# A New Kind of Shift Operators for Infinite Circular and Spherical Wells

Authors: Guo-Hua Sun; K. D. Launey; T. Dytrych; Shi-Hai Dong; J. P. Draayer
Journal: Advances in Mathematical Physics
 (2014)

---

## Abstract

A new kind of shift operators for infinite circular and spherical wells is identified. These shift operators depend on all spatial variables of quantum systems and connect some eigenstates of confined systems of different radiiR sharing energy levels with a common eigenvalue. In circular well, the momentum operators P±=Px±iPy play the role of shift operators. The Px and Py operators, the third projection of the orbital angular momentum operator Lz, and the Hamiltonian H form a complete set of commuting operators with the SO(2) symmetry. In spherical well, the shift operators establish a novel relation between ψlm(r) and ψ(l±1)(m±1)(r).


---

## Body

## 1. Introduction

It is well-known that algebraic methods have become the subject of interest in different fields. Systems with a dynamical symmetry can be treated algebraically [1–7]. Using the factorization method, we have established the ladder operators for some solvable potentials like the Morse potential, the Pöschl-Teller potential, the square-well potential, and others [8]. Contrary to traditional approaches [9], where an auxiliary variable was introduced, the ladder operators constructed in the framework of our approach are expressed only in terms of spatial coordinates. It should be pointed out that the ladder operators for those quantum systems depend on only one variable, that is, one-dimensional problem [8].Motivated by the recently proposed factorization method [8], we generalize it to construct ladder operators that depend on all spatial variables. For this purpose, we derive shift operators for the circular [10–13] and spherical wells [14] directly from the normalized wave functions [15, 16]. The difference between the ladder and shift operators is that the former correspond to the same potential, but the latter correspond to the same energy of different potential groups. Therefore, the latter method is named potential group approach. We start with a brief review, with the purpose of introducing the general formalism, and then we present the new shift operators that provide a novel insight into the problem. In particular, we include specific illustrations to show the generation of a series of potentials with different radii and the same energy.This work is organized as follows. In Section2 we review the exact solutions of the two quantum systems, namely, in the circular- and spherical-well cases. In Section 3 we present general formalism to establish the shift operators and illustrate how to generate a group of different potentials with same energy levels. Some concluding remarks are given in Section 4.


## 2. Exact Solutions

### 2.1. Circular Well

The circular and spherical wells are defined asV(r)=V(r)=0 for r<R and V(r)=∞ for r≥R, where r denotes (r,φ) and (r,θ,ϕ) for circular and spherical wells, respectively. The Schrödinger equation in relevant polar coordinates can be written as
(1)-ħ22μ(∂2∂r2+1r∂∂r+1r2∂2∂φ2)ψ(r,φ)=Eψ(r,φ).By takingψ(r,φ)=R(r)eimφ/2π (m=0,±1,±2,…) and substituting it into (1), we have
(2)d2R(r)dr2+1rdR(r)dr-m2r2R(r)=-k2R(r),k=2μEħ2.By definingz=kr, it becomes the Bessel differential equation
(3)d2R(z)dz2+1zdR(z)dz+(1-m2z2)R(z)=0.
It has two linearly independent solutions for each value of |m|. One is the regular Jm(z) and the other is the singular Ym(z). Only the well-behaved Jm(z) are physically acceptable, with the eigenvalues determined by Jm(kR)=0. That is, kR is the nth root αmn of the Bessel function Jm(x) (see Table 1). In terms of this condition, the eigenvalues are then given by
(4)Emn=ħ2kmn22μ=ħ22μR2αmn2.
The corresponding eigenfunctions for the nth root are given by
(5)ψm(r,φ)=NmJm(kr)eimφ,Nm=1Rπ1/2|Jm+1(αmn)|.Table 1
Zeros of the Bessel functionsJm(x).
nth zero
J
0
(
x
)
J
1
(
x
)
J
2
(
x
)
J
3
(
x
)
J
4
(
x
)
J
5
(
x
)
1
2.4048
3.8317
5.1356
6.3802
7.5883
8.7715
2
5.5201
7.0156
8.4172
9.7610
11.0647
12.3386
3
8.6537
10.1735
11.6198
13.0152
14.3725
15.7002
4
11.7915
13.3237
14.7960
16.2235
17.6160
18.9801
5
14.9309
16.4706
17.9598
19.4094
20.8269
22.2178Obviously, the complete spectrum for the full circular well corresponds to one set ofm=0 solutions but is doubly degenerate for each value of |m|>0, since ±m give the same contributions. This corresponds physically to the equivalence of clockwise versus counterclockwise motions.


