaps energy E(3

, 1,D). This can be

explained well by Eq. (13.28). When D>1, the energies

E(3,l,D) (l = 2, 1, 0) are almost overlapped. The red dotted,

blue dashed and green solid lines correspond to quantum

numbers l =2, 1, 0, respectively ..................165

Fig. 13.7 The energy E(1, 0,ξ) decreases with the increasing potential

parameter ξ ≤1...........................165

Fig. 13.8 The energies E(2,l,ξ) (l =0, 1) decrease with the increasing

potential parameter ξ . Notice that ξ ≤1forl =0(green dotted

line), while ξ ≤2forl =1(red dashed line) ...........166

Fig. 13.9 The energies E(3,l,ξ) (l =0, 1, 2) decrease with the increasing

potential parameter ξ . Note that ξ ≤1forl =0(blue solid line),

while ξ

≤2forl =1(green dotted line) and ξ ≤3forl =2(red

dashed line). Generally, the energies E(n,l,ξ) decrease with the

potential parameter ξ ≤l +1 for a given l .............166

Fig. 13.10 The energy E(1, 0,D) decreases with the increasing dimension

D ∈ (0, 1), but increases with the increasing dimension D ≥1.

The parameters v = 0.2 and s =0.3 are taken here and also in

Figs. 13.11 and 13.12 ........................172

Fig. 13.11 The variation of energy E(2, 0,D) (red dashed line)onthe

dimension D is similar to E(1, 0,D). The energy E(2, 1,D)

(blue solid line) increases with the dimension D.ForD>2the

energy E(2, 1,D) almost overlaps E(2, 0

,D) ...........173

Fig. 13.12 The variations of energy levels E(3,l,D) (l = 2, 1, 0) on

dimension D are similar to E(2,l,D) (l = 1, 0). The energy

E(3, 2,D) almost overlaps energy E(3, 1,D).ForD>2, the

energies E(3,l,D) (l = 2, 1, 0) are almost overlapped. The blue

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

respectively .............................173

Fig. 13.13 The energy E(1, 0,D) decreases with the increasing dimension

D ∈ (0, 0.55], but increases with the increasing dimension

D ≥ 1.45. There are no bound states for D ∈ (0.55, 1.45)

The parameters v = 0.3 and s =0.2 are taken here and also in

Figs. 13.15 and 13.16 ........................174

List of Figures xxi

Fig. 13.14 The plot of energy spectra E(1, 0,D) as a function of

dimension D. Two signs “±” are considered. Note that their

variations are absolutely different from each other ........174

Fig. 13.15 The variation of energy E(2, 0,D) (green solid line)onthe

dimension D is similar to E(1, 0,D). The energy E(2, 1,D)

(red dashed line) increases with dimension D. The energy

E(2, 1,D) almost overlaps E(2, 0,D) for D ≥2 .........175

Fig. 13.16 The variations of energy levels E(3,l,D) on the dimension

D are similar to E(2,l,D) (l = 1, 0). For D ≥2 the energies

E(3,l,D) (l =2, 1, 0) are almost overlapped. The red dashed,

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

respectively

.............................175

Fig. 13.17 The energy E(1, 0,D) decreases with the increasing dimension

D ∈ (0, 1), but increases with the increasing dimension D ≥1.

The parameters v =s =0.2 are taken here and also in Figs. 13.18

and 13.19 ..............................176

Fig. 13.18 The variation of energy E(2, 0,D) (green solid line)onthe

dimension D is similar to E(1, 0,D). The energy E(2, 1,D)

(red dashed) increases with the dimension D. The energy

E(2, 1,D) completely overlaps E(2, 0,D) for D ≥1.......176

Fig. 13.19 The variation of energy E(3, 0,D) on the dimension D is

similar to E(n,0,D) (n =2, 1). The energy E(3, 2,D) overlaps

E(3

, 1,D) completely. For D ≥ 1 the energies E(3,l,D)

(l = 2, 1, 0) are completely overlapped. The red dashed, green

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

respectively .............................177

Fig. 13.20 The plot of energy levels E(3,l,v) (l = 0, 1, 2) as a function

of parameter v. They decrease with the parameter v.The

v ≤



(l +1)

for D = 3. The red dashed, green dotted

and blue solid lines correspond to l = 2, 1, 0, respectively. The

parameters D =3 and s = 0.2aretaken..............178

Fig. 13.21 The plot of energy levels E(3,l,s) (l =2, 1, 0) as a function of

parameter s. The energy decreases with s.Thered dashed, green

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

The parameters D =3 and v =0.2 are chosen . . .........178

Fig. 14.1 For D ∈ (0, 1.8), the energy difference E(1, 0,D) decreases

with the increasing dimension D, while increases with the

increasing dimension D ∈[1.8, 2.8) and then decreases again

with the increasing dimension D

≥3.2. There exist two singular

points around D = 2 and D = 3. Note that E(1, 0,D) is

symmetrical with respect to point (2.5, 0). The parameter

ξ =11/137 is taken here and also in Figs. 14.2–14.6 .......185

Fig. 14.2 Note that the energy difference E(2, 1,D) increases with the

increasing dimension D ∈(0, 0.8), and then decreases with the

increasing dimension D ≥1.2 ...................185

xxii List of Figures

Fig. 14.3 The variation of energy difference E(3,l,D) on the

dimension D. It is noted that E(3, 2,D) (red solid line)

decreases monotonically with the increasing dimension D>0.

