Dong S.-H. Wave Equations in Higher Dimensions

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64 5 Klein-Gordon Equation in Higher Dimensions

4 Concluding Remarks

In this Chapter we have presented the D-dimensional Klein-Gordon equation and

applied it to the Coulomb-like potential as an illustration. The eigenfunctions are

analytically obtained and expressed by the conﬂuent hypergeometric functions. The

eigenvalues as well as their ﬁne structure are also studied.

Part III

Applications in Non-relativistic Quantum

Mechanics

Chapter 6

Harmonic Oscillator

1 Introduction

It is well known that the quantum harmonic oscillator is analog of the classical

harmonic oscillator. It is one of the most important model systems in quantum me-

chanics. There are several reasons for its pivotal role. First, it represents one of few

quantum mechanical systems for which the simple exact solutions are known. Sec-

ond, as in classical mechanics, a wide variety of physical situations can be reduced

to it either exactly or approximately. In particular, more complicated quantum sys-

tems can always be analyzed in terms of normal modes—formally equivalent to

harmonic oscillators—of motion whenever the interaction forces are linear func-

tions of the relative displacements. Therefore, it is not surprising that the harmonic

oscillator has become very important for the quantum mechanical treatment of such

physical problems as the vibrations of individual atoms in molecules and in crystals,

in which the linear harmonic oscillator describes vibrations in molecules and their

counterparts in solids, the phonons. Third, the most eminent role of the harmonic

oscillator is its linkage to the boson, one of the conceptual building blocks of mi-

croscopic physics. For example, bosons describe the modes of the electromagnetic

ﬁeld, providing the basis for its quantization. Even though the linear harmonic os-

cillator may represent rather non-elementary objects like a solid and a molecule, it

provides a window into the most elementary structure of the physical world. The

most likely reason for this connection with fundamental properties of matter is that

the harmonic oscillator Hamiltonian is symmetric in momentum and position, both

operators appearing as quadratic terms. On the other hand, the harmonic oscillator

also provides the key to the quantum theory of the electromagnetic ﬁeld, whose vi-

brations in a cavity can be analyzed into harmonic normal modes, each of which has

energy levels of the harmonic oscillator type.

This Chapter is organized as follows. In Sect. 2 we ﬁrst study exact solutions of

harmonic oscillator in arbitrary dimensions. Section 3 is devoted to the recurrence

relations for the radial wavefunction. We shall show the realization of dynamic al-

gebra su(1, 1) in Sect. 4. In Sect. 5 we carry out the generalized harmonic oscillator

named the pseudoharmonic oscillator, whose exact solutions, ladder operators and

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

DOI 10.1007/978-94-007-1917-0_6, © Springer Science+Business Media B.V. 2011

68 6 Harmonic Oscillator

recurrence relations are presented. The position and momentum information entropy

is reviewed brieﬂy in Sect. 6. Some concluding remarks are given in Sect. 7.

2 Exact Solutions of Harmonic Oscillator

Based on the results presented in Chap. 3, we are going to study the exact solu-

tions of harmonic oscillator. Consider a D-dimensional Schrödinger equation with

a harmonic potential

V(r)=

,k=Mω

, (6.1)

where M and ω denote the mass and vibration frequency of the particle, respectively.

Let

[λ]

(r) =e

−αξ

R(ξ), ξ =r

, (6.2)

where α and β are the parameters to be determined. Substituting Eqs. (6.1) and (6.2)

into the radial equation (3.66) where λ =l given in Chap. 3,wehave( =1)

R(ξ)

dξ



2β +

−2αξ



dR(ξ)

dξ



−1



β(2β +D −2)

−

l(l +D −2)



+ξ



−





−2α



β +



R(ξ) =0. (6.3)

Take the parameters α and β to remove the terms of ξ

−1

and ξ ,

α =

√

,β=

. (6.4)

Further deﬁning a variable by

ρ = 2αξ, (6.5)

we obtain the conﬂuent hypergeometric differential equation [192]:

R(ρ)

dρ



l +

−ρ



dR(ρ)

dρ





−

(2l +D)



R(ρ) =0. (6.6)

Therefore, the solutions to the D-dimensional Schrödinger equation with a har-

monic potential are given by

D−1

,...,l

(x) =N

E,l

−ρ/2

l/2

(−n

;l +D/2;ρ)Y

D−1

,...,l

(

x),

ρ =r

√

Mk, n



−

(2l +D)

=0, 1,...,

(6.7)

3 Recurrence Relations for the Radial Function 69

where N

E,l

is the normalization factor. The consideration of the ﬁniteness of the

solutions at inﬁnity leads to a quantum condition that n

has to be a non-negative

integer. Hence, the energy E can be quantized by a principal quantum number n:

n,l



k/M(n +D/2), n =2n

+l = 0, 1, 2,..., (6.8)

from which we note that

l =



0, 2, 4,...,n, when n is even,

1, 3, 5,...,n, when n is odd.

