Dong S.-H. Wave Equations in Higher Dimensions
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54 4 Dirac Equation in Higher Dimensions
The product of Y
[λ]
m
(
ˆ
x) and ξ(m
) belongs to the direct product of two represen-
tations [λ] and [s], which is a reducible representation:
[λ]×[s][λ, 0,...,0, 1]⊕[λ −1, 0,...,0, 1]. (4.13)
Besides, in order to construct a wavefunction belonging to the representation
[j]≡[λ,0,...,0, 1] there are two different ways: the combination of Y
[λ]
m
(
ˆ
x)ξ(m
)
and that of Y
[λ+1]
m
(
ˆ
x)ξ(m
). They are different in eigenvalues of κ. Since the system
is spherically symmetric, we only need to calculate the highest weight state for the
representation [j] in terms of the CGCs
φ
|K|,[j ]
(
ˆ
x) = Y
[λ]
[λ]
(
ˆ
x)ξ([s]) =N
D,λ
r
−λ
(x
1
+ix
2
)
λ
ξ([s]),
|K|=C
2
([j]) −C
2
([λ]) −C
2
([s]) +N =λ +N.
(4.14)
φ
−|K|,[j ]
(
ˆ
x) =
m
Y
[λ+1]
m
(
ˆ
x)ξ([j]−m)
×[λ +1], m, [s], [j ]−m|[j], [j ]
= N
2N+1,λ
r
−λ−1
(x
1
+ix
2
)
λ
{x
2N+1
ξ([s])
+(x
2N−1
+ix
2N
)ξ([0,...,0, 1, 1])
+(x
2N−3
+ix
2N−2
)ξ([0,...,0, 1, 1, 1]) +···
+(x
3
+ix
4
)ξ([1, 1, 0,...,0, 1])
+(x
1
+ix
2
)ξ([1, 0,...,0, 1])},
−|K|=C
2
([j]) −C
2
([λ +1]) −C
2
([s]) +N =−λ −N.
(4.15)
The wavefunction
K,[j ]
(x) of the total angular momentum belonging to the
irreducible representation [j ] can be expressed as the wave function
K,[j ]
(x) of
the total angular momentum belonging to the irreducible representation [j] can be
expressed as
K,[j ]
(x,t)=r
−N
e
−iEt
F(r)φ
K,[j ]
(
ˆ
x)
iG(r)φ
−K,[j ]
(
ˆ
x)
,
H
1
(J )
K,[j ]
(x) =l
K,[j ]
(x),
H
N
(J )
K,[j ]
(x) =
K,[j ]
(x),
H
ν
(J )
K,[j ]
(x) =0,ν∈[2,N −1],
κ
K,[j ]
(x) =K
K,[j ]
(x), K =±(l +N).
(4.16)
Their partners can be calculated by the lowering operators F
ν
.
The radial equation depends on the explicit forms of β
a
(γ
a
) matrices. Therefore,
by considering two introduced operators
β ·
ˆ
x =
1
r
2N+1
i=1
β
i
x
i
, β ·∇=
2N+1
i=1
β
i
∂
∂x
i
, (4.17)
3 The Radial Equations 55
we obtain
(
β ·
ˆ
x)φ
K,[j ]
(
ˆ
x) =
1
r
2N+1
a=1
β
a
x
a
φ
K,[j ]
(
ˆ
x) = φ
−K,[j ]
(
ˆ
x),
(
β ·
p)
1
r
N
φ
K,[j ]
(
ˆ
x) =
2N+1
a=1
β
a
p
a
1
r
N
φ
K,[j ]
(
ˆ
x) = iK
1
r
N+1
φ
−K,[j ]
(
ˆ
x).
(4.18)
Substituting
K,[j ]
(x) into the Dirac equation (4.3) allows us to obtain the radial
equations
dG(r)
dr
+
K
r
G(r) =[E −V(r)−M]F(r),
−
dF(r)
dr
+
K
r
F(r)=[E −V(r)+M]G(r).
