Symmetry 2012, 4(3), 452-473; doi:10.3390/sym4030452

Review
Supersymmetric Quantum Mechanics and Solvable Models
Jonathan Bougie 1,*, Asim Gangopadhyaya 1, Jeffry Mallow 1 and Constantin Rasinariu 2
1
Department of Physics, Loyola University Chicago, 1032 W. Sheridan Rd., Chicago, IL 60660, USA; Email: agangop@gmail.com (A.G.); jvmallow@gmail.com (J.M.)
2
Department of Science and Mathematics, Columbia College Chicago, 600 S. Michigan Ave., Chicago, IL 60605, USA; Email: crasinariu@colum.edu
*
Author to whom correspondence should be addressed; Email: jbougie@luc.edu; Tel.: +1-773-508-3543; Fax: +1-773-508-3534.
Received: 29 June 2012; in revised form: 20 July 2012 / Accepted: 31 July 2012 /
Published: 14 August 2012

Abstract

: We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of Symmetry 04 00452 i001-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on Symmetry 04 00452 i001.
Keywords:
supersymmetry; quantum mechanics; shape invariance

Classification: PACS 03.65.-w, 47.10.-g, 11.30.Pb

1. Introduction

Supersymmetric quantum mechanics (SUSYQM) is a generalization of the factorization method commonly used for the harmonic oscillator. The factorization technique begun by Darboux [1] about one hundred years ago, and used by Schrödinger [2,3,4] in the 1940’s and Infeld and Hull in the 1950’s [5], could be considered a precursor of SUSYQM.

The current form of SUSYQM appeared in 1981 [6] as a model of dynamical symmetry breaking. It was developed further by the authors of [8,7] among others. This simplified model turned out to have important applications in quantum mechanics.

In the next section, we will describe the general formalism of SUSYQM and in Section 3 we introduce the shape invariance condition that makes a potential solvable. Section 4 is dedicated to a description of potential algebra and its connection to shape invariance. In Section 5 we will describe a method for determining solutions of the translational shape invariance condition. We will then conclude with the analysis of recently discovered shape invariant potentials that are inherently functions of Symmetry 04 00452 i002.

2. Supersymmetric Quantum Mechanics

Throughout this paper, we use units such that Symmetry 04 00452 i003. For the harmonic oscillator, the Hamiltonian [9] Symmetry 04 00452 i004 is factorized into Symmetry 04 00452 i005 and Symmetry 04 00452 i006. Similarly, in the SUSYQM formalism [10,11] a general Hamiltonian Symmetry 04 00452 i007 is written as a product of Symmetry 04 00452 i008 and Symmetry 04 00452 i009, where the function Symmetry 04 00452 i010 is known as the superpotential. The product Symmetry 04 00452 i011 is then given by

Symmetry 04 00452 i012

Thus, the product Symmetry 04 00452 i011 indeed reproduces the Hamiltonian Symmetry 04 00452 i013 above, provided the potential Symmetry 04 00452 i014 is related to the superpotential Symmetry 04 00452 i010 by Symmetry 04 00452 i015. The product Symmetry 04 00452 i016 produces another Hamiltonian Symmetry 04 00452 i017 with Symmetry 04 00452 i018. To see the underlying supersymmetry of this formalism, let us construct a generator Symmetry 04 00452 i019 and its adjont Symmetry 04 00452 i020 by:

Symmetry 04 00452 i021

Operators Symmetry 04 00452 i019 and Symmetry 04 00452 i020 generate the following supersymmetry algebra:

Symmetry 04 00452 i022

The groundstate energy of this Hamiltonian is then given by

Symmetry 04 00452 i023

Thus, the non-vanishing of the groundstate energy implies that either Symmetry 04 00452 i024 or Symmetry 04 00452 i025, and hence signals the spontaneous breaking of the supersymmetry. We therefore require that Symmetry 04 00452 i026 to preserve unbroken supersymmetry.

The Hamiltonians Symmetry 04 00452 i027 and Symmetry 04 00452 i013 are intertwined; i.e., Symmetry 04 00452 i028 and Symmetry 04 00452 i029. This leads to the following relationships among their eigenvalues and eigenfunctions [12]

Symmetry 04 00452 i030

Since H± are products of the operator Symmetry 04 00452 i031 and its adjoint Symmetry 04 00452 i032, their eigenvalues are either zero or positive [13]. The ground state eigenvalue of one of these Hamiltonians must be zero in order to have unbroken supersymmetry. Without loss of generality, we choose that Hamiltonian to be Symmetry 04 00452 i013. Thus, we have

Symmetry 04 00452 i033

where Symmetry 04 00452 i034 is an arbitrary point in the domain and Symmetry 04 00452 i035 is the normalization constant that depends on the choice of Symmetry 04 00452 i034. Thus, if Symmetry 04 00452 i036 is a normalizable groundstate, we have a system with unbroken supersymmetry. Its groundstate Symmetry 04 00452 i037 is zero and the operator Symmetry 04 00452 i031 annihilates the corresponding eigenstate Symmetry 04 00452 i036. For all higher states of Symmetry 04 00452 i013, as indicated in Equation (5), there is an one-to-one correspondence with the states of Symmetry 04 00452 i027.

