# Exact treatment of states

###### Abstract

Using the basic ingredient of supersymmetry, a general procedure for the treatment of quantum states having nonzero angular momenta is presented.

Over the years the Schrödinger equation has been studied extensively regarding its exact solvability. Many advances have been made in this area by classifying quantum mechanical potentials according to their symmetry properties. For instance, various algebra which reveal the underlying symmetry as well as facilitating obtaining the solutions have been found. In this respect, the application of supersymmetry ideas to nonrelativistic quantum mechanics has revived fresh interest in the problem of obtaining algebraic solutions of exactly solvable nonrelativistic potentials and provided a deeper understanding of analytically solvable Hamiltonians, as well as a set of powerful approximate schemes for dealing with problems admitting no exact solutions. The concept of shape invariance [1] has played a fundamental role in these developments.

In this letter, a novel method within the frame of supersymmetric quantum mechanics [1] is introduced, using the spirit of perturbation theory, for the treatment states assuming that the potential of interest is exactly solvable for state.

For the consideration of spherically symmetric potentials, the corresponding Schrödinger equation for the radial wave function reads

(1) |

Now, assuming that the term of angular momentum barrier can be treated like a perturbing potential, which will be discussed below, one writes the wave function as

(2) |

in which is the known normalized eigenfunction of the unperturbed Schrödinger equation corresponding to state whereas is a moderating function due to the angular momentum barrier. Substituting (2) into (1) yields

(3) |

With the new definitions,

(4) |

one arrives at

(5) |

where is the eigenvalue for the exactly solvable potential for case, and

(6) |

in which is the correction term to the energy due to the barrier term, and . Subsequently, Eq. (3) reduces to

(7) |

from which it is clear that the whole superpotential, , should address exactly solvable potentials. In other words, Eq. (7) is a novel sophisticated and extended treatment of Eq. (5). That is why the present model works well for all analytically solvable potentials. For a recent application of the technique used here regarding perturbed Coulomb interactions the reader is referred to [2].

In principle as one knows explicitly the solution of (5), namely the whole spectrum and corresponding eigenfunctions of the potential , the goal here is to solve only Eq. (6), which is the main result of this letter, leading to the full corrections to the energy and wave functions for all quantum states with .

To test the effectiveness of the model we consider through this short letter three well known problems of quantum mechanics as illustrative examples. We first apply the model to the three dimensional harmonic oscillator problem

(8) |

Starting with its exact solutions [1] for ,

(9) |

and introducing a superpotential for the treatment of quantum states, via Eq. (6), with nonzero angular momenta

(10) |

one easily obtains the full corrections to the wave function and energy in (9) due to the effect of angular momentum barrier,

(11) |

From which, the exact solution of the potential in (8) reads

(12) |

These are indeed the exact results [1] which can be checked out also by the use of Eq. (7).

We proceed with another example,

(13) |

having in mind, as in the previous example, that analytical solutions of (13) for is known [1]

(14) |

Setting the superpotential

(15) |

and substituting it, together with in (14), into (6) yields

(16) |

from which, the correct energy is

(17) |

and from (15), the moderating function

(18) |

which leads to the full wavefunction

(19) |

Eqs. (17) and (19) justify once more the reliability of the present model since they are exact [1].

Finally, we consider the corrected form of a super family potential [3] which is known in the literature as the approximate Hulthen effective potential introduced by Greene and Aldrich [4] in their method to generate pseudo-Hulthen wave function for states,

(20) |

that (considering ) reduces to

(21) |

In (20), denotes the partner number with leading to the supersymmetric partner potentials. It is noted that for the potentials in (20) and (21) lead to the well known Hulthen potential.

Though we know the potential in (20) is exactly solvable [3], to illustrate again the elegancy of the present treatment, we assume for a while that the Schrödinger equation with this potential has an analytic solution for only , and try to calculate corrections to the solution within the framework Eq. (6) due to the other states with in the same system.

The superpotential, corresponding ground state wavefunction and energy expressions for the Hulthen potential are

(22) |

For the present example, the unique superpotential leading to the exact corrections should read

(23) |

which can be tested through Eq. (6), from which

(24) |

Thus, one readily sees that

(25) |

and

(26) |

which are exact [3] for the potential in (20).

Although we have considered here only the ground solutions for the sake of clarity, the application of our simple approach to excited states does not cause any problem. As the entire spectrum wave functions for the unperturbed part of the potential of we interest are known explicitly, what one needs is just to set the corresponding superpotential via to use in (6), together with a properly chosen superpotential leading to the barrier term with . In this respect, Eq. (4) in our work put forward a new perspective when compared to the usual treatment of supersymmetric quantum mechanics in which generally the superpotentail is related to the ground state wavefunction. This fresh idea in (4) would also be helpful in treating excited states within the frame of supersymmetric perturbation theory [5] where one needs to deal with nasty integrals and tedious calculation procedures. Furthermore, Eq. (4) allows us having a unique but closed analytical expression for the energy corrections involving all states. Examples of such treatmens, together with a new look at perturbation problems, are discussed in detail by the present author, which will appear elsewhere.

We believe that the present technique would find a wide application in the related area. In particular, such treatments would shed a light in the search of analytical solutions of Morse, Rosen-Morse, Eckart, Pöschl-Teller and Scarf potentials in case . Along this line the works are in progress.

Finally, from the whole discussion presented in this letter it is obvious that and in case , which means that here may be treated like a perturbation parameter. A brief discussion behind this observation is given below, which leads to more understanding the frame of Eq. (6).

Here, as a further knowledge to the reader, a general procedure is outlined for the solution of Riccati equation in (6),

from which the exact solution for reads

where is a special solution of (A.1). More specifically, Eq. (A.2) can be rewritten as

or

which is obvious that due to the present consideration of exactly solvable potentials, the full superpotential in the above equations should involve the whole of correction terms in a perturbation series discussed in detail below.

As is known explicitly, we briefly discuss here the physics behind leading to to find an exact analytical solution for . Using the spirit of recently introcuded supersymmetric perturbation theory [5], we expand the related functions in terms of that is treated here like a perturbation parameter,

Substitution of the above expansion into (A.1) by equating terms with the same power of on both sides yields up to , one arrives at

From this short discussion, one sees that or corresponding to the first and second order, are the candidates for a special solution () of (A.1) or (A.3). For simplicity, we choose and, from (A.6), give its explicit form as

in which is evaluated by

For instance, if one performs the calculations for the first example discussed in this letter, that is the harmonic oscillator potential with the angular momentum barrier worked out for , then it is not difficult to see that and higher order corrections are