(4.19)
3.2 The SO(2N) Case
As is well known, the spinor representation of the SO(2N) group is reducible and
can be reduced to two non-equivalent fundamental spinor representations [+s]≡
[1, 0,...,0, 1] and [−s]≡[1, 0,...,0, 1, 0].FromEq.(4.7) the Casimir for both
spinor representations are calculated as C
2
([±s]) = (2N
2
−N)/4. In terms of the
β
a
matrices, we define the γ
μ
matrices for N =2l:
γ
0
=β
2N+1
,γ
a
=β
2N+1
β
a
,a∈[1, 2N]. (4.20)
γ
0
is a diagonal matrix where half of the diagonal elements are equal to +1 and the
remaining to −1. Because the spinor operator S
ab
and the operator κ commutes with
γ
0
, each of them becomes a direct sum of two matrices, referring to the rows with
the eigenvalues +1 and −1oftheγ
0
, respectively. The fundamental spinors ξ
±
(m)
belong to the fundamental spinor representations [+s] and [−s], respectively, and
satisfy
γ
0
ξ
±
(m) =±ξ
±
(m). (4.21)
The product of Y
[λ]
m
(
ˆ
x) and ξ
±
(m
) belongs to the direct product of two repre-
sentation [λ] and [±s], which is a reducible representation:
[λ]×[+s][λ,0,...,0, 1]⊕[λ −1, 0,...,0, 1, 0],
[λ]×[−s][λ,0,...,0, 1, 0]⊕[λ −1, 0,...,0, 1].
(4.22)
There are two kinds of representations for the total angular momentum: the rep-
resentation [j
1
]≡[λ, 0,...,0, 1] and the representation [j
2
]≡[λ, 0,...,0, 1, 0].
Their Casimirs are the same:
C
2
([j
1
]) = C
2
([j
2
]) =λ(λ +2N −1) +
2N
2
−N
4
. (4.23)
There are two different ways to construct wavefunction belonging to the repre-
sentation [j
1
]: the combination of Y
[λ]
m
(
ˆ
x)ξ
+
(m
) and that of Y
[λ+1]
m
(
ˆ
x)ξ
−
(m
).Due
56 4 Dirac Equation in Higher Dimensions
to the spherical symmetry, we only calculate the highest weight state for the repre-
sentation [j
1
] by the CGCs:
φ
K,[j
1
]
(
ˆ
x) = Y
[λ]
[λ]
(
ˆ
x)ξ
+
([+s]) =N
2N,λ
r
−λ
(x
1
+ix
2
)
λ
ξ
+
([+s]),
φ
−K,[j
1
]
(
ˆ
x) =
m
Y
[λ+1]
m
(
ˆ
x)ξ
−
([j
1
]−m)
×[λ +1], m, [+s], [j
1
]−m|[j
1
], [j
1
]
= N
2N,λ
r
−λ−1
(x
1
+ix
2
)
λ
{x
2N−1
+ix
2N
ξ
−
([−s])
+(x
2N−3
+ix
2N−2
)ξ
−
([0,...,0, 1, 1, 0])
+(x
2N−5
+ix
2N−4
)ξ
−
([0,...,0, 1, 1, 0, 1]) +···
+(x
3
+ix
4
)ξ
−
([1, 1, 0,...,0, 1])
+(x
1
+ix
2
)ξ
−
([1, 0,...,0, 1])},
K =C
2
([j
1
]) −C
2
([λ +1]) −C
2
([+s]) +N −
1
2
= λ +N −
1
2
.
(4.24)
For the representation [j
2
]≡[λ, 0,...,0, 1, 0],wehave
φ
K,[j
2
]
(
ˆ
x) =
m
Y
[λ+1]
m
(
ˆ
x)ξ
+
([j
2
]−m)
×[λ +1], m, [−s], [j
2
]−m|[j
2
], [j
2
]
= N
2N,λ
r
−λ−1
(x
1
+ix
2
)
l
{x
2N−1
−ix
2N
ξ([+s])
+(x
2N−3
+ix
2N−2
)ξ
+
([0,...,0, 1, 0, 1])
+(x
2N−5
+ix
2N−4
)ξ
+
([0,...,0, 1, 1, 1, 0]) +···
+(x
3
+ix
4
)ξ
+
([1, 1, 0,...,0, 1, 0])
+(x
1
+ix
2
)ξ
+
([1, 0,...,0, 1, 0])},
φ
−K,[j
2
]
(
ˆ
x) = Y
[λ]
[λ]
(
ˆ
x)ξ
−
([−s]) =N
D,λ
r
−λ
(x
1
+ix
2
)
λ
ξ
−
([−s]),
K =C
2
([j
2
]) −C
2
([λ +1]) −C
2
([+s]) +N −
1
2
=−λ −N +
1
2
.