### 2.2. Infinite Spherical Well

Whenr≤R, the Schrödinger equation in spherical coordinates is given by
(6)-ħ22μ[+1r2sin2θ∂2∂ϕ21r2∂∂r(r2∂∂r)+1r2sinθ∂∂θ(sinθ∂∂θ)+1r2sin2θ∂2∂ϕ2]ψ(r,θ,ϕ)=Eψ(r,θ,ϕ).For symmetrically central force fields, we can always express the wave functions asψ(r,θ,ϕ)=R(r)Ylm(θ,ϕ)=eimϕR(r)Θ(θ)/2π and substitute this into (6). The polar angle dependence θ leads to the associated Legendre equation
(7)1sinθddθ(sinθdΘ(θ)dθ)+[l(l+1)-m2sin2θ]Θ(θ)=0,
which is satisfied by the normalized associated Legendre functions Plm(cos⁡θ):
(8)Θlm(θ)=Plm(cos⁡θ)=2l+12(l-m)!(l+m)!Plm(cos⁡θ),lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllm∈[-l,l].
Thus, we may define the spherical harmonics as
(9)Ylm(θ,ϕ)≡(-1)m2l+14π(l-m)!(l+m)!Plm(cos⁡θ)eimϕ,
where the factor (-1)m is the so-called Condon-Shortley phase.By separating the spherical harmonics, we obtain the radial equation as(10)-ħ22μ(d2dr2+2rddr-l(l+1)r2)R(r)=ER(r).
By defining z=kr, where k is given in (2), (10) is rearranged as spherical Bessel differential equation
(11)d2R(z)dz2+2zdR(z)dz+(1-l(l+1)z2)R(z)=0
with two linearly independent solutions jl(z) and nl(z). Only the well-behaved jl(z) are acceptable in physics. The eigenvalues are determined by the boundary condition jl(kR)=0 (see Table 2) as
(12)Eln=ħ2kln22μ=ħ22μR2αln2,
where kR is the nth root αln of the spherical Bessel function jl(x). The eigenfunctions of the quantum system for the nth root are given by
(13)ψlm(r,θ,ϕ)=Nljl(kr)Ylm(θ,ϕ),Nl=2R3/2|jl+1(αln)|.Table 2
Zeros of the spherical Bessel functionsjl(x).
nth zero
j
0
(
x
)
j
1
(
x
)
j
2
(
x
)
j
3
(
x
)
j
4
(
x
)
j
5
(
x
)
1
3.1416
4.4934
5.7635
6.9879
8.1826
9.3558
2
6.2832
7.7253
9.0950
10.4171
11.7049
12.9665
3
9.4248
10.9041
12.3229
13.6980
15.0397
16.3547
4
12.5664
14.0662
15.5146
16.9236
18.3013
19.6532
5
15.7080
17.2208
18.6890
20.1218
21.5254
22.9046


## 2.1. Circular Well

The circular and spherical wells are defined asV(r)=V(r)=0 for r<R and V(r)=∞ for r≥R, where r denotes (r,φ) and (r,θ,ϕ) for circular and spherical wells, respectively. The Schrödinger equation in relevant polar coordinates can be written as
(1)-ħ22μ(∂2∂r2+1r∂∂r+1r2∂2∂φ2)ψ(r,φ)=Eψ(r,φ).By takingψ(r,φ)=R(r)eimφ/2π (m=0,±1,±2,…) and substituting it into (1), we have
(2)d2R(r)dr2+1rdR(r)dr-m2r2R(r)=-k2R(r),k=2μEħ2.By definingz=kr, it becomes the Bessel differential equation
(3)d2R(z)dz2+1zdR(z)dz+(1-m2z2)R(z)=0.
It has two linearly independent solutions for each value of |m|. One is the regular Jm(z) and the other is the singular Ym(z). Only the well-behaved Jm(z) are physically acceptable, with the eigenvalues determined by Jm(kR)=0. That is, kR is the nth root αmn of the Bessel function Jm(x) (see Table 1). In terms of this condition, the eigenvalues are then given by
(4)Emn=ħ2kmn22μ=ħ22μR2αmn2.
The corresponding eigenfunctions for the nth root are given by
(5)ψm(r,φ)=NmJm(kr)eimφ,Nm=1Rπ1/2|Jm+1(αmn)|.Table 1
Zeros of the Bessel functionsJm(x).
nth zero
J
0
(
x
)
J
1
(
x
)
J
2
(
x
)
J
3
(
x
)
J
4
(
x
)
J
5
(
x
)
1
2.4048
3.8317
5.1356
6.3802
7.5883
8.7715
2
5.5201
7.0156
8.4172
9.7610
11.0647
12.3386
3
8.6537
10.1735
11.6198
13.0152
14.3725
15.7002
4
11.7915
13.3237
14.7960
16.2235
17.6160
18.9801
5
14.9309
16.4706
17.9598
19.4094
20.8269
22.2178Obviously, the complete spectrum for the full circular well corresponds to one set ofm=0 solutions but is doubly degenerate for each value of |m|>0, since ±m give the same contributions. This corresponds physically to the equivalence of clockwise versus counterclockwise motions.