2.1. Example

Consider a system described by the superpotential Symmetry 04 00452 i038, defined over the domain Symmetry 04 00452 i039. Corresponding partner potentials are given by Symmetry 04 00452 i040. This is a rather complicated potential, and rarely analyzed in quantum mechanics courses. However, for Symmetry 04 00452 i041, the potential Symmetry 04 00452 i042 reduces to the infinite square well, with the bottom of the potential at Symmetry 04 00452 i043 and zero groundstate energy. We know that the corresponding eigenstates are given by Symmetry 04 00452 i044 and eigenvalues Symmetry 04 00452 i045. Thus, using the familiarity with the relatively simpler potential Symmetry 04 00452 i046, we are able to derive all of the eigenvalues and eigenfunctions of Symmetry 04 00452 i047. Then the inter-relations expressed through Equation (5) enable us to determine eigenvalues and eigenfunctions of Symmetry 04 00452 i048.

Although we assume that our Hamiltonians are hermitian, hermiticity is not necessary to generate real eigenvalues. Replacing the sufficient but not necessary condition of hermiticity with the weaker condition of PT symmetry, has led to the discovery of new potentials with real energy eigenvalues [14,15,16]. Considerable work has been done on the study of SI potentials with PT symmetry. In reference [17], the author examined the shape invariant hyperbolic Rosen-Morse potential as a case of exact solvability with PT invariance. In reference [18], the Scarf II potential was shown as an example of spontaneous PT-symmetry-breaking. The case of a square well with discrete PT symmetry but with intervals of non-hermiticity was shown in reference [19] to produce real eigenvalues. Authors of [20] constructed the spectrum of a square well of imaginary strength, to obtain the hierarchy of SI potentials, while authors of [21] constructed a set of solvable rational potentials, and related the PT symmetry condition to the condition that they be free of singularities. However, in this paper we limit ourselves to the case of hermitian Hamiltonians, i.e., real Symmetry 04 00452 i010.

The remainder of this manuscript is devoted to the study of the shape invariance condition and its solutions.

3. Shape Invariance in Supersymmetric Quantum Mechanics

If the superpotential Symmetry 04 00452 i049 of a system obeys the condition

Symmetry 04 00452 i050

where Symmetry 04 00452 i051, the system is called shape invariant [5,22,23,24]. Various forms of the function Symmetry 04 00452 i052 define classes of shape invariance. The most commonly discussed classes are:

Symmetry 04 00452 i053

From Equation (7) it follows that for a shape invariant system, the partner potentials Symmetry 04 00452 i054 and Symmetry 04 00452 i055 differ only by values of parameter Symmetry 04 00452 i056 and additive constants Symmetry 04 00452 i057. In particular,

Symmetry 04 00452 i058

In terms of operators A±, the shape invariance condition becomes

Symmetry 04 00452 i059

As we will see in Section 4, Equation (9) implies that for every shape invariant system, there is always an underlying potential algebra [28,30,29,32,31] that guarantees its solvability. In the rest of this section, we will show how shape invariance enables us to find the eigenvalues and eigenfunctions of the system.

3.1. Determination of Eigenvalues

From Equation (9), we see that Hamiltonians Symmetry 04 00452 i060 and Symmetry 04 00452 i061 differ only by the constant Symmetry 04 00452 i062. We already know that Symmetry 04 00452 i063. Let us determine the first excited state of Symmetry 04 00452 i013; i.e., Symmetry 04 00452 i064. Hence, using Symmetry 04 00452 i065[33], we find that the energy Symmetry 04 00452 i064 of the first excited state of Symmetry 04 00452 i066 and Symmetry 04 00452 i067 of the groundstate of Symmetry 04 00452 i060 both are given by Symmetry 04 00452 i062. Similarly, to determine Symmetry 04 00452 i068, we use the isospectrality condition (5) to relate it to Symmetry 04 00452 i069. But by the shape invariance condition (9), Symmetry 04 00452 i069= Symmetry 04 00452 i070. Following the method used for determining Symmetry 04 00452 i064, we find Symmetry 04 00452 i071, and hence, Symmetry 04 00452 i072. Extending this argument to higher excited states of Symmetry 04 00452 i066, we get

Symmetry 04 00452 i073

3.2 Determination of Eigenfunctions

Again from Equation (9), we see that since Symmetry 04 00452 i060 and Symmetry 04 00452 i061 only differ by the constant Symmetry 04 00452 i062, they must have common eigenfunctions. Hence, from Equation (6), we have Symmetry 04 00452 i074. Then the isospectrality condition (5), requires Symmetry 04 00452 i075. Iterating this procedure, we obtain

Symmetry 04 00452 i076

Thus, for a system with a given shape invariant superpotential, the eigenvalues and eigenfunctions can be determined analytically. This result makes it very important to find all such potentials. In the past, researchers had found a list of additive shape invariant potentials [10,11], mostly by trial and error [34,35]. In Section 5, we will discuss how to find solutions of Equation (9) for the additive case. Before that, however, in the next section we will show why the shape invariance condition leads to solvability.

4. Shape Invariance and Potential Algebra

We will now show that the symmetry behind the shape invariance is essential in building the algebraic structures known as potential algebras. As we will show below, a potential algebra is in general a deformation of a three-dimensional Lie algebra, whose representations yield the spectrum of the corresponding shape invariant system.

4.1. Building the Algebra

The starting point of our construct is the shape invariance condition given by Equation (9), which we rewrite as

Symmetry 04 00452 i077

where Symmetry 04 00452 i078, and Symmetry 04 00452 i052 is a function that models the change of parameter Symmetry 04 00452 i079. For example, if the change of parameter is a translation, Symmetry 04 00452 i080.

