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A New Kind of Shift Operators for Infinite Circular and Spherical Wells
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 radii sharing energy levels with a common eigenvalue. In circular well, the momentum operators play the role of shift operators. The and operators, the third projection of the orbital angular momentum operator , and the Hamiltonian form a complete set of commuting operators with the SO(2) symmetry. In spherical well, the shift operators establish a novel relation between and .
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 . Contrary to traditional approaches , 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 .
Motivated by the recently proposed factorization method , 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  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 Section 2 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 as for and for , where denotes and for circular and spherical wells, respectively. The Schrödinger equation in relevant polar coordinates can be written as
By taking () and substituting it into (1), we have
By defining , it becomes the Bessel differential equation It has two linearly independent solutions for each value of . One is the regular and the other is the singular . Only the well-behaved are physically acceptable, with the eigenvalues determined by . That is, is the th root of the Bessel function (see Table 1). In terms of this condition, the eigenvalues are then given by The corresponding eigenfunctions for the th root are given by
Obviously, the complete spectrum for the full circular well corresponds to one set of solutions but is doubly degenerate for each value of , since give the same contributions. This corresponds physically to the equivalence of clockwise versus counterclockwise motions.
2.2. Infinite Spherical Well
When , the Schrödinger equation in spherical coordinates is given by
For symmetrically central force fields, we can always express the wave functions as and substitute this into (6). The polar angle dependence leads to the associated Legendre equation which is satisfied by the normalized associated Legendre functions : Thus, we may define the spherical harmonics as where the factor is the so-called Condon-Shortley phase.
By separating the spherical harmonics, we obtain the radial equation as By defining , where is given in (2), (10) is rearranged as spherical Bessel differential equation with two linearly independent solutions and . Only the well-behaved are acceptable in physics. The eigenvalues are determined by the boundary condition (see Table 2) as where is the th root of the spherical Bessel function . The eigenfunctions of the quantum system for the th root are given by
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 , 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, and .
For this purpose, we start by acting with the following operator on the wave functions of (5): where we have used the recurrence relations among the Bessel functions: as well as the coordinate transformations and .
Let us calculate matrix elements for some related functions. It follows from (14) that from which we have
Let us study this problem from another point of view. From the definitions of the momentum operator and orbital angular momentum operator , we can define the following operators: from which we have the following commutation relations: The Hamiltonian is given by from which we have This means that the linear momentum operators , and the Hamiltonian form a complete set of commuting operators with the SO symmetry. The play the role of the shift operators as given by .
If we define , then for fixed energy and an initial radius (chosen without loss of generality) we have The different radii correspond to different circular wells for the same energy level. This group of circular disks with different radii is thus constructed for a given . For example, when , we have , , , , and so forth. This has been illustrated in Figure 1. For different , we can construct many different groups of circular rings.
3.2. Spherical Well
Let us discuss the infinite spherical well case. Likewise, we construct the shift operators by considering all variables, , , and , in . 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 . 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  we have with the properties
Now, let us derive the shift operators for the spherical harmonics . For fixed orbital quantum number , the ladder operators for are well-known :
Acting with them on leads to
Now, our key issue is to find the shift operators for when is fixed. According to (7), the equation can be modified as
Referring to the technique in , we obtain the following ladder operators for fixed : where is the ratio coefficients for . The introduction of “−” agrees with the standard phase conventions for spherical harmonics.
To calculate , we can calculate by stepping from to along two different paths, and (see Figure 2), in the parameter space. Path 1 steps first from to via the step-up operator (27) and then steps from to with the step-up operator (29). After some algebraic manipulations, one has In the calculation, we used the following identity together with the equation (28):
In a similar way, following path 2, that is, we step first from to via the step-up operator (29) and then step from to with the step-up operator (27). As a result, one has Comparing (30) and (32), we have , which implies that the ratio is independent of the quantum number . This is because is fixed but changes in (28). Consequently, we can calculate it by setting . Substituting this into (30) or (32), we have
The corresponding relations (29) become
Thus, relation (32) can be written as where
Similarly, let us look for the relation between and . To this end, we can calculate by stepping from to along two different paths, and , in the parameter space (see Figure 1).
The relation yields . Thus, based on this result we have with
Using the operators and , we can express the shift operators as follows: with . Consequently, the relation between and is established for the same but different-radius spherical wells.
As treated in the circular well, define . For fixed energy and an initial radius , we have The different radii 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 . For example, when , we have , , , , and so forth. This is a special case. The reason is that for , the zeros of the spherical Bessel function are equal to as illustrated in Figure 3. For different , we can construct many different groups of spherical wells.
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 cases (for circular wells) and (for spherical wells), we have illustrated the construction of the different-radius wells for the same energy level. For different and , we have obtained a series of shift operators with a given quantum number or . 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees for positive and invaluable suggestions which have improved the paper greatly. This work is partially supported by SIP-20140772-IPN, Mexico, as well as in part by US National Science Foundation (Grant no. OCI-0904874), and US Department of Energy (Grant no. DE-SC0005248). This work is dedicated to the 75th anniversary of Professor Eugenio Ley-koo.
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