## 2.2. Infinite Spherical Well

Whenr≤R, the Schrödinger equation in spherical coordinates is given by
(6)-ħ22μ[+1r2sin2θ∂2∂ϕ21r2∂∂r(r2∂∂r)+1r2sinθ∂∂θ(sinθ∂∂θ)+1r2sin2θ∂2∂ϕ2]ψ(r,θ,ϕ)=Eψ(r,θ,ϕ).For symmetrically central force fields, we can always express the wave functions asψ(r,θ,ϕ)=R(r)Ylm(θ,ϕ)=eimϕR(r)Θ(θ)/2π and substitute this into (6). The polar angle dependence θ leads to the associated Legendre equation
(7)1sinθddθ(sinθdΘ(θ)dθ)+[l(l+1)-m2sin2θ]Θ(θ)=0,
which is satisfied by the normalized associated Legendre functions Plm(cos⁡θ):
(8)Θlm(θ)=Plm(cos⁡θ)=2l+12(l-m)!(l+m)!Plm(cos⁡θ),lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllm∈[-l,l].
Thus, we may define the spherical harmonics as
(9)Ylm(θ,ϕ)≡(-1)m2l+14π(l-m)!(l+m)!Plm(cos⁡θ)eimϕ,
where the factor (-1)m is the so-called Condon-Shortley phase.By separating the spherical harmonics, we obtain the radial equation as(10)-ħ22μ(d2dr2+2rddr-l(l+1)r2)R(r)=ER(r).
By defining z=kr, where k is given in (2), (10) is rearranged as spherical Bessel differential equation
(11)d2R(z)dz2+2zdR(z)dz+(1-l(l+1)z2)R(z)=0
with two linearly independent solutions jl(z) and nl(z). Only the well-behaved jl(z) are acceptable in physics. The eigenvalues are determined by the boundary condition jl(kR)=0 (see Table 2) as
(12)Eln=ħ2kln22μ=ħ22μR2αln2,
where kR is the nth root αln of the spherical Bessel function jl(x). The eigenfunctions of the quantum system for the nth root are given by
(13)ψlm(r,θ,ϕ)=Nljl(kr)Ylm(θ,ϕ),Nl=2R3/2|jl+1(αln)|.Table 2
Zeros of the spherical Bessel functionsjl(x).
nth zero
j
0
(
x
)
j
1
(
x
)
j
2
(
x
)
j
3
(
x
)
j
4
(
x
)
j
5
(
x
)
1
3.1416
4.4934
5.7635
6.9879
8.1826
9.3558
2
6.2832
7.7253
9.0950
10.4171
11.7049
12.9665
3
9.4248
10.9041
12.3229
13.6980
15.0397
16.3547
4
12.5664
14.0662
15.5146
16.9236
18.3013
19.6532
5
15.7080
17.2208
18.6890
20.1218
21.5254
22.9046