The left hand side of Equation (12) resembles a commutation relation. This suggests that we use Symmetry 04 00452 i031 and Symmetry 04 00452 i032 to build the generators of the potential algebra. To transform the above shape invariance condition into an exact commutator, we replace operators Symmetry 04 00452 i081 and Symmetry 04 00452 i082 by

Symmetry 04 00452 i083

where Symmetry 04 00452 i084 is a constant parameter, Symmetry 04 00452 i085 is an auxiliary variable, and Symmetry 04 00452 i086. The function Symmetry 04 00452 i087 will be appropriately chosen to emulate the relationship between parameters Symmetry 04 00452 i088 and Symmetry 04 00452 i089. Thus, to generate Symmetry 04 00452 i090, we multiplied the operator Symmetry 04 00452 i081 from right by Symmetry 04 00452 i091 and replaced the parameter Symmetry 04 00452 i088 by the differential operator Symmetry 04 00452 i092. If we now compute the commutator between operators Symmetry 04 00452 i093 and Symmetry 04 00452 i090, we find [36]

Symmetry 04 00452 i094

Observe now that the right hand side of Equation (14) matches the left hand side of Equation (12) provided that we make the following mappings

Symmetry 04 00452 i095

Since we know that Symmetry 04 00452 i079, these mappings require that the function Symmetry 04 00452 i096 satisfy

Symmetry 04 00452 i097

Let us look at some examples to illustrate this procedure.

• Translation: Symmetry 04 00452 i080

If the change of parameters is a translation, then the function Symmetry 04 00452 i087 that models it is the identity function

Symmetry 04 00452 i098

We have Symmetry 04 00452 i099, which gives the desired change of parameters.

• Scaling: Symmetry 04 00452 i100

For shape invariant potentials characterized by a scaling change of parameters, the corresponding function Symmetry 04 00452 i087 is the exponential

Symmetry 04 00452 i101

Indeed Symmetry 04 00452 i102 where we denoted Symmetry 04 00452 i103.

• Cyclic: Symmetry 04 00452 i104

Cyclic potentials form a series of shape invariant potentials that repeats after a cycle of Symmetry 04 00452 i105 iterations. These potentials appear also in connection with the dressing chain formalism [37]. To satisfy the cyclic parameter change, the function Symmetry 04 00452 i052 should obey Symmetry 04 00452 i106. The projective map Symmetry 04 00452 i107 with specific constraints [38] on the parameters Symmetry 04 00452 i108, and Symmetry 04 00452 i109 satisfies such a condition [27]. The function Symmetry 04 00452 i087 satisfying Equation (16) in this case is given by

Symmetry 04 00452 i110

where Symmetry 04 00452 i111 are solutions of the equation Symmetry 04 00452 i112 and Symmetry 04 00452 i113 is an arbitrary periodic function of Symmetry 04 00452 i114 with period Symmetry 04 00452 i105.

• Other choices of parameters follow from more complicated choices for Symmetry 04 00452 i115. For example taking

Symmetry 04 00452 i116

we obtain the change of parameters: Symmetry 04 00452 i117.

Now, let us get back to the building of the potential algebra. In terms of operators Symmetry 04 00452 i093 and Symmetry 04 00452 i090, the shape invariance condition (12) becomes:

Symmetry 04 00452 i118

Thus, the commutation relation of operators Symmetry 04 00452 i119 and Symmetry 04 00452 i120 generates an operator Symmetry 04 00452 i121 that has no Symmetry 04 00452 i122-dependence. If we now define a third operator Symmetry 04 00452 i123 in terms of the operator Symmetry 04 00452 i124, the shape invariance condition becomes simply one of the commutation relations of the potential algebra. In particular, if we define

Symmetry 04 00452 i125

where Symmetry 04 00452 i105 is an arbitrary constant, the three commutators among these generators are given by [39]

Symmetry 04 00452 i126

Putting together the above results, we arrive at the following

Lemma: To any shape invariant system characterized by

Symmetry 04 00452 i127

for which we can find a function Symmetry 04 00452 i087such that Symmetry 04 00452 i128for arbitrary parameters Symmetry 04 00452 i114and Symmetry 04 00452 i084, we can associate an algebra [40]generated by

Symmetry 04 00452 i129

Symmetry 04 00452 i130

Symmetry 04 00452 i131

satisfying the commutation relations

Symmetry 04 00452 i132

where

Symmetry 04 00452 i133

The function Symmetry 04 00452 i134in Equation (29) is given by the shape invariance condition (24), while the function Symmetry 04 00452 i087satisfies the compatibility equation: Symmetry 04 00452 i135, where Symmetry 04 00452 i052models the change of parameter Symmetry 04 00452 i079 of Equation (24).