(4.25)
In terms of the explicit forms of β
a
we obtain
(
β ·
ˆ
x)φ
K,[j
ω
]
(
ˆ
x) =
1
r
2N
a=1
β
a
x
a
φ
K,[j
ω
]
(
ˆ
x) = φ
−K,[j
ω
]
(
ˆ
x),
(
β ·
p)r
−N+1/2
φ
K,[j
ω
]
(
ˆ
x) =
2N
a=1
β
a
p
a
r
−N+1/2
φ
K,[j
ω
]
(
ˆ
x)
= iKr
−N−1/2
φ
−K,[j
ω
]
(
ˆ
x),
ω =1or2.
(4.26)
4 Application to Hydrogen Atom 57
The wavefunction
K,[j
ω
]
(x) of the total angular momentum belonging to the
irreducible representation [j
ω
] can be expressed as
|K|,[j
1
]
(x,t)=r
−N+1/2
e
−iEt
{F(r)φ
|K|,[j
1
]
(
ˆ
x) +iG(r)φ
−|K|,[j
1
]
(
ˆ
x)},
−|K|,[j
2
]
(x,t)=r
−N+1/2
e
−iEt
{F(r)φ
−|K|,[j
2
]
(
ˆ
x) +iG(r)φ
|K|,[j
2
]
(
ˆ
x)},
κ
K,[j
ω
]
(x) =K
K,[j
ω
]
(x), |K|=λ +N −1/2,ω=1or2,
H
1
(J )
K,[j
ω
]
(x) =λ
K,[j
1
]
(x),
H
N−1
(J )
K,[j
1
]
(x) =0,H
N
(J )
K,[j
1
]
(x) =
K,[j
1
]
(x),
H
N−1
(J )
K,[j
2
]
(x) =
K,[j
2
]
(x), H
N
(J )
K,[j
2
]
(x) =0,
H
ν
(J )
K,[j
ω
]
(x) =0,ν∈[2,N −2].
(4.27)
Their partners can be calculated by the lowering operators F
ν
.
By substituting
K,[j
ω
]
(x) into the Dirac equation (4.3), we obtain the radial
equations
d
dr
G(r) +
K
r
G(r) =(E −V(r)−M)F(r),
−
d
dr
F(r)+
K
r
F(r)=(E −V(r)+M)G(r),
(4.28)
which are the same as those in D =2N +1 case.
4 Application to Hydrogen Atom
Although the wavefunctions and the eigenvalues K are different for the cases D =
2N +1 and D =2N, the radial equations are unified as
d
dr
G
KE
(r) +
K
r
G
KE
(r) =[E −V(r)−M]F
KE
(r),
−
d
dr
F
KE
(r) +
K
r
F
KE
(r) =[E −V(r)+M]G
KE
(r),
K =±
1
2
(2l +D −1).
(4.29)
For definiteness we discuss the attractive Coulomb potential
V(r)=−
ξ
r
,ξ=Zα > 0, (4.30)
where and later α =1/137 is the fine structure constant. It is easy to see that the so-
lution for the repulsive potential can be obtained from that for the attractive potential
by interchanging
F
KE
←→ G
−K−E
,V(r)←→ − V(r). (4.31)
From the Sturm-Liouville theorem [184], there are bound states with the energy less
than and near M for the attractive Coulomb potential and with the energy larger than
and near −M for the repulsive potential, if the interaction is not too strong.
58 4 Dirac Equation in Higher Dimensions
For convenience, we introduce a new variable ρ in Eq. (4.29) for bound states:
ρ = 2r
M
2
−E
2
,E/M∈(0, 1]. (4.32)
Solving F(ρ) from Eq. (4.29),
F
KE
(ρ) =
−
1
2
M −E
M +E
+
ξ
ρ
−1
d
dρ
G
KE
(ρ) +
K
ρ
G
KE
(ρ)
, (4.33)
we obtain a second-order differential equation of G
KE
(ρ):
d
2
dρ
2
G
KE
(ρ) +
−
1
4
−
K
2
−ξ
2
+K
ρ
2
+
Eξ
ρ
√
M
2
−E
2
G
KE
(ρ)
+
ρ −
ρ
2
2ξ
M −E
M +E
−1
d
dρ
G
KE
(ρ) +
K
ρ
G
KE
(ρ)
=0. (4.34)
From the behavior of G
KE
(ρ) at the origin and at infinity, define
G
KE
(ρ) =ρ
λ
e
−ρ/2
R(ρ), λ =
K
2
−ξ
2
> 0,
ω =
1
2ξ
M −E
M +E
,τ=
Eξ
√
M
2
−E
2
.