## 3. Shift Operators

### 3.1. Circular Well

We now address the problem of finding the creation and annihilation operators for the wave functions (5) by the factorization method. As shown in [8], the shift operators can be constructed directly from the wave functions using only one variable, while in the present study they depend on all variables, r and φ.For this purpose, we start by acting with the following operator on the wave functions of (5):
(14)(∂∂x±i∂∂y)ψm(r,φ)=Nme±iφ[∂Jm(kr)∂r∓mrJm(kr)]eimφ=∓NmNm±1Nm±1kJm±1(kr)ei(m±1)φ=∓C±kψm±1(r,φ),C±=NmNm±1,
where we have used the recurrence relations among the Bessel functions:
(15)dJm(z)dz=±Jm∓1(z)∓mzJm(z),
as well as the coordinate transformations x=rcos⁡φ and y=rsinφ.Let us calculate matrix elements for some related functions. It follows from (14) that
(16)∂∂xψm=k2(C-ψm-1-C+ψm+1),2i∂∂yψm=-k(C+ψm+1+C-ψm-1),
from which we have
(17)2〈ψn′|∂∂x|ψm〉=k(C-δn′,m-1-C+δn′,m+1),2i〈ψn′|∂∂y|ψm〉=-k(C+δn′,m+1+C-δn′,m-1).Let us study this problem from another point of view. From the definitions of the momentum operatorP=-iħ∇ and orbital angular momentum operator L=r×P, we can define the following operators:
(18)P±=Px±iPy=-iħe±iφ[∂∂r±(i/r)∂∂φ],Lz=xPy-yPx=-iħ∂∂φ,
from which we have the following commutation relations:
(19)[Lz,Px]=iħPy,[Lz,Py]=-iħPx,[Lz,P±]=±ħP±,[P+,P-]=0,P+P-=Px2+Py2=P-P+.
The Hamiltonian is given by
(20)H=12μ(Px2+Py2)=12μP+P-=12μP-P+,
from which we have
(21)[Lz,H]=[Px,H]=[Py,H]=0.
This means that the linear momentum operators Px,y, Lz and the Hamiltonian H form a complete set of commuting operators with the SO(2) symmetry. The P± play the role of the shift operators as given by P±ψm(r,φ)=∓iħC±kψm±1(r,φ).If we defineεmn=2μEmn/ħ2=(αmn/Rmn)2, then for fixed energy and an initial radius Rm1 (chosen without loss of generality) we have
(22)Rm2=αm2αm1Rm1,Rm3=αm3αm1Rm1,Rm4=αm4αm1Rm1,Rm5=αm5αm1Rm1,….
The different radii Rm1,m2,m3,… correspond to different circular wells for the same energy level. This group of circular disks with different radii Rmn is thus constructed for a given m. For example, when m=0, we have R02=2.29545R01, R03=3.59851R01, R04=4.90331R01, R05=6.20879R01, and so forth. This has been illustrated in Figure 1. For different m, we can construct many different groups of circular rings.Figure 1
Different-radius circular wells constructed by shift operators form=0 case.