As an example let us build the potential algebra corresponding to the Pöschl-Teller II potential. The potential

Symmetry 04 00452 i136

is generated by the superpotential Symmetry 04 00452 i137, through Symmetry 04 00452 i138, where Symmetry 04 00452 i139. Its supersymmetric partner Symmetry 04 00452 i140 is given by

Symmetry 04 00452 i141

The shape invariance is now evident if we observe that Symmetry 04 00452 i142. Therefore, in terms of Symmetry 04 00452 i032 and Symmetry 04 00452 i031 operators, the shape invariance (24) for the Pöschl-Teller II potential reads

Symmetry 04 00452 i143

Now we can identify the main objects of our model and build the corresponding algebra:

1. The parameters of the model are Symmetry 04 00452 i144 and Symmetry 04 00452 i145, where Symmetry 04 00452 i146 is an arbitrary positive constant greater than Symmetry 04 00452 i147 so that Symmetry 04 00452 i088 is a positive quantity;

2. The change of parameter Symmetry 04 00452 i079 is thus given by Symmetry 04 00452 i148. This is a translational change of parameter Symmetry 04 00452 i149 with the translation parameter Symmetry 04 00452 i150. Translation implies that the function Symmetry 04 00452 i087 satisfying (16) is the identity function Symmetry 04 00452 i098;

3. From the concrete shape invariance condition (32) of this potential we get Symmetry 04 00452 i151. Then, the function Symmetry 04 00452 i152 is given by Symmetry 04 00452 i153 if we choose the arbitrary constant Symmetry 04 00452 i154;

4. Defining Symmetry 04 00452 i155 and Symmetry 04 00452 i123 as prescribed by Equations (25) and (27), we obtain the differential realization of the algebra’s generators:

Symmetry 04 00452 i156

satisfying the commutation relations Symmetry 04 00452 i157, and Symmetry 04 00452 i158. Thus, shape invariance of this system implies that the system has a potential algebra given by Symmetry 04 00452 i159[?, 41, 42].

4.2. Obtaining the Energy Spectrum from Algebra Representations

Once we know the potential algebra for a given potential, we can use its representations to obtain the energy spectrum for the Hamiltonian. Using Equations (25) and (26), we observe that

Symmetry 04 00452 i160

From the reciprocal of the mapping Equation (13), we obtain Symmetry 04 00452 i161. Consequently, the spectrum of the operator Symmetry 04 00452 i162 gives the spectrum of the Hamiltonian. To find the concrete values for the energy, we need to know the action of individual operators Symmetry 04 00452 i093 and Symmetry 04 00452 i090 respectively on a set of eigenvectors of the operator Symmetry 04 00452 i123. In this general case, Symmetry 04 00452 i093, Symmetry 04 00452 i090 and Symmetry 04 00452 i123 commute with the Casimir operator given by

Symmetry 04 00452 i163

with the function Symmetry 04 00452 i164(defined up to an additive constant) such that

Symmetry 04 00452 i165

It can be explicitly checked that Symmetry 04 00452 i166 does indeed commute with Symmetry 04 00452 i093, Symmetry 04 00452 i090 and Symmetry 04 00452 i123[41]. In a basis in which Symmetry 04 00452 i123 and Symmetry 04 00452 i166 are diagonal, Symmetry 04 00452 i155 play the role of raising and lowering operators, respectively. Operating on an arbitrary eigenstate Symmetry 04 00452 i167 we have

Symmetry 04 00452 i168

where we have used Symmetry 04 00452 i169.

Keeping in mind that Symmetry 04 00452 i170, and observing that Symmetry 04 00452 i171, we see that in order to find the energies of the system one must find the coefficients Symmetry 04 00452 i172. If we apply the third commutation relation of Equation (23) to Symmetry 04 00452 i167, we obtain using Equation (36)

Symmetry 04 00452 i173

Next, we will determine the allowed values of Symmetry 04 00452 i174 and the corresponding values Symmetry 04 00452 i172. Let us say Symmetry 04 00452 i175 corresponds to the lowest state in a given representation of the algebra. This implies that Symmetry 04 00452 i176, which means Symmetry 04 00452 i177. From Equation (38) we get

Symmetry 04 00452 i178

Iterating this procedure we can generate a general formula for Symmetry 04 00452 i172

Symmetry 04 00452 i179

Substituting Symmetry 04 00452 i180 leads to

Symmetry 04 00452 i181

The profile of Symmetry 04 00452 i182 determines the dimension of the representation. For example, let us consider the two cases presented in Figure 1.

Symmetry 04 00452 g001 200
Figure 1. Generic behaviors of Symmetry 04 00452 i182. Case (a) corresponds to a finite, and (b) to an infinite representation of the potential algebra.

Click here to enlarge figure

Figure 1. Generic behaviors of Symmetry 04 00452 i182. Case (a) corresponds to a finite, and (b) to an infinite representation of the potential algebra.
Symmetry 04 00452 g001 1024

One obtains the finite dimensional representations of Figure 1a, by starting from a point on the Symmetry 04 00452 i182vs. Symmetry 04 00452 i174 graph corresponding to Symmetry 04 00452 i175, and moving in integer steps parallel to the Symmetry 04 00452 i174-axis to the point corresponding to Symmetry 04 00452 i185. At the end points, Symmetry 04 00452 i186, and we get a finite representation. If Symmetry 04 00452 i182 is decreasing monotonically, as in Figure 1b, there exists only one end point at Symmetry 04 00452 i175. Starting from Symmetry 04 00452 i187 the value of Symmetry 04 00452 i174 can be increased in integer steps up to infinity. In this case we have an infinite dimensional representation. As in the finite case, Symmetry 04 00452 i187 labels the representation. The difference is that here Symmetry 04 00452 i187 takes continuous values. Similar arguments apply for a monotonically increasing function Symmetry 04 00452 i182. Having the representation of the algebra associated with a characteristic model, we obtain, using Equation (41), the complete spectrum of the system.

