Novel Superpotentials for Supersymmetric Quantum Mechanics: A New Mathematical Investigation and Study ()
1. Introduction
In this work, referring to the articles about supersymmetric quantum mechanics (SUSY QM) or about potentials having accurate solution, we utilize some of them such as Cooper’s review article published in 1995 [1] that through assuming:
(1)
and by means of superpotential 
and its derivative
, two self-consistent potentials 
and 
are obtained as follows:
(2)
If for all the 
and
, there are x-independent 
s satisfying the following shape-invariance equation:
(3)
The energy levels are obtained through the following important equation:
(4)
On the other hand, assuming that 
is the wave function of n-th level and
, the Schrö- dinger equation for this level is as follows:
(5)
If the ground energy level for the potential 
is formulated by the known function 
on zero (
), by considering Equation (5) we acquire:
(6)
However, by comparing Equation (6) with Equation (2), the superpotential W is:
(7)
Conversely, if the superpotential 
is a characteristic function, 
can be:
(8)
where 
is the normalizing factor.
Afterwards, through the operator method and by virtue of the raising operator below:
(9)
other energy levels’ characteristic functions are as follows:
(10)
For such potentials, since the Schrödinger equation’s 
s and 
s can be accurately obtained by Equations (4), (8), and (10), they are termed potentials with accurate solution.
Through this method, a few numbers of potentials have been hitherto discovered. Twelve cases of these potentials are well known for which 
s and 
s are accurately computed. After the publication of Cooper’s review article in 1995 [1], although the scientists have sought to propose newer potentials using the abovementioned method for many years, any new or important potential has not been heretofore introduced.
2. The Classification of Well-Known Superpotentials
With regard to the supersymmetric quantum mechanics, twelve superpotentials [1-47] have been hitherto proposed whose Schrödinger equation and its resulting potentials possess accurate solution.
In terms of parameters’ type, the well-known superpotentials are divided into three main categories.
2.1. A Superpotential with Two Constant Parameters A and B
(11)
is the generator of transferred oscillating potential with the energies
.
2.2. Superpotentials with a Variable Parameter A and a Constant Parameter B
(12)
is the generator of three-dimensional oscillating potential with the energies
.
(13)
is the generator of Coulomb potential with the energies
.
(14)
is the generator of Morse potential with the energies
.
(15)
is the generator of Rosen-Morse I potential with the energies
.
(16)
is the generator of Rosen-Morse II potential with the energies
.
(17)
is the generator of Eckart potential with the energies
.
(18)
is the generator of Scarf I potential with the energies
.
(19)
is the generator of Scarf II potential with the energies
.
(20)
is the generator of generalized Pöschl-Teller potential with the energies
.
2.3. Superpotentials with Two Variable Parameters A and B
(21)
is the generator of Pöschl-Teller I potential with the energies
.
(22)
is the generator of Pöschl-Teller II potential with the energies
.
3. Creating the Intended Superpotentials
Considering the form of functions W in Equations (14) to
(22), and through utilizing two constant parameters β and γ, and four general parameters C, D, F, and G, the following general superpotential is considered:
(23)
By applying the shape-invariance condition of Equation (3) to the above superpotential, a number of constraint equations of parameters C, D, F, G, 
, 
, 
, and 
are found whose general solutions are not simply possible; however, if the special cases are considered, three cases of them lead to the following three categories.
3.1. A Superpotential with a Variable Parameter
For the special case
, the constraints resulting from Equation (3) for the first step (
) are as follows:
(24)
The designation below is appropriate for the above equations:
(25)
It means that our intended superpotential for the n-th level in the state 
should be written as the following form:
(26)
and the necessary equations for the shape invariance between n-th level and n + 1-th level are as follows:
(27)
(28)
Then, through exploiting Equations (4) and (28), the energy levels are:
(29)
By means of these two transforms 
and
, the four equations above are transformed into the forms below:
(30)
(31)
(32)
(33)
In special cases of Equations (26), (29), (30), and (33), a number of previously known superpotentials and their respective energies can be achieved of which the main ones are:
1) Selecting 
from Equation (26) results in 
which is the Morse superpotential (Equation (14)) with the energies 
by assuming
,
, and
.
