fulltext

On the radial-part equation of the wave function

in N dimensions (*)

S. M. AL-JABER

Department of Physics, An-Najah National University - Nablus, West Bank, via Israel (ricevuto il 19 Agosto 1996; approvato il 20 Novembre 1996)

Summary. — We consider the equation for the radial part of the wave function of the

Schrödinger equation in the N-dimensional space. A new effective potential is derived when the equation for the radial part of the wave function is written in the form of a one-dimensional Schrödinger equation. As a constructive example, we find and discuss the solution, the orthonormality, and the energy eigenvalues of the radial part of the wave function for an infinite spherical potential well in N dimensions.

PACS 03.65.Ca – Formalism.

PACS 03.65.Bz – Foundations, theory of measurements, miscellaneous theories (including Aharonov-Bohm effect, Bell inequalities, Berry’s phase).

1. – Introduction

Recently, problems in the N-dimensional space are becoming increasingly important: the formulation of path integrals and its quantization [1, 2], in the study of the dependence of the Wentzel-Kramers-Brillouin (WKB) approximations in connection with the hyperspherical quantum billiards [3], the use of the path integrals on curved manifolds [4], in the theory of zero-range potentials [5], in the generalization of Fermi pseudopotentials to higher dimensions [6], in the study of the Ising limit of quantum field theory [7, 8], in Casimir effect [9], and just recently by the present author on the quantization of angular momentum in N dimensions [10].

It is the purpose of this paper to investigate some aspects of the radial-part equation of the wave function in the N-dimensional space. In sect. 2, we derive a new effective potential when the equation for the radial part of the wave function is reduced to a one-dimensional problem. In sect. 3, a simple, but constructive, example is discussed, namely an infinite spherical potential well in N dimensions. There, we find and discuss the solution and the orthonormality of the radial

(*) The author of this paper has agreed to not receive the proofs for correction.

part of the wave function and then we examine the energy eigenvalues of the problem. The final section is devoted to results and discussion.

2. – The effective potential in N dimensions

The equation for the radial part of the wave function in N dimensions (r , u1, u2, R , uN 22, f) has the form

d2_R dr2 1 N 21 r dR dr 1

k

l

where b is a separation constant whose values are given by [11]

b 4n(n1N22) ,

( 2 )

with n 40, 1, 2, R, and k24 2 M

(

)

, where M is the mass of the particle. It is tempting to reduce eq. (1) to a one-dimensional Schrödinger equation. This can be achieved as follows: Let

R(r) 4rau(r) ,

(3)

where a is a constant to be determined. The substitution of eq. (3) into eq. (1) yields d2u dr2 1 ( 2 a 1N21) r du dr 1 a(a 1N22) r2 u 1

k

l

In order that the second term of eq. (4) vanishes, we should have a 4 (12N)O2 and thus the latter equation becomes

d2_u dr2 1

y

z

Equation (5) is the analogue of the one-dimensional Schrödinger equation if one introduces an effective potential given by

Veff4 V(r) 1 b r2 1 (N 21)(N23) 4 r2 . (6)

This is the general form of the effective potential in N dimensions. The second term on the right-hand side of eq. (6) is well known as the centrifugal barrier and its origin is well understood [12]. The third term of eq. (6) is an additional repulsive potential (for

N D3), which pushes the particle further away from the origin. For N42, this term

becomes an attractive one. In the usual three-dimensional space (N 43) this third term vanishes and one recovers the usual effective potential. The presence of this remarkable additional potential term in Veff(r) is another example of the role of the

topological structure of the configuration space of a physical system in the quantum nature of the system. The emphasis on the role of the topology of the configuration space in the behaviour of the quantum system has been reported by the present author and others [13, 14].

3. – An infinite spherical potential well in N dimensions

We consider an infinite spherical potential well, in the N-dimensional space, defined as

V(r) 40 , for rEa ; V(r) 4Q , for rDa .

The differential equation for the radial part of the wave function in the region of the well is given by eq. (1) with k now given by k2_{4 2 MEOˇ}2.

The solution for the differential equation d2_Y dx2 1

g

h

y

z

where Jp and Np are the ordinary Bessel and Neumann functions, respectively, and a,

b, c and p are constants. Comparing eq. (1) with eq. (7) yields a 4 (22N)O2, c41, b 4k, and p4n1 (N22)O2. Therefore, with the help of eq. (8), the solution can be

readily written down:

R(r) 4r2(N 2 2 ) O2_[AJ

n 1 (N22)O2(kr) 1BNn 1 (N22)O2(kr) ] , (9)

where A and B are constants. Because of the constraint that R(r) must be finite at

r 40, we see that the N(kr) term in eq. (9) must be rejected due to its singular

behaviour at the origin. Thus we have

R(r) 4 A

r(N 22)O2Jn 1 (N22)O2(kr) .

(10)

Note that the order of the Bessel function is integer for even N and half-odd integer for odd N. For the usual three-dimensional case, N 43, eq. (10) gives the well-known spherical Bessel function jn(kr) which is found in most quantum mechanics textbooks [16].

In order to discuss the orthonormality of the radial-part solution, R(kr), we adopt the Sturm-Liouville theory. The differential equation (1) can be written in the form

d dr

g

h

The Sturm-Liouville differential equation has the form d

dx

k

dx

l

It is known [17] that two different eigenfunctions, yj(x) and yk(x), corresponding to different eigenvalues, lj and lk, respectively, are orthogonal in the interval a GxGb with respect to the weight factor (or weight function) W(x). Comparing eqs. (11) and

(12) yields

P(r) 4W(r) 4rN 21_{, Q(r) 42br}N 23_, (13)

and therefore the eigenfunctions R(kr) are orthogonal in the interval 0 GrGa with respect to the weight function W(r) 4rN 21_{. The condition that R(kr) vanishes at}

r 4a implies that

Jn(ka) 40 , (14)

where n 4n1 (N22)O2. If we let knm and knm be the m-th and the m-th roots of Jn, respectively, then the orthogonality of the corresponding eigenfunctions reads



0

a

Rn(knmrOa) Rn(knmrOa) rN 21dr 40 . (15)

The normalization of Rn(kr) can be achieved if one recalls the normalization of the ordinary Bessel functions which is [18]



The substitution of eq. (10) into eq. (16) yields



which is the normalization relation for the radial part of the wave function.

The energy eigenvalues of stationary states are readily obtained with the help of eq. (14) which gives ka 4knm and therefore the energy eigenvalues are

Enm4 ˇ2 2 M k 2 nm. (18)

One should note that [19] for given m, knmD kn 8 mwhenever n Dn8 and thus, by eq. (18),

EnmD En 8 m. This means that for a given n, the higher the dimension N the higher the energy

(

)

4. – Results and discussion

In this paper, a new effective potential was derived when the equation for the radial part of the wave function is written in a form which is analogous to the one-dimensional Schrödinger equation. This effective potential contains, in addition to the usual three-dimensional centrifugal term, an extra term which is repulsive for N D3 and attractive for N E3. This extra term vanishes for N43 and therefore one recovers the usual three-dimensional effective potential. We considered, in sect. 3, the infinite spherical potential well in N dimensions and found the radial wave function, the orthonormality, and the energy eigenvalues. We have shown in an explicit way that the

topological structure of the configuration space of a physical system can determine the quantum nature of an observable of the system.

R E F E R E N C E S

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[13] AL-JABERS. M. and HENNEBERGER W. C., J. Phys. A, 23 (1990) 2939. [14] HO VU B., J. Phys. A, 27 (1994) 6237.

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