For example, let us consider the scaling change of parameters Symmetry 04 00452 i188. Consider the simple choice Symmetry 04 00452 i189, where Symmetry 04 00452 i190 is a constant. This choice generates self-similar potentials studied in references [44,26,25]. In this case, combining Equation (18) with Equation (28) yields:

Symmetry 04 00452 i191

which is a deformation of the standard Symmetry 04 00452 i192 Lie algebra.

For this case, from Equations (42) and (36) one gets

Symmetry 04 00452 i193

Note that for scaling problems [25], one requires Symmetry 04 00452 i194, which leads to Symmetry 04 00452 i195. From the monotonically decreasing profile of the function Symmetry 04 00452 i182, it follows that the unitary representations of this algebra are infinite dimensional. If we label the lowest weight state of the operator Symmetry 04 00452 i123 by Symmetry 04 00452 i187, then Symmetry 04 00452 i177. Without loss of generality we can choose the coefficients Symmetry 04 00452 i172 to be real. Then one obtains from (38) for an arbitrary Symmetry 04 00452 i196

Symmetry 04 00452 i197

The spectrum of the Hamiltonian Symmetry 04 00452 i061 is given by

Symmetry 04 00452 i198

Therefore, the eigenenergies are

Symmetry 04 00452 i199

in agreement with the known results [25].

5. How Do We Find Additive Shape Invariant Superpotentials?

Since we have demonstrated the value of shape-invariant superpotentials, the question now becomes how to find such superpotentials. This question is equivalent to asking how to solve the difference-differential Equation (9) to find the list of desired superpotentials Symmetry 04 00452 i200. For this section, we will restrict ourselves to considering cases of translational shape invariance. Before we embark on solving this equation, let us first note that quantum mechanical potentials generally have terms of two very different orders: One “large" and another “small". For example, the classical and quantum potentials for the radial oscillator system are Symmetry 04 00452 i201 and Symmetry 04 00452 i202 respectively. To make the transition from the quantum to the classical system, one takes the limit Symmetry 04 00452 i203 with the constraint that Symmetry 04 00452 i204. Thus, the quantum Hamiltonian can be written as Symmetry 04 00452 i205. This shows that in quantum mechanics, the potential generally has one small term that depends on Symmetry 04 00452 i002[43]. In SUSYQM, since the potential is given by Symmetry 04 00452 i206, the derivative term always brings in a factor of Symmetry 04 00452 i002, even if the superpotential is independent of Symmetry 04 00452 i002. In the following analysis, as we determine how to solve Equation (9) to find shape invariant superpotentials, we will consider the cases of Symmetry 04 00452 i002-independent and Symmetry 04 00452 i002-dependent superpotentials separately.

5.1. Known Symmetry 04 00452 i001-Independent Shape Invariant Superpotentials

We begin our discussion of known shape invariant systems by considering only superpotentials that do not depend explicitly on Symmetry 04 00452 i002, which we call “conventional” superpotentials. In Table 1 we list the known “conventional" superpotentials that meet this criterion.

Previous work [45,46,47] has proven that this list of conventional shape-invariant superpotentials is complete. We now show a new proof of this completeness which has the advantage of being significantly more straightforward and elegant than it predecessors.

Table 1. The complete family of Symmetry 04 00452 i001-independent additive shape-invariant superpotentials.

Click here to display table

Table 1. The complete family of Symmetry 04 00452 i001-independent additive shape-invariant superpotentials.
Name Superpotential
Harmonic Oscillator Symmetry 04 00452 i208
Coulomb Symmetry 04 00452 i209
-D oscillator Symmetry 04 00452 i210
Morse Symmetry 04 00452 i211
Rosen-Morse I Symmetry 04 00452 i212
Rosen-Morse II Symmetry 04 00452 i213
Eckart Symmetry 04 00452 i214
Scarf I Symmetry 04 00452 i215
Scarf II Symmetry 04 00452 i216
Gen. Pöschl-Teller Symmetry 04 00452 i217

5.2. New Proof of Completeness of the Conventional Shape-Invariant Superpotentials

Because of additive shape invariance, the dependence of Symmetry 04 00452 i200 on Symmetry 04 00452 i218 and Symmetry 04 00452 i002 is through the linear combination Symmetry 04 00452 i219; therefore, the derivatives of Symmetry 04 00452 i200 with respect to Symmetry 04 00452 i218 and Symmetry 04 00452 i002 are related by: Symmetry 04 00452 i220. Since Equation (7) must hold for an arbitrary value of Symmetry 04 00452 i002, if we assume that Symmetry 04 00452 i200 does not depend explicitly on Symmetry 04 00452 i002, we can expand in powers of Symmetry 04 00452 i002, and the coefficient of each power must separately vanish. Expanding the right hand side in powers of Symmetry 04 00452 i002 yields

Symmetry 04 00452 i221

Symmetry 04 00452 i222

Symmetry 04 00452 i223

Thus, all conventional additive shape invariant superpotentials are solutions of the above set of non-linear partial differential equations [46,47]. Although this represents an infinite set, note that if equations of Symmetry 04 00452 i224 and Symmetry 04 00452 i225 are satisfied, all others automatically follow. Therefore, we proceed to find all possible solutions to the two partial differential equations:

Symmetry 04 00452 i226

and

Symmetry 04 00452 i227

In doing so, we derive a new proof that the superpotentials shown in Table 1 are the only possible solutions.