2) Selecting 
from Equation (26) results in 
which is the generalized Pöschl-Teller superpotential (Equation (20)) with the energies 
by assuming
, 
, and
.
3) Selecting 
from Equation (26) results in 
which is the Scarf II superpotential (Equation (19)) with the energies 
by assuming
, 
, and
.
4) Selecting 
from Equation (30) results in 
which is the Scarf I superpotential (Equation (18)) with the energies of Equation (33) 
by assuming
, 
, and
.
5) Selecting
, 
, 
, and 
, Equation (30) is transformed into 
which is the Pöschl-Teller I superpotential; considering Equation (31), 
, and
, there should be 
and
; the energies of Pöschl-Teller I (
) are derived from Equation (33).
6) Selecting
, 
, 
, and 
, Equation (26) is transformed into 
which is the Pöschl-Teller II superpotential; considering Equation (27), 
, and
, there should be 
and
; the energies of Pöschl-Teller II (
) are derived from Equation (29).
3.2. A Superpotential with Two Variable Parameters
For the special case
, through assuming 
, Equation (23) is transformed into the following form:
(34)
for which, through applying the shape-invariance condition of Equation (3) for n = 0, it becomes necessary that:
(35)
(36)
After removing 
from the functions of Equation (35), it can be written as follows:
(37)
in which the following solution satisfies:
(38)
By inputting Equation (38) to
:
(39)
Also, by applying the shape-invariance condition of Equation (3) for the general level n, we obtain:
(40)
(41)
Then, through Equations (4), (40), and (41), the energy levels are achieved as follows:
(42)
Also, through transforming 
and
, Equations (34), (40), and (42) are transformed into the following forms:
(43)
(44)
(45)
For the special case
, in which the superpotential of Equation (34) is transformed into 
which is the Morse superpotential 
by assuming
, 
, and
; the transforms 
and
, and the energies of Morse superpotential of Equation (14), 
, are obtained through Equations (40) and (42). Furthermore, with
, for the special cases
, the superpotentials of Scarf II, Rosen-Morse I and II, and Pöschl-Teller I and II result from the superpotentials of Equations (34) and (43). This indicates the results’ correctness.
3.3. A Superpotential with Three Variable Parameters
For the special case
, in which Equation (23) is transformed into the form below:
(46)
through applying the shape-invariance condition of Equation (3) for the first step (
), the following constraints should be necessarily established:
(47)
(48)
(49)
Now, by achieving 
through Equations (47) and (48), and equalizing them, we obtain the result below:
(50)
in which the following equations fulfill:
(51)
(52)
Through inputting Equations (51) and (52) to Equation (47), 
is acquired as follows:
(53)
In the second step, by applying the shape-invariance condition of Equation (3), the following functions are obtained:
(54)
Subsequently, in the n-th step, we will have:
(55)
(56)
(57)
(58)
Through utilizing Equation (4), its energies are as follows:
(59)
By transforming
,
,
, and
, Equations (46) and (59) are converted into the forms below:
(60)
(61)
But, Equations (55)-(57) are not changed.
Here again, to insure the above equations’ accuracy, some special cases of Equations (46) and (60) are obtained in the following steps:
1) For the special case
, Equation (46) is converted to
, and Equations (55)-(57) are respectively transformed into Fn = F, 
, and
; by assuming
, 
, and
, the equations of Morse superpotential (
,
, and
) are acquired; the Morse energies, 
, are derived from Equation (59).
2) For the special case
, in which 
and
, it can be assumed that
. Then by assuming
, Equation (46) is transformed into 
where
and
; by assuming 
and
, the Eckart superpotential 
is achieved. The Eckart energies,
are obtained through Equation (59).
3) For the special case
, by assuming
, 
, and
, Equation (60) is transformed into the Rosen-Morse I superpotential (
) and the energies of Equation (61) are converted to those of Rosen-Morse I
(
).
4) For the special case
, by assuming
,
, and
, Equation (46) is transformed into the Rosen-Morse II superpotential (
) and the energies of Equation (59) are converted to those of Rosen-Morse II
(
).
4. The Application in Fundamental Particles
In the articles on the quarkonium spectroscopy using the Klein-Gordon equation [48-52] and the coherent state of no-spin relativistic particle [53-57], some applications of Schrödinger equation’s energies are employed to obtain relativistic particles’ energy spectra, the importance of equally-spaced Klein-Gordon relativistic equation’s energy levels is investigated, and the characteristic values of Klein-Gordon equation 
according to the characteristic values of Schrödinger equation
are achieved as follows:
(62)
where m is the particle’s rest mass.
On the other hand, if the energies of Schrödinger equation for a potential like 
are 
where δ is an n-independent constant, the energies of Schrödinger equation for the potential 
is
; based on Equation (62), the energies of Klein-Gordon equation for the potential 
should be obtained through the following equation:
(63)
Therefore, if 
is obtained as a first-order function of n, the Klein-Gordon energy levels will have equal distances. In this section, we utilize some of energies (
) resulting from the superpotentials achieved in the Sections 3.1, 3.2, and 3.3 as follows:
1) Regarding Equations (30) and (33), if through using:
(64)
we form the potential below:
(65)
its 
will be as the following form:
(66)
Subsequently, by exploiting Equations (63) and (66), the Klein-Gordon equation’s energies include:
(67)
The equally-spaced energy levels are
.
2) Considering Equation (45), if the potential is obtained using Equation (43) and following equation:
(68)
we will have:
(69)
Consequently, with regard to Equations (63) and (69), its Klein-Gordon equation’s energies must be achieved using the following equation:
(70)
The special case 
can be utilized to form equally-spaced levels, in which case we have:
(71)
(72)
By considering Equation (68), for the potential 
, the Klein-Gordon equation’s energies are acquired using Equation (71) having equallyspaced energy levels (
).
3) Regarding Equation (61), if the potential is obtained using Equation (60) and the equation below:
(73)
we will have:
(74)
Therefore, through Equation (63), the
’s include:
(75)
which is transformed into the following form using Equations (56) and (57):
(76)
If the special case 
is used, Equation (76) is converted to:
(77)
As a result, when the potential of Equation (73) is obtained by the following superpotential:
(78)
the Klein-Gordon equation’s energies with this potential are derived from Equation (77) having equaled-spaced energy levels (
).
It is evident that there is not a fundamental difference between Equations (72) and (78), but Equation (64) is more general than Equations (72) and (78).
5. Conclusions
Briefly, it can be said that we discovered three new superpotentials through these calculations. The first set includes the superpotential (26), transforms (27), and energies (29) or superpotential (30), transforms (31), and energies (33) having one variable parameter, and being transformed into the forms of Morse, generalized PöschlTeller, Pöschl-Teller I and II, and Scarf I and II superpotentials for specific values of parameters.
The second set includes the superpotential (34), transforms (40), and energies (42) or superpotential (43), transforms (44), and energies (45) having two variable parameters, and being transformed into the forms of Morse, Scarf II, Rosen-Morse I and II, and Pöschl-Teller I and II superpotentials for specific values of parameters.
The third set includes the superpotentials (46) and (60), transforms (55), (56), and (57), and energies (59) and (61) having three variable parameters, and being transformed into the forms of Morse, Eckart , and Rosen-Morse I and II superpotentials for specific values of parameters.
6. Acknowledgements
The work described in this paper was fully supported by grants from the Institute for Advanced Studies of Iran. The authors would like to express genuinely and sincerely thanks and appreciated and their gratitude to Institute for Advanced Studies of Iran.
NOTES