### 3.2. Spherical Well

Let us discuss the infinite spherical well case. Likewise, we construct the shift operators by considering all variables,r, θ, and ϕ, in ψlm(r,θ,ϕ). In order to construct the shift operators, we proceed in two steps. First, we obtain the shift operators for the radial part. Second, we find the shift operators for the spherical harmonics Ylm(θ,ϕ). The combination of the two results provides the final solution.Let us perform the first step. Based on the recurrence relations among the spherical Bessel functions [17]
(23)djl(z)dz={jl-1(z)-l+1zjl(z),-jl+1(z)+lzjl(z),
we have
(24)A+=-(ddr-lr),A-=ddr+l+1r,
with the properties
(25)A+ψlm(r,θ,ϕ)=Nlkjl+1(kr)Ylm(θ,ϕ),A-ψlm(r,θ,ϕ)=Nlkjl-1(kr)Ylm(θ,ϕ).Now, let us derive the shift operators for the spherical harmonicsYlm(θ,ϕ). For fixed orbital quantum number l, the ladder operators for m are well-known [14]:
(26)L±=Lx±iLy=±e±iϕ(∂∂θ±icotθ∂∂ϕ).Acting with them onYlm(θ,ϕ) leads to
(27)Yl(m∓1)(θ,ϕ)=e∓iϕ(l±m)(l∓m+1)×(∓∂∂θ+icotθ∂∂ϕ)Ylm(θ,ϕ).Now, our key issue is to find the shift operators forl when m is fixed. According to (7), the θ equation can be modified as
(28)-sin2θd2Θ(θ)dθ2-sinθcos⁡θdΘ(θ)dθ-l(l+1)sin2θΘ(θ)=-m2Θ(θ).Referring to the technique in [4], we obtain the following ladder operators for fixed m:
(29)Θ(l+1)m(θ)=-c(l+1)mclm1(l+1+m)(l+1-m)×[-sinθddθ-(l+1)cos⁡θ]Θlm(θ),Θ(l-1)m(θ)=-c(l-1)mclm1(l+m)(l-m)×[sinθddθ-lcos⁡θ]Θlm(θ),
where clm is the ratio coefficients for Θlm(θ). The introduction of “−” agrees with the standard phase conventions for spherical harmonics.To calculateclm, we can calculate Θ(l+1)(m+1)(θ) by stepping from (l,m) to (l+1,m+1) along two different paths, 1 and 2 (see Figure 2), in the (l,m) parameter space. Path 1 steps first from (l,m) to (l,m+1) via the m step-up operator (27) and then steps from (l,m+1) to (l+1,m+1) with the l step-up operator (29). After some algebraic manipulations, one has
(30)Θ(l+1)(m+1)(θ)=c(l+1)(m+1)cl(m+1)1(l+m+1)(l+m+2)×[cos⁡θddθ-msinθ-(l+1)sinθ]Θlm(θ).
In the calculation, we used the following identity together with the θ equation (28):
(31)d2dθ2+cotθddθ=m2sin2θ-l(l+1).Figure 2
Connecting paths between the states(l,m) and (l±1,m±1).In a similar way, following path 2, that is, we step first from(l,m) to (l+1,m) via the l step-up operator (29) and then step from (l+1,m) to (l+1,m+1) with the m step-up operator (27). As a result, one has
(32)Θ(l+1)(m+1)(θ)=c(l+1)mclm1(l+m+1)(l+m+2)×[cos⁡θddθ-msinθ-(l+1)sinθ]Θlm(θ).
Comparing (30) and (32), we have c(l+1)m/clm=c(l+1)(m+1)/cl(m+1), which implies that the ratio is independent of the quantum number m. This is because m is fixed but l changes in (28). Consequently, we can calculate it by setting m=l. Substituting this into (30) or (32), we have
(33)Θ(l+1)(l+1)(θ)=Nl+1sin(l+1)θ=c(l+1)lcll1(2l+1)(2l+2)×[cos⁡θddθ-lsinθ-(l+1)sinθ]Nlsinlθ=-c(l+1)lcll2l+12l+2Nlsin(l+1)θ.According to the normalization factorNl given in [14]
(34)Nl=(-1)l(2l)!2ll!2l+14π,
we have Nl+1/Nl=-(2l+3)/(2l+2). Therefore, in terms of (33) we obtain c(l+1)l/cll=(2l+3)/(2l+1)=c(l+1)m/clm.The corresponding relations (29) become
(35)Θ(l+1)m(θ)=(2l+3)/(2l+1)(l+1+m)(l+1-m)×[sinθddθ+(l+1)cos⁡θ]Θlm(θ),Θ(l-1)m(θ)=(2l-1)/(2l+1)(l+m)(l-m)×[-sinθddθ+lcos⁡θ]Θlm(θ).Thus, relation (32) can be written as
(36)Θ(l+1)(m+1)(θ)=B+Θlm(θ),
where
(37)B+=(2l+3)/(2l+1)(l+1+m)(l+2+m)×[cos⁡θddθ-msinθ-(l+1)sinθ].Similarly, let us look for the relation betweenΘ(l-1)(m-1)(θ) and Θlm(θ). To this end, we can calculate Θ(l-1)(m-1)(θ) by stepping from (l,m) to (l-1,m-1) along two different paths, 3 and 4, in the (l,m) parameter space (see Figure 1).Following path 3, we step first from(l, m) to (l-1, m) via the l step-down operator (29) and then step from (l-1, m) to (l-1, m-1) with the m step-down operator (27). After some algebraic manipulations, one has
(38)Θ(l-1)(m-1)(θ)=c(l-1)mclm1(l-m)(l+m)(l+m-1)×[sinθ(d2dθ2+cotθddθ)-(l-m)cos⁡θddθ×-mlcos⁡2θsinθ+lsinθ]=-c(l-1)mclm1(l+m)(l+m-1)×[cos⁡θddθ+msinθ+lsinθ]Θlm(θ).Following path 4, we step first from(l, m) to (l, m-1) via the m step-down operator (27) and then step from (l, m-1) to (l-1, m-1) with the l step-down operator (29). Similarly, one has
(39)Θ(l-1)(m-1)(θ)=c(l-1)(m-1)cl(m-1)1(l-m+1)(l+m)(l+m-1)×[sinθd2dθ2-msinθ+mcos⁡θddθ×-lcos⁡θddθ-mlcos⁡2θsinθ]=-c(l-1)(m-1)cl(m-1)1(l+m)(l+m-1)×[cos⁡θddθ+msinθ+lsinθ]Θlm(θ).Comparing (38) and (39), we have c(l-1)(m-1)/cl(m-1)=c(l-1)m/clm. Likewise, setting m=l and substituting it into (38) yield
(40)Θ(l-1)(l-1)(θ)=Nl-1sin(l-1)θ=-c(l-1)lcll1(2l)(2l-1)×[cos⁡θddθ+lsinθ+lsinθ]Nlsinlθ=-c(l-1)lcll2l2l-1Nlsin(l-1)θ.The relationNl-1/Nl=-2l/(2l+1) yields cll/c(l-1)l=(2l+1)/(2l-1)=clm/c(l-1)m. Thus, based on this result we have
(41)Θ(l-1)(m-1)(θ)=B-Θlm(θ),
with
(42)B-=(2l-1)/(2l+1)(l+m)(l+m-1)[-cos⁡θddθ-msinθ-lsinθ].Using the operatorsB± and A±, we can express the shift operators as follows:
(43)B±A±ψlm(r,θ,ϕ)=C±kψ(l±1)(m±1)(r,θ,ϕ),
with C±=Nl/Nl±1. Consequently, the relation between ψlm(r,θ,ϕ) and ψ(l±1)(m±1)(r,θ,ϕ) is established for the same k but different-radius spherical wells.As treated in the circular well, defineεln=2μEln/ħ2=(αln/Rln)2. For fixed energy and an initial radius Rl1, we have
(44)Rl2=αl2αl1Rl1,Rl3=αl3αl1Rl1,Rl4=αl4αl1Rl1,Rl5=αl5αl1Rl1,….
The different radii Rl1,l2,l3,… correspond to different-radius spherical wells with the same energy levels. This group of spherical wells is thus constructed via the potential group approach for fixed l. For example, when l=0, we have R02=2R01, R03=3R01, R04=4R01, R05=5R01, and so forth. This is a special case. The reason is that for l=0, the zeros of the spherical Bessel function j0n(x) are equal to nπ as illustrated in Figure 3. For different l, we can construct many different groups of spherical wells.Figure 3
Different-radius spherical wells constructed by shift operators forl=0 case illustrated in cross section.