The E(3, 1,D) (green dashed line) and E(3, 0,D) (blue

dotted line) are very similar to E(2, 1,D) and E(1, 0,D),

respectively .............................186

Fig. 14.4 Note that the energy E(1, 0,D) decreases with the increasing

dimension D ∈ (0, 1.8), but increases with the increasing

dimension D ≥ 2.2. It should be stressed that there exists a

singular point around D ∼2. Note that the energy E(1, 0,D) is

symmetric with respect to axis D =2forD ∈(0, 4), so are those

energy levels E(n,0,D)

......................186

Fig. 14.5 The variation of energy E(2, 0,D) (green dotted line)on

the dimension D is very similar to E(1, 0,D). The energy

E(2, 1,D) (red dashed line) increases with the increasing

dimension D. Specially, it is found for D>2 that the energy

E(2, 1,D) almost overlaps E(2, 0,D) ...............187

Fig. 14.6 The variation of E(3,l,D) (l =2, 1, 0) on the dimension D is

similar to E(2,l,D) (l =1, 0). Note that the energy E(3, 2,D)

(red dashed line) almost overlaps that of the energy E(3, 1,D)

(green dotted line). When D>2 the energies E(3,l,D)

(l =2, 1, 0) are almost overlapped .................187

Fig. 14.7 The energy E(1

, 0,D) decreases with the dimension D ∈(0, 2),

but increases with the increasing dimension D ≥ 2. The

parameters v = 0.2 and s =0.3 are taken here and also in

Figs. 14.8 and 14.9. Note that the energy E(1, 0,D)is symmetric

with respect to axis D =2 .....................192

Fig. 14.8 The variation of energy E(2, 0,D) (green dashed line)on

the dimension D is very similar to E(1, 0,D). The energy

E(2, 1,D) (red solid line) increases with the increasing

dimension D.ForD>2 the energy E(2, 1,D) almost overlaps

E(2, 0,D) .............................192

Fig. 14.9 The variations of the E(3,l,D) (l =2, 1, 0) on the dimension

D are similar to E(2

,l,D) (l = 1, 0). The energy E(3, 2,D)

(red solid line) almost overlaps the energy E(3, 1,D) (green

dashed line) for a large D.ForD>2, the energies E(3,l,D)

(l =2, 1, 0) are almost overlapped. The red solid, green dashed

and blue dotted lines correspond to l =2, 1, 0, respectively ....193

Fig. 14.10 The energy E(1, 0,D) decreases with the increasing dimension

D ∈ (0, 1.6], but increases with the increasing dimension

D ≥ 2.4. There are no bound states for D ∈[1.6, 2.4].The

parameters v = 0.3 and s =0.2 are taken here and also in

Figs. 14.11 and 14.12

........................193

List of Figures xxiii

Fig. 14.11 The variation of the energy E(2, 0,D) (green solid line)on

the dimension D is very similar to E(1, 0,D). The energy

E(2, 1,D) (red dashed line) increases with the dimension

D ≥0.4. The energy E(2, 1,D) almost overlaps E(2, 0,D) for

D ≥2.6...............................194

Fig. 14.12 The variation of the E(3,l,D) (l = 2, 1, 0) on the dimension

D is similar to E(2,l,D) (l = 1, 0). For D ≥ 2.6 the energies

E(3,l,D) (l =2, 1, 0) are almost overlapped. The red dashed,

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

respectively

.............................194

Fig. 14.13 The energy E(1, 0,D) decreases with the increasing dimension

D ∈ (0, 2), but increases with the increasing dimension D ≥2.

The parameters v = 0 and s = 0.2 are taken and also in

Fig. 14.14. Note that its variation is very similar to that illustrated

in Fig. 14.7 except for a different amplitude . . . .........195

Fig. 14.14 The variation of energy E(2, 0,D) (green solid line)on

the dimension D is very similar to E(1, 0,D). The energy

E(2, 1,D) (red dashed line) increases with dimension D.The

energy E(2, 1,D) almost overlaps E(2, 0,D) for D>2.6....195

Fig. 14.15 The variations of the E(3,l,D) (l =2, 1, 0) on the dimension D

are similar to E(2,l,D) (l = 1

, 0). The energy E(3, 2,D) (red

dashed line) almost overlaps energy E(3, 1,D) (green dotted

line). For D>2.6, the energies E(3,l,D) (l = 2, 1, 0) are

almost overlapped .........................196

Fig. 14.16 The energy E(1, 0,D) decreases with the increasing dimension

D ∈ (0, 2), but increases with the increasing dimension D ≥2.