The general solution to Equation (51) is

Symmetry 04 00452 i228

Therefore, to generate all shape invariant superpotentials, we need to determine all possible combinations of Symmetry 04 00452 i229, Symmetry 04 00452 i230, and Symmetry 04 00452 i231 that satisfiy Equation (50). We will ignore the case when both Symmetry 04 00452 i230, and Symmetry 04 00452 i231 are constants, as this corresponds to a flat potential with no Symmetry 04 00452 i122-dependence. We will also ignore the case in which Symmetry 04 00452 i230, and Symmetry 04 00452 i231 are linearly dependent on each other; i.e., Symmetry 04 00452 i232. In this case, Symmetry 04 00452 i233. If we redefine Symmetry 04 00452 i234, this case becomes equivalent to a superpotential with a shifted parameter and constant Symmetry 04 00452 i235 which will be considered shortly. We can therefore assume that Symmetry 04 00452 i236 and Symmetry 04 00452 i235 are linearly independent of each other without loss of generality. Note that from here onward, we will use lower case Greek letters to denote constants that are independent of both Symmetry 04 00452 i218 and Symmetry 04 00452 i122.

To determine Symmetry 04 00452 i010, we first focus on determining Symmetry 04 00452 i229. To do so, we take two derivatives of (50) with respect to Symmetry 04 00452 i218. This leads to the following differential equation:

Symmetry 04 00452 i237

where dots and primes represent derivatives taken with respect to Symmetry 04 00452 i218 and Symmetry 04 00452 i122 respectively. Since Symmetry 04 00452 i227, this simplifies to:

Symmetry 04 00452 i238

Inserting the form of the general solution (52) into (53) yields

Symmetry 04 00452 i239

where Symmetry 04 00452 i240 is a function of Symmetry 04 00452 i218, and is independent of Symmetry 04 00452 i122. Since Symmetry 04 00452 i236 and Symmetry 04 00452 i235 are linearly independent, we find that there are only three possible ways for the left-hand-side of Equation (54) to be independent of Symmetry 04 00452 i122:

• Case 1: Symmetry 04 00452 i236 is a constant and Symmetry 04 00452 i241;

• Case 2: Symmetry 04 00452 i235 is a constant and Symmetry 04 00452 i242;

• Case 3: Neither Symmetry 04 00452 i236 nor Symmetry 04 00452 i235 are not constants, but Symmetry 04 00452 i242 and Symmetry 04 00452 i241.

For each of these cases we can determine the form of Symmetry 04 00452 i229. Then we can determine Symmetry 04 00452 i230 and Symmetry 04 00452 i231 for these three cases. This we do by taking two derivatives of (50), this time one with respect to Symmetry 04 00452 i218 and another with respect to Symmetry 04 00452 i122. This yields:

Symmetry 04 00452 i243

Inserting the form of the general solution (52) into (55) yields

Symmetry 04 00452 i244

Now, we will analyze each of the three cases in detail.

5.2.1. Case 1: X1 Is a Constant and Symmetry 04 00452 i245

Let Symmetry 04 00452 i246. Since Symmetry 04 00452 i235 cannot be a constant as well, Equation (54) requires Symmetry 04 00452 i241. This leads to Symmetry 04 00452 i247 for some arbitrary constants Symmetry 04 00452 i146, Symmetry 04 00452 i248, Symmetry 04 00452 i249, and Symmetry 04 00452 i250. Inserting Symmetry 04 00452 i236 and Symmetry 04 00452 i251 into Equation (52) yields Symmetry 04 00452 i252 where Symmetry 04 00452 i253.

We now find Symmetry 04 00452 i235 by inserting the above Symmetry 04 00452 i200 into Equation (50). This yields

Symmetry 04 00452 i254

or equivalently,

Symmetry 04 00452 i255

where Symmetry 04 00452 i256.

Since Symmetry 04 00452 i231 is independent of Symmetry 04 00452 i218, and the left side of (57) is a sum of four linearly independent functions of Symmetry 04 00452 i218 Symmetry 04 00452 i257, and the term Symmetry 04 00452 i258 on the right-hand-side is independent of Symmetry 04 00452 i122, the coefficient of each power of Symmetry 04 00452 i218 must separately be independent of Symmetry 04 00452 i122. The linear term in Symmetry 04 00452 i218 therefore requires that Symmetry 04 00452 i259 be independent of Symmetry 04 00452 i122. Since a constant Symmetry 04 00452 i235 leads to a trivial solution, we must have Symmetry 04 00452 i260 The remaining Symmetry 04 00452 i122-dependent terms on the left side of (57), Symmetry 04 00452 i261 must be a constant:

Symmetry 04 00452 i262

The solution depends on the value of the constants Symmetry 04 00452 i052 and Symmetry 04 00452 i263.

• Case 1A: Symmetry 04 00452 i264 Symmetry 04 00452 i265 In this case, Symmetry 04 00452 i266, which is a trivial solution;

• Case 1B: Symmetry 04 00452 i264 Symmetry 04 00452 i267 In this case, Symmetry 04 00452 i268 so Symmetry 04 00452 i269 Defining Symmetry 04 00452 i270 yields the harmonic oscillator superpotential;

• Case 1C: Symmetry 04 00452 i271 The solution is then Symmetry 04 00452 i272, Symmetry 04 00452 i273 Symmetry 04 00452 i246. Therefore, Symmetry 04 00452 i274 For Symmetry 04 00452 i275, this yields Symmetry 04 00452 i276, where Symmetry 04 00452 i277 and Symmetry 04 00452 i278. This is the Morse superpotential. Note that Symmetry 04 00452 i279 decreases as Symmetry 04 00452 i218 increases, and hence signals a finite number of eigenstates [48].