## 3.1. Circular Well

We now address the problem of finding the creation and annihilation operators for the wave functions (5) by the factorization method. As shown in [8], the shift operators can be constructed directly from the wave functions using only one variable, while in the present study they depend on all variables, r and φ.For this purpose, we start by acting with the following operator on the wave functions of (5):
(14)(∂∂x±i∂∂y)ψm(r,φ)=Nme±iφ[∂Jm(kr)∂r∓mrJm(kr)]eimφ=∓NmNm±1Nm±1kJm±1(kr)ei(m±1)φ=∓C±kψm±1(r,φ),C±=NmNm±1,
where we have used the recurrence relations among the Bessel functions:
(15)dJm(z)dz=±Jm∓1(z)∓mzJm(z),
as well as the coordinate transformations x=rcos⁡φ and y=rsinφ.Let us calculate matrix elements for some related functions. It follows from (14) that
(16)∂∂xψm=k2(C-ψm-1-C+ψm+1),2i∂∂yψm=-k(C+ψm+1+C-ψm-1),
from which we have
(17)2〈ψn′|∂∂x|ψm〉=k(C-δn′,m-1-C+δn′,m+1),2i〈ψn′|∂∂y|ψm〉=-k(C+δn′,m+1+C-δn′,m-1).Let us study this problem from another point of view. From the definitions of the momentum operatorP=-iħ∇ and orbital angular momentum operator L=r×P, we can define the following operators:
(18)P±=Px±iPy=-iħe±iφ[∂∂r±(i/r)∂∂φ],Lz=xPy-yPx=-iħ∂∂φ,
from which we have the following commutation relations:
(19)[Lz,Px]=iħPy,[Lz,Py]=-iħPx,[Lz,P±]=±ħP±,[P+,P-]=0,P+P-=Px2+Py2=P-P+.
The Hamiltonian is given by
(20)H=12μ(Px2+Py2)=12μP+P-=12μP-P+,
from which we have
(21)[Lz,H]=[Px,H]=[Py,H]=0.
This means that the linear momentum operators Px,y, Lz and the Hamiltonian H form a complete set of commuting operators with the SO(2) symmetry. The P± play the role of the shift operators as given by P±ψm(r,φ)=∓iħC±kψm±1(r,φ).If we defineεmn=2μEmn/ħ2=(αmn/Rmn)2, then for fixed energy and an initial radius Rm1 (chosen without loss of generality) we have
(22)Rm2=αm2αm1Rm1,Rm3=αm3αm1Rm1,Rm4=αm4αm1Rm1,Rm5=αm5αm1Rm1,….
The different radii Rm1,m2,m3,… correspond to different circular wells for the same energy level. This group of circular disks with different radii Rmn is thus constructed for a given m. For example, when m=0, we have R02=2.29545R01, R03=3.59851R01, R04=4.90331R01, R05=6.20879R01, and so forth. This has been illustrated in Figure 1. For different m, we can construct many different groups of circular rings.Figure 1
Different-radius circular wells constructed by shift operators form=0 case.