(4.35)
Substitution of them into Eq. (4.34) leads to
(ρ −ωρ
2
)
d
2
dρ
2
R(ρ) +[ωρ
2
−(2λω +1)ρ +2λ +1]
d
dρ
R(ρ)
+[ω(λ −τ)ρ +ω(K +λ) +τ −λ −1/2]R(ρ) =0. (4.36)
Equation (4.36) can be solved by the power series expansion method, which was
used in [175, 176]for(3 +1) dimensions. The results are calculated as
F
KE
(ρ)
G
KE
(ρ)
=
(M
2
−E
2
)
1/4
(2λ +1)
(M ±E)E(n
+2λ +1)
2M
2
τ(j +τM/E)n
!
×ρ
λ
e
−ρ/2
{(K +τM/E)
1
F
1
(−n
;2λ +1;ρ)
∓n
1
F
1
(1 −n
;2λ +1;ρ)}
×
∞
0
(|F
KE
(ρ)|
2
+|G
KE
(ρ)|
2
)dr =1, (4.37)
n
=τ −λ =0, 1, 2,.... (4.38)
When n
=0, K has to be positive. Introduce a principal quantum number
n =|K|−(D −3)/2 +n
=|K|−(D −3)/2 +τ −λ =1, 2,.... (4.39)
The n can be equal to 1 only for K =(D −1)/2 and equal to other positive integers
for both signs of K. The energy E can be calculated as
E =M
1 +
ξ
2
(
K
2
−ξ
2
+n −|K|+(D −3)/2)
2
−1/2
. (4.40)
5 Concluding Remarks 59
Expanding Eq. (4.40)inpowersofξ
2
,wehave
E M
1 −
ξ
2
2[n +(D −3)/2]
2
−
ξ
4
2[n +(D −3)/2]
4
×
n +(D −3)/2
|K|
−
3
4
, (4.41)
where the first term on the right hand side is the rest energy M (c
2
=1), the second
one coincides with the energy 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.
5 Concluding Remarks
In this Chapter we have generalized the Dirac equation to (D + 1)-dimensional
space-time. The conserved angular momentum operators and their quantum num-
bers are discussed. The eigenfunctions of the total angular momentums are calcu-
lated for both odd D = 2N +1 and even D = 2N cases, respectively. The unified
radial equations for a spherically symmetric system are obtained. As an illustration,
we have dealt with Coulomb potential problem by the series approach. The exact
solutions are expressed by the confluent hypergeometric functions. The eigenvalues
as well as their fine structure energy have also been studied.
Chapter 5
Klein-Gordon Equation in Higher Dimensions
1 Introduction
It is known that the exact solutions of non-relativistic and relativistic equations in the
spherically central fields have become an important subject in quantum mechanics.
As illustrated above, the main contributions have been made to the Schrödinger
and Dirac equations. During the past several decades, however, the Klein-Gordon
equation with the Coulomb potential has been studied in three dimensions such as
the operator analysis [185], in an intense laser field [186], in two dimensions [187]
and in one dimension [188–191]. On the other hand, the Klein-Gordon equation
with a Coulomb potential in (D +1) dimensions has been discussed by the different
approaches like the large-N expansion approximate method [93]. The purpose of
this Chapter is to present the Klein-Gordon equation in arbitrary dimensions and
solve the hydrogen-like atom problem.
2 The Radial Equations
For a particle moving in a spherically symmetric potential V(r), the symmetric
group of the quantum system is the SO(D) group. The time-independent Klein-
Gordon equation ( =c =1) is written as
(−
2
+M
2
)(r) =[E −V(r)]
2
(r), (5.1)
where M and E denote the mass and the energy of the particle, respectively. As
shown in Eq. (3.69), we take wavefunction as
(r) =r
−
D−1
2
R
l
(r)Y
l
l
D−2
...l
1
(
ˆ
x). (5.2)
Substitution of this into Eq. (5.1) allows us to obtain the D-dimensional radial
Klein-Gordon equation
d
2
dr
2
−
κ
2
−1/4
r
2
R
l
(r) =−{[E −V(r)]
2
−M
2
}R
l
(r), (5.3)
S.-H. Dong, Wave Equations in Higher Dimensions,
DOI 10.1007/978-94-007-1917-0_5, © Springer Science+Business Media B.V. 2011
61
62 5 Klein-Gordon Equation in Higher Dimensions
where
κ =|l −1 +D/2| (5.4)
depends on the angular momentum l and the spatial dimension D. Y
l
l
N−2
...l
1
(
ˆ
x) in
Eq. (5.2) is called the generalized spherical harmonics.
