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
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 [
Motivated by the recently proposed factorization method [
This work is organized as follows. In Section
The circular and spherical wells are defined as
By taking
By defining
Zeros of the Bessel functions
|
|
|
|
|
|
|
---|---|---|---|---|---|---|
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 | 22.2178 |
Obviously, the complete spectrum for the full circular well corresponds to one set of
When
For symmetrically central force fields, we can always express the wave functions as
By separating the spherical harmonics, we obtain the radial equation as
Zeros of the spherical Bessel functions
|
|
|
|
|
|
|
---|---|---|---|---|---|---|
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 |
We now address the problem of finding the creation and annihilation operators for the wave functions (
For this purpose, we start by acting with the following operator on the wave functions of (
Let us calculate matrix elements for some related functions. It follows from (
Let us study this problem from another point of view. From the definitions of the momentum operator
If we define
Different-radius circular wells constructed by shift operators for
Let us discuss the infinite spherical well case. Likewise, we construct the shift operators by considering all variables,
Let us perform the first step. Based on the recurrence relations among the spherical Bessel functions [
Now, let us derive the shift operators for the spherical harmonics
Acting with them on
Now, our key issue is to find the shift operators for
Referring to the technique in [
To calculate
Connecting paths between the states
In a similar way, following path 2, that is, we step first from
According to the normalization factor
The corresponding relations (
Thus, relation (
Similarly, let us look for the relation between
Following path 3, we step first from
Following path 4, we step first from
Comparing (
The relation
Using the operators
As treated in the circular well, define
Different-radius spherical wells constructed by shift operators for
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
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.