## 3.2. Spherical Well

Let us discuss the infinite spherical well case. Likewise, we construct the shift operators by considering all variables,r, θ, and ϕ, in ψlm(r,θ,ϕ). In order to construct the shift operators, we proceed in two steps. First, we obtain the shift operators for the radial part. Second, we find the shift operators for the spherical harmonics Ylm(θ,ϕ). The combination of the two results provides the final solution.Let us perform the first step. Based on the recurrence relations among the spherical Bessel functions [17]
(23)djl(z)dz={jl-1(z)-l+1zjl(z),-jl+1(z)+lzjl(z),
we have
(24)A+=-(ddr-lr),A-=ddr+l+1r,
with the properties
(25)A+ψlm(r,θ,ϕ)=Nlkjl+1(kr)Ylm(θ,ϕ),A-ψlm(r,θ,ϕ)=Nlkjl-1(kr)Ylm(θ,ϕ).Now, let us derive the shift operators for the spherical harmonicsYlm(θ,ϕ). For fixed orbital quantum number l, the ladder operators for m are well-known [14]:
(26)L±=Lx±iLy=±e±iϕ(∂∂θ±icotθ∂∂ϕ).Acting with them onYlm(θ,ϕ) leads to
(27)Yl(m∓1)(θ,ϕ)=e∓iϕ(l±m)(l∓m+1)×(∓∂∂θ+icotθ∂∂ϕ)Ylm(θ,ϕ).Now, our key issue is to find the shift operators forl when m is fixed. According to (7), the θ equation can be modified as
(28)-sin2θd2Θ(θ)dθ2-sinθcos⁡θdΘ(θ)dθ-l(l+1)sin2θΘ(θ)=-m2Θ(θ).Referring to the technique in [4], we obtain the following ladder operators for fixed m:
(29)Θ(l+1)m(θ)=-c(l+1)mclm1(l+1+m)(l+1-m)×[-sinθddθ-(l+1)cos⁡θ]Θlm(θ),Θ(l-1)m(θ)=-c(l-1)mclm1(l+m)(l-m)×[sinθddθ-lcos⁡θ]Θlm(θ),
where clm is the ratio coefficients for Θlm(θ). The introduction of “−” agrees with the standard phase conventions for spherical harmonics.To calculateclm, we can calculate Θ(l+1)(m+1)(θ) by stepping from (l,m) to (l+1,m+1) along two different paths, 1 and 2 (see Figure 2), in the (l,m) parameter space. Path 1 steps first from (l,m) to (l,m+1) via the m step-up operator (27) and then steps from (l,m+1) to (l+1,m+1) with the l step-up operator (29). After some algebraic manipulations, one has
(30)Θ(l+1)(m+1)(θ)=c(l+1)(m+1)cl(m+1)1(l+m+1)(l+m+2)×[cos⁡θddθ-msinθ-(l+1)sinθ]Θlm(θ).
In the calculation, we used the following identity together with the θ equation (28):
(31)d2dθ2+cotθddθ=m2sin2θ-l(l+1).Figure 2
Connecting paths between the states(l,m) and (l±1,m±1).In a similar way, following path 2, that is, we step first from(l,m) to (l+1,m) via the l step-up operator (29) and then step from (l+1,m) to (l+1,m+1) with the m step-up operator (27). As a result, one has
(32)Θ(l+1)(m+1)(θ)=c(l+1)mclm1(l+m+1)(l+m+2)×[cos⁡θddθ-msinθ-(l+1)sinθ]Θlm(θ).
Comparing (30) and (32), we have c(l+1)m/clm=c(l+1)(m+1)/cl(m+1), which implies that the ratio is independent of the quantum number m. This is because m is fixed but l changes in (28). Consequently, we can calculate it by setting m=l. Substituting this into (30) or (32), we have
(33)Θ(l+1)(l+1)(θ)=Nl+1sin(l+1)θ=c(l+1)lcll1(2l+1)(2l+2)×[cos⁡θddθ-lsinθ-(l+1)sinθ]Nlsinlθ=-c(l+1)lcll2l+12l+2Nlsin(l+1)θ.According to the normalization factorNl given in [14]
(34)Nl=(-1)l(2l)!2ll!2l+14π,
we have Nl+1/Nl=-(2l+3)/(2l+2). Therefore, in terms of (33) we obtain c(l+1)l/cll=(2l+3)/(2l+1)=c(l+1)m/clm.The corresponding relations (29) become
(35)Θ(l+1)m(θ)=(2l+3)/(2l+1)(l+1+m)(l+1-m)×[sinθddθ+(l+1)cos⁡θ]Θlm(θ),Θ(l-1)m(θ)=(2l-1)/(2l+1)(l+m)(l-m)×[-sinθddθ+lcos⁡θ]Θlm(θ).Thus, relation (32) can be written as
(36)Θ(l+1)(m+1)(θ)=B+Θlm(θ),
where