(6.9)

The degeneracy of the states with the energy E

n,l

is calculated as

m =



(D) =

(n +D −1)!

n!(D −1)!

. (6.10)

We now determine the normalization factor N

E,l

from the normalization condi-

tion



|ψ

D−1

,...,l

(x)|



a=1

=1. (6.11)

Notice that when n

is a non-negative integer, the conﬂuent hypergeometric func-

tions

(−n

;β +1;ρ) can be expressed by the associated Laguerre polynomial

(ρ).UsingEq.(5.21), through a direct calculation we obtain the normalization

factor

E,l

(Mk)

D/8

(l +D/2)



2(n

+l +D/2)



1/2

. (6.12)

3 Recurrence Relations for the Radial Function

It is shown from Eq. (3.66) that the Hamiltonian of the D-fold degenerate oscillator

for a given angular momentum as follows ( =1):

H =−



D −1

−

l(D +l −2)



Mω

, (6.13)

with the property

HR(n,l)=E

R(n,l), E

=ω(n +D/2). (6.14)

For convenience, deﬁne a new variable  =

√

Mωr. As a result, the Hamiltonian

is modiﬁed as

H(l)=−



d

D −1



d

−

l(D +l −2)



−



, (6.15)

with the property

H(l)R(n,l) =εR(n, l), ε =(n +D/2) (6.16)

70 6 Harmonic Oscillator

and

H =ωH (l), E

=ωε. (6.17)

To obtain the recurrence relations for the radial function, we deﬁne the following

operators [14]

=−

d



+,

−

d

(l +D −2)



+,

d

−



+,

−

=−

d

−

(l +D −2)



+.

(6.18)

Based on these operators, we may obtain the following identities on the Hamiltonian

2H(l)=

⎧

⎪

⎨

⎪

⎩

−

(l +1)A

(l) −(2l +D),

(l −1)A

−

(l) −(2l +D −4),

−

(l +1)a

(l) +(2l +D),

(l −1)a

−

(l) +(2l +D −4).

(6.19)

By multiplying A

and a

appropriately on both sides of Eqs. (6.19), we are able

to obtain

H(l±1)A

(l) −A

(l)H (l) =±A

(l),

H(l±1)a

(l) −a

(l)H (l) =∓a

(l).

(6.20)

Operating on the radial function R(n,l) allows us to obtain

H(l±1)A

(l)R(n, l) =(ε ±1)A

(l)R(n, l),

H(l±1)a

(l)R(n, l) =(ε ±1)a

(l)R(n, l),

(6.21)

from which we have

(l)R(n, l) =B

R(n ±1,l±1),

(l)R(n, l) =C

R(n ∓1,l±1),

(6.22)

where

=−

√

2(n +l +D),

−

=−

√

2(n +l +D −2),

√

2(n −l),

−

√

2(n −l +2).

(6.23)

In terms of these relations, one is able to derive a few useful recurrence relations

as follows:

3 Recurrence Relations for the Radial Function 71

R(n,l) =

⎧

⎪

⎨

⎪

⎩

−



n+l+D

R(n +1,l+1)



n−l

R(n −1,l+1),

−



n+l+D−2

R(n −1,l−1)



n−l+2

R(n +1,l−1),

(6.24)

D +2l −2



R(n,l) =

⎧

⎪

⎨

⎪

⎩

−

√

2(n +l +D)R(n +1,l+1)

−

√

2(n −l +2)R(n +1,l−1),

−

√

2(n +l +D −2)R(n −1,l −1)

−

√

2(n −l)R(n −1,l +1),

(6.25)

(D +2l −2)

dR(n,l)

d

⎧

⎪

⎨

⎪

⎩

(D +l −2){



n+l+D

R(n +1,l+1)



n−l

R(n −1,l+1)},

−l{



n+l+D−2

R(n −1,l−1)



n−l+2

R(n +1,l−1)}.

(6.26)

Before ending this section, let us review the results given by Coulson and Joseph

[89]. In a D-dimensional Euclidean space we deﬁne the Hamiltonian

H =

), (6.27)

where the atomic units  =M =ω are used. Denote its eigenfunctions by ψ

, where

l is the total orbital angular momentum quantum number. If σ

(D+1)

is a vector in

D-space with components σ

i(D+1)

(i =1,...,D), then we let

(D+1)

=r ·σ

(D+1)

=−p ·σ

(D+1)

, (6.28)

where σ are the generalized Pauli spin operators with the properties [88]

=−σ

,σ

=σ

†

,σ

=1,

=iσ

, [σ

,σ

]=0.