3 Application to Hydrogen-like Atom
As an illustration, we take the symmetric potential V(r) as the Coulomb-type po-
tential
V(r)=−
ξ
r
,ξ=Zα. (5.5)
We will discuss the weak potential, say |ξ | < 1/2. The radial equation (5.3) thus
becomes
d
2
dr
2
R
l
(r) +
ξ
2
−κ
2
+1/4
r
2
R
l
(r) +
2Eξ
r
R
l
(r) +(E
2
−M
2
)R
l
(r) =0. (5.6)
Take a new variable ρ for the bound states:
ρ =2r
M
2
−E
2
, |E|<M. (5.7)
Equation (5.6) thus changes to
d
2
dρ
2
R
l
(ρ) +
1
ρ
2
ξ
2
−κ
2
+
1
4
R
l
(ρ) +
τ
ρ
R
l
(ρ) −
1
4
R
l
(ρ) =0, (5.8)
with
τ =
Eξ
√
M
2
−E
2
. (5.9)
From the behaviors of radial function at the origin and at infinity, we take the
wavefunction of the form
R
l
(ρ) =ρ
λ+1/2
e
−ρ/2
φ(ρ), λ =
κ
2
−ξ
2
> 0, (5.10)
whereweassumeκ
2
>ξ
2
.Theφ(ρ) satisfies the following confluent hypergeomet-
ric equation
ρ
d
2
dρ
2
φ(ρ)+(2λ +1 −ρ)
d
dρ
φ(ρ)+
τ −λ −
1
2
φ(ρ)=0. (5.11)
Thus, the radial function can be written out
R
l
(ρ) =N
l
ρ
λ+1/2
e
−ρ/2
1
F
1
(λ −τ +1/2;2λ +1;ρ), (5.12)
where N
l
is the normalization factor to be determined.
We now discuss the eigenvalues. From the finiteness of the solutions at infinity,
the general quantum condition is obtained from Eq. (5.12)
τ −λ −
1
2
=n
=0, 1, 2,.... (5.13)
3 Application to Hydrogen-like Atom 63
By introducing a principal quantum number
n =n
+κ −
N
2
+2 =n
+l +1, (5.14)
we obtain
Eξ
√
M
2
−E
2
=τ =n −l −
1
2
+
κ
2
−ξ
2
> 0. (5.15)
Therefore, we can obtain E with the same sign as ξ
E(n,l,D) =M
ξ
|ξ|
1 +
ξ
2
(n −l −
1
2
+
κ
2
−ξ
2
)
2
−1/2
, (5.16)
which essentially coincides with that of [87] except that the factor ξ/|ξ| was not
considered there.
For a large D,wehave
E(n,D) =M
ξ
|ξ|
[1 −2ξ
2
D
−2
+4ξ
2
(2n −3)D
−3
−···]. (5.17)
For a small ξ ,wehave
E(n,l,D) =M
ξ
|ξ|
1 −
ξ
2
2[n +(D −3)/2]
2
+
ξ
4
(D +6l −4n)
4(2l −2 +D)[n +(D −3)/2]
4
, (5.18)
where the first term on the right hand side is the rest energy M, the second one is
from the solutions of the Schrödinger equation with this potential, and the third one
is the fine structure energy, which removes the degeneracy between the states with
same n.
We now calculate the normalization factor N
l
from the normalization condition
∞
0
R
l
(ρ)
2
dr =1. (5.19)
Since n
=τ −λ −1/2 is a non-negative integer, we can express the confluent hy-
pergeometric functions
1
F
1
(−n
;β +1;ρ) by the associated Laguerre polynomial
L
β
n
(ρ). In terms of the following formulas [192]:
ρ
1
F
1
(α +1;β +1;ρ) =β
1
F
1
(α +1;β;ρ) −β
1
F
1
(α;β;ρ), (5.20)
L
β
n
(ρ) =
(β +n +1)
n!(β +1)
1
F
1
(−n;β +1;ρ),
∞
0
ρ
β
e
−ρ
L
β
n
(ρ)L
β
m
(ρ)dρ =
(n +β +1)
n!
δ
nm
,
(5.21)
we obtain through a direct calculation
N
l
=
(M
2
−E
2
)
1/4
(2λ +1)
2(n
+2λ +1)
n
!(2n
+2λ +1)
1/2
,
λ =[(l −1 +D/2)
2
−ξ
2
]
1/2
.
(5.22)