5.2.2. Case 2: Symmetry 04 00452 i280 Is Constant

In this case, let Symmetry 04 00452 i281; then Equation (54) requires Symmetry 04 00452 i242. This yields Symmetry 04 00452 i282. We now insert this form of Symmetry 04 00452 i251 and Symmetry 04 00452 i281 into (56) to get an ordinary differential equation in Symmetry 04 00452 i122 for Symmetry 04 00452 i236:

Symmetry 04 00452 i283

or equivalently,

Symmetry 04 00452 i284

Integrating it once, we get

Symmetry 04 00452 i285

This equation can be simplified by setting Symmetry 04 00452 i286. This leads to

Symmetry 04 00452 i287

The solutions for Symmetry 04 00452 i288 depend on the constant Symmetry 04 00452 i289.

• Case 2A: Symmetry 04 00452 i290 In this case, Symmetry 04 00452 i291 The whole superpotential is then given by

Symmetry 04 00452 i292

Setting Symmetry 04 00452 i293 and Symmetry 04 00452 i294, and identifying Symmetry 04 00452 i295, Symmetry 04 00452 i296 and Symmetry 04 00452 i297, we get Symmetry 04 00452 i298: the superpotential for Coulomb [49];

• Case 2B: Symmetry 04 00452 i299 In this case, we have either Symmetry 04 00452 i300(Eckart) or Symmetry 04 00452 i301(Rosen-Morse II). In the first case, the superpotential is given by Symmetry 04 00452 i302, where we have set Symmetry 04 00452 i303 and Symmetry 04 00452 i304. This is the well known Eckart potential. Similarly, the other solution with Symmetry 04 00452 i305 generates Rosen-Morse II;

• Case 2C: Symmetry 04 00452 i306 In this case, we obtain Symmetry 04 00452 i307. An analysis similar to the previous case generates the superpotential for Rosen-Morse I.

5.2.3. Case 3: Symmetry 04 00452 i308 and Symmetry 04 00452 i280 Are not Constant, but Symmetry 04 00452 i309 and Symmetry 04 00452 i245

In this case, since Symmetry 04 00452 i241, and Symmetry 04 00452 i242, we have Symmetry 04 00452 i310. Therefore Symmetry 04 00452 i311. In this case, Equation (56) yields

Symmetry 04 00452 i312

Integrating,

Symmetry 04 00452 i285b

Thus, again we have

Symmetry 04 00452 i287b

Note that this is the same differential equation as (60) and will therefore give the same solutions for Symmetry 04 00452 i236 as Case 2. However, in this case, Symmetry 04 00452 i313(this is equivalent to choosing Symmetry 04 00452 i314 in Case 2) and Symmetry 04 00452 i235 is not constant. Instead, in each case we must plug our solutions for Symmetry 04 00452 i229 and Symmetry 04 00452 i230 into Equation (50), which yields

Symmetry 04 00452 i315

This equation is again simplified by setting Symmetry 04 00452 i286, which yields

Symmetry 04 00452 i316

Since Symmetry 04 00452 i288 and Symmetry 04 00452 i235 are independent of Symmetry 04 00452 i218, the terms linear in Symmetry 04 00452 i218 and the terms independent of Symmetry 04 00452 i218 on the left side of this equation must each separately be independent of Symmetry 04 00452 i122. Therefore,

Symmetry 04 00452 i317

Symmetry 04 00452 i318

For different values of Symmetry 04 00452 i289, we get different superpotentials:

• Case 3A: Symmetry 04 00452 i319. We again get Symmetry 04 00452 i320, where with an appropriate choice for the origin we have set Symmetry 04 00452 i294. Equation (65) for Symmetry 04 00452 i235 becomes

Symmetry 04 00452 i321

Its solution is Symmetry 04 00452 i322. With the identification Symmetry 04 00452 i323, Symmetry 04 00452 i324, Symmetry 04 00452 i325, we get Symmetry 04 00452 i326, the superpotential for the 3D-harmonic oscillator;

• Case 3B: Symmetry 04 00452 i299 As seen before, Symmetry 04 00452 i327 implies that Symmetry 04 00452 i300 or Symmetry 04 00452 i301. By translation and scaling of Symmetry 04 00452 i122, we can simplify the first solution to Symmetry 04 00452 i328. Substituting Symmetry 04 00452 i288 in Equation (65), we get

Symmetry 04 00452 i329

where we have set Symmetry 04 00452 i330 The solution to the homogeneous equation is Symmetry 04 00452 i331, and the particular solution is Symmetry 04 00452 i332. Hence Symmetry 04 00452 i333. Thus, the superpotential is given by Symmetry 04 00452 i334, the General Pöschl-Teller potential given in Table 1. The second solution generates the Scarf II potential;

• Case 3C: Symmetry 04 00452 i306 A similar analysis for this case leads to Scarf I as the corresponding shape invariant superpotential.

Thus, we have generated all the superpotentials of Table 1 and shown that these are the only possible Symmetry 04 00452 i002-independent solutions to the additive shape invariant condition.