Note that it is symmetric with respect to the axis D = 2. The

parameters v = s =0.2 are chosen here and also in Figs. 14.17

and 14.18 ..............................196

Fig. 14.17 The variation of the energy E(2, 0,D) (green solid line)on

the dimension D is very similar to E(1, 0,D). The energy

E(2, 1,D) increases with the dimension D. The energy

E(2, 1

,D) (red dashed line) completely overlaps E(2, 0,D) for

D ≥2................................197

Fig. 14.18 The variation of the E(3, 0,D)(blue solid line) on the dimension

D is similar to E(n,0,D) (n = 2, 1). The energy E(3, 2,D)

(red dashed line) overlaps the E(3, 1,D) (green dotted line)

completely. For D ≥ 2 the energies E(3,l,D) (l = 2, 1, 0) are

completely overlapped .......................197

Fig. 14.19 The variation of the energy levels E(3,l,v) (l =0, 1, 2) on the

parameter v. The energy decreases with the parameter v.The

red dashed, green dotted and blue solid lines correspond to

l =2, 1, 0, respectively. The parameters D =3 and s =

0.2are

taken ................................198

xxiv List of Figures

Fig. 14.20 The variation of the energy levels E(3,l,s) on the parameter s.

The energy decreases with the v.Thered dashed, green dotted

and blue solid lines correspond to the l = 2, 1, 0, respectively.

The parameters D =3 and v =0.2aretaken ...........198

Fig. 14.21 The plot of energy levels E(2,l,D) as a function of dimension

D for energy (14.38) with sign “−”. The red solid and green

dashed lines correspond to l = 1, 0, respectively. The v =0.3

and s =0.2aretaken........................199

Fig. 14.22 The plot of energy levels E(3,l,D) as a function of

dimension D.Thered solid, green dashed and blue dotted lines

correspond to l = 2, 1, 0, respectively. Same parameters are

takenasthoseinFig.14.21 .....................199

Fig. 14.23 The plot of energy levels E(3,l,D) as a function of

dimension D.Thered solid, green dashed and blue dotted lines

correspond to l =2, 1, 0, respectively. The parameters v =0.2

and s =0.3aretaken........................200

Fig. 14.24 The plot of energy levels E(3,l,D) as a function of

dimension D.Thered solid, green dashed and blue dotted lines

correspond to l = 2, 1, 0, respectively. Here the parameters

v =0.6 and s =0.3 are chosen ...................200

List of Tables

Table11.1 Someusefulintegralformulae ..................131

Table11.2 Abbreviatedsymbolsandexactlysolvablepotentials.......131

Table 11.3 Solvable potentials, Maslov index μ and eigenvalues E

....132

xxv

Part I

Introduction

Chapter 1

Introduction

1 Basic Review

The exact solutions of wave equations with a spherically symmetric potential have

become an important subject in quantum mechanics [1–6]. It should be noticed that

many works along this line have been carried out in the usual three dimensional

space. However, what extra dimensions could there possibly be if we never see

them? It turns out that we do not really know yet how many dimensions our world

has. Nevertheless, all that our current observations tell us is that the world around

us is at least (3 +1) dimensional space-time as illustrated in general relativity.

The idea of extra dimensions has a rich history, dating back at least as far as

the middle of 1910s and earlier 1920s when the Nordstrom-Kaluza-Klein theory

—

usually named as the Kaluza-Klein theory—was proposed [8–11]. This theory is a

physical model that seeks to unify two fundamental forces of gravitation and electro-

magnetism. More precisely, the idea of additional spatial dimensions is from string

theory, the only self-consistent quantum theory of gravity so far. For a consistent

description of gravity, scientist needs more than (3 +1) dimensions, and the world

could have up to 11 or more spatial dimensions. The reason why we do not feel these

additional spatial dimensions in our life is because they are very different from the

The theory was ﬁrst proposed by the Finnish physicist Gunnar Nordström in 1914. Before Ein-

stein’s general relativity theory was presented, Nordström proposed a relativistic theory for grav-

ity. He uniﬁed his gravity theory with Maxwell’s electromagnetism through introducing a 5-vector

gauge ﬁeld where the ﬁrst four components are identiﬁed with Maxwell’s vector potential A

and the 5th component with the scalar gravity ﬁeld. After that in 1919 a German mathematician

Theodor Kaluza performed similar calculations but with Einstein’s gravity theory and Maxwell’s

electromagnetism. In terms of a circular extra dimension Kaluza obtained a 4-dimensional action

from a 5-dimensional one. The 4-action contained a graviton, an Abelian gauge boson identiﬁed

as the photon and a scalar ﬁeld that Kaluza put to be constant. The resulting equations can be sep-

arated out into further sets of equations, one of which is equivalent to Einstein’s ﬁeld equations,

another set equivalent to Maxwell’s equations for electromagnetic ﬁeld and the ﬁnal part an extra

scalar ﬁeld now termed the “radian”. In 1926, it was Swedish physicist Oskar Klein who focused

on the resulting higher modes of the particles and the size of the extra dimension [7].

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