(37)B+=(2l+3)/(2l+1)(l+1+m)(l+2+m)×[cos⁡θddθ-msinθ-(l+1)sinθ].Similarly, let us look for the relation betweenΘ(l-1)(m-1)(θ) and Θlm(θ). To this end, we can calculate Θ(l-1)(m-1)(θ) by stepping from (l,m) to (l-1,m-1) along two different paths, 3 and 4, in the (l,m) parameter space (see Figure 1).Following path 3, we step first from(l, m) to (l-1, m) via the l step-down operator (29) and then step from (l-1, m) to (l-1, m-1) with the m step-down operator (27). After some algebraic manipulations, one has
(38)Θ(l-1)(m-1)(θ)=c(l-1)mclm1(l-m)(l+m)(l+m-1)×[sinθ(d2dθ2+cotθddθ)-(l-m)cos⁡θddθ×-mlcos⁡2θsinθ+lsinθ]=-c(l-1)mclm1(l+m)(l+m-1)×[cos⁡θddθ+msinθ+lsinθ]Θlm(θ).Following path 4, we step first from(l, m) to (l, m-1) via the m step-down operator (27) and then step from (l, m-1) to (l-1, m-1) with the l step-down operator (29). Similarly, one has
(39)Θ(l-1)(m-1)(θ)=c(l-1)(m-1)cl(m-1)1(l-m+1)(l+m)(l+m-1)×[sinθd2dθ2-msinθ+mcos⁡θddθ×-lcos⁡θddθ-mlcos⁡2θsinθ]=-c(l-1)(m-1)cl(m-1)1(l+m)(l+m-1)×[cos⁡θddθ+msinθ+lsinθ]Θlm(θ).Comparing (38) and (39), we have c(l-1)(m-1)/cl(m-1)=c(l-1)m/clm. Likewise, setting m=l and substituting it into (38) yield
(40)Θ(l-1)(l-1)(θ)=Nl-1sin(l-1)θ=-c(l-1)lcll1(2l)(2l-1)×[cos⁡θddθ+lsinθ+lsinθ]Nlsinlθ=-c(l-1)lcll2l2l-1Nlsin(l-1)θ.The relationNl-1/Nl=-2l/(2l+1) yields cll/c(l-1)l=(2l+1)/(2l-1)=clm/c(l-1)m. Thus, based on this result we have
(41)Θ(l-1)(m-1)(θ)=B-Θlm(θ),
with
(42)B-=(2l-1)/(2l+1)(l+m)(l+m-1)[-cos⁡θddθ-msinθ-lsinθ].Using the operatorsB± and A±, we can express the shift operators as follows:
(43)B±A±ψlm(r,θ,ϕ)=C±kψ(l±1)(m±1)(r,θ,ϕ),
with C±=Nl/Nl±1. Consequently, the relation between ψlm(r,θ,ϕ) and ψ(l±1)(m±1)(r,θ,ϕ) is established for the same k but different-radius spherical wells.As treated in the circular well, defineεln=2μEln/ħ2=(αln/Rln)2. For fixed energy and an initial radius Rl1, we have
(44)Rl2=αl2αl1Rl1,Rl3=αl3αl1Rl1,Rl4=αl4αl1Rl1,Rl5=αl5αl1Rl1,….
The different radii Rl1,l2,l3,… correspond to different-radius spherical wells with the same energy levels. This group of spherical wells is thus constructed via the potential group approach for fixed l. For example, when l=0, we have R02=2R01, R03=3R01, R04=4R01, R05=5R01, and so forth. This is a special case. The reason is that for l=0, the zeros of the spherical Bessel function j0n(x) are equal to nπ as illustrated in Figure 3. For different l, we can construct many different groups of spherical wells.Figure 3
Different-radius spherical wells constructed by shift operators forl=0 case illustrated in cross section.


## 4. Conclusions

We have provided a brief review and a new insight on constructing the shift operators for infinite circular and spherical wells using the potential group approach. The name of this method relates to the fact that the shift operators connect those quantum systems with different potentials but with the same energy spectrum. In particular, we have constructed shift operators that depend on all spatial variables. By considering two special casesm=0 (for circular wells) and l=0 (for spherical wells), we have illustrated the construction of the different-radius wells for the same energy level. For different m and l, we have obtained a series of shift operators with a given quantum number m or l. Before ending this work, we give a useful remark on (22) and (44). That is, these two equations define the corresponding eigenstates and energy eigenvalue for the potentials with the successive radii but do not hold for the entire spectrum.



---

*Source: 987376-2014-05-22.xml*