(6.29)

From Eqs. (6.27) and (6.28) as well as the properties of σ

(D+1)

, it may be veriﬁed

that

[H,A

(D+1)

]=iB

(D+1)

, [H,iB

(D+1)

]=A

(D+1)

, (6.30)

which are the Fourier transforms of each other apart from an irrelevant constant.

Combining them leads to an important result

[H,(A

(D+1)

±iB

(D+1)

)]=±[A

(D+1)

±iB

(D+1)

], (6.31)

which means that (A

(D+1)

± iB

(D+1)

) are the ladder operators for the Hamilto-

nian H . We should point out that the ladder operators given by Coulson and Joseph

are self-adjoint ones unlike those (6.18), which cannot be self-adjoint. This is be-

cause the adjoint of either ladder operator must be that operator which steps the

eigenfunctions in the opposite sense.

72 6 Harmonic Oscillator

Up to this stage the discussion of the ladder operators has been purely algebraic

without reference to any particular set of coordinates. Undoubtedly, it shall be very

interesting to determine the analytical form assumed by A

(D+1)

and B

(D+1)

when

expressed in terms of spherical polar coordinates. The results are given by [89, 193]

(D+1)

±iB

(D+1)

[r ∓(∂/∂r −r

−1

)], (6.32)

where

(D+1)

A

(D+1)



(D+1)



i=1

i(D+1)



m=2

(D+1)



i=1

i(D+1)

(6.33)

This means that the operator may be decomposed into two parts. One of these

(D+1)

steps the angular part of wavefunction. The other steps the radial part.

4 Realization of Dynamic Group SU(1, 1)

In this section, we are going to realize the dynamic group SU(1, 1) for the radial

Schrödinger equation [194]. For simplicity, we may write down the D-dimensional

radial function as follows:

(r) =



(n

+l +D/2)

−r

l+(D−2)/2

). (6.34)

Making use of the following relations for the Laguerre functions [192]

(x) =



(x) −(n +β)L

n−1

(x),

(n +1)L

n+1

(x) −(n +β +1 −x)L

(x),

(6.35)

as well as the relation (C.29) for arbitrary operators A and B (see Appendix C), we

may obtain the raising and lowering operators for the quantum number n



±r

−r

+H(l)±



, (6.36)

where H(l) is the radial Hamiltonian (6.15).

By deﬁning L

=H(l)/2, it is found that L

and L

satisﬁes the commutation

relations of an su(1, 1) algebra

]=±L

, [L

−

]=−2L

, (6.37)

5 Generalization to Pseudoharmonic Oscillator 73

with the following properties

(r) =



+1)



+l +



+1)l

(r),

−

(r) =





+l +

−1



−1)l

(r),

(r) =





(r).

(6.38)

The Casimir operator is calculated as

C = L

−L

−

) (6.39)

with eigenvalue k(k −1). After calculating we ﬁnd the Casimir eigenvalue

(r) =

2l +D



2l +D

−1



(r), (6.40)

from which we have two possible solutions k = (2l + D)/4ork =−[(2l +D)/

4 −1]. However, we are interested only in the positive discrete representations of

the su(1, 1), D

(k). Thus, we take the former k =(2l +D)/4.

5 Generalization to Pseudoharmonic Oscillator

5.1 Introduction

It is well known that the real molecular vibrations are anharmonic even though the

harmonic oscillator model is widely used as mentioned above. For instance, some

anharmonic oscillator molecular potentials such as the Morse potential and Pöschl-

Teller potential represent two typical model potentials to describe the molecular

vibrations. In this section we want to study the pseudoharmonic oscillator pro-

posed by Goldman and Krivchenkov in the early 1960s and expressed as V(x)=

(x/a −a/x)

[195], where V

and a represent two potential parameters. It is ob-

vious to see that this potential is the sum of the harmonic oscillator and inversely

quadratic potential. For the inverse squared interaction potential, Post proposed it in

1956 when he studied the one-dimensional many-identical-particle problem for pair-

force interaction between the particles [196]. Since 1961 such a quantum system

has been studied by many authors [2, 195–209]. For example, Landau and Lifshitz

studied its exact solutions in three dimensions [2]. Hurley found that this kind of

pseudoharmonic oscillator interaction between the particles can be exactly solved

when he studied three-body problem in one dimension [198]. Several years later,

Calogero studied the one-dimensional three- and many-body problems interacting

pairwise via harmonic and inverse square (centrifugal) potential [199, 200]. Also,

this potential was generalized by Camiz and Dodonov et al. to the non-stationary

(varying frequency) pseudoharmonic oscillator potential [201–203]. In addition to