5.3. Symmetry 04 00452 i001-Dependent Superpotentials

In the previous section we generated the complete list of additive shape-invariant superpotentials that do not depend explicitly on Symmetry 04 00452 i002. However, a new class of superpotentials was discovered by Quesne [50,51]. It has been shown [46] that these “extended" superpotentials obey the shape invariance condition in the form of Equation (7) only when Symmetry 04 00452 i200 is allowed to depend explicitly on Symmetry 04 00452 i002. While this dependence is frequently ignored by the conventional notation that sets Symmetry 04 00452 i335, we will show that this constraint results in important consequences for the energy spectrum of the resulting potentials. In each case, the new potential is isospectral with a potential that arises from one of the “conventional" superpotentials listed in Table 1. Authors of [52,53,54] have added an infinite number of potentials that belong in this class, and extended shape invariant potentials continue to be objects of intense research [55,56].

We now extend our formalism to include “extended" superpotentials that contain Symmetry 04 00452 i002 explicitly. To do so, we expand the superpotentials in powers of Symmetry 04 00452 i002. Hence, we define

Symmetry 04 00452 i336

We will now substitute Equation (68) into the shape invariance condition given in Equation (7), for which we compute Symmetry 04 00452 i337 and Symmetry 04 00452 i338. We obtain

Symmetry 04 00452 i339

and

Symmetry 04 00452 i340

Since Symmetry 04 00452 i148, Symmetry 04 00452 i341 Expanding in powers of Symmetry 04 00452 i342

Symmetry 04 00452 i343

Similarly,

Symmetry 04 00452 i344

We substitute these into Equation (7) and stipulate that the equation hold for any value of Symmetry 04 00452 i002. After some significant algebraic manupulation we find that the following equation must be true separately for each positive integer value of Symmetry 04 00452 i345:

Symmetry 04 00452 i346

For Symmetry 04 00452 i347, we obtain

Symmetry 04 00452 i348

This equation is identical to Equation (50) for Symmetry 04 00452 i002-independent Symmetry 04 00452 i200’s. We have already found a set of solutions for Equation (70) that includes all known Symmetry 04 00452 i002-independent superpotentials. The extended cases [50,51] are solutions to (69) as well, as shown in [46]. Note that Equation (69) provides a consistency condition for all Symmetry 04 00452 i002-dependent potentials; however, these are not easy to solve to determine new potentials.

Additionally, Equation (70) provides a constraint for the possible energy spectra of the extended potentials. from Equation (70), the function Symmetry 04 00452 i349 is given by

Symmetry 04 00452 i350

Symmetry 04 00452 i351

Thus, the function Symmetry 04 00452 i349, and hence the energy of the system, is given entirely in terms of the Symmetry 04 00452 i002-independent part of the superpotential. Hence, the eigenvalues are not affected by the Symmetry 04 00452 i002-dependent extension of the superpotential.

Thus far, each of the known extended potentials contains a solution from Table 1 as the Symmetry 04 00452 i002-independent term of the superpotential Symmetry 04 00452 i352. Therefore, each of the expanded potentials is isospectral with its corresponding conventional potential. Future possibilities for finding new shape-invariant superpotentials fall into one of two categories:

1. Further extended superpotentials may be found based on the conventional superpotentials. In this case, the potentials derived from the extended superpotential will be isospectral with the potentials derived from the corresponding conventional superpotential;

2. While Symmetry 04 00452 i352 is required to satisfy Equation (70), which is equivalent to Equation (50) for Symmetry 04 00452 i002-independent Symmetry 04 00452 i200’s, Symmetry 04 00452 i352 is not required to satisfy Equation (48). Rather, the additional constraints for an extended Symmetry 04 00452 i200 are supplied by Equation (69). It therefore may be possible to find an Symmetry 04 00452 i002-dependent superpotential whose Symmetry 04 00452 i002-independent term Symmetry 04 00452 i352 is not equivalent to a conventional superpotential. Intriguingly, it therefore may still be possible to discover shape-invariant systems with new energy spectra.

6. Summary and Conclusions

While supersymmetric quantum mechanics began as a simplified model to account for dynamical symmetry breaking, the application of this formalism to quantum mechanics has become an important field in its own right. In this manuscript we have reviewed research on supersymmetric quantum mechanics with a particular emphasis on the property of shape invariance. As we have shown, shape invariance is a sufficient condition for exact solvability of quantum mechanical problems; i.e., given a superpotential with shape invariance, all its eigenvalues and eigenfunctions can be determined analytically.

However, in its traditional form, the shape invariance condition Equation (7) is a difference-differential equation and is difficult to solve. It has recently been established that for additive shape-invariant superpotentials that do not explicitly depend on Symmetry 04 00452 i002, this condition can be written as a set of local partial differential equations [45,46,47]. The solution to these equations showed that the list of such superpotentials was indeed complete. In this manuscript, we have presented a more straightforward proof of this result.

Since 2008, new sets of additive shape invariant potentials have been discovered [50,51,52,53,54]. We have reviewed the development of these “extended" shape-invariant systems and have provided an infinite set of partial differential equations that all extended potentials (where superpotentials are inherently functions of Symmetry 04 00452 i002) must obey [46,47]. We have also discussed the constraints placed on the energy spectra of these extended potentials as well as possibilities for finding additional as-yet-undiscovered cases of additive shape invariance.

It may also be possible to extend this method to other forms of shape invariance such as multiplicative or cyclic. For these, the potentials are generally not available in terms of known functions, except in very special cases ( Symmetry 04 00452 i353 for cyclic and limiting cases for multiplicative). It remains to be shown whether the shape invariance condition for these classes can be transformed from a difference-differential equation into a set of partial differential equations and be subjected to similar analysis.

Acknowledgements

This research was supported by an award from Research Corporation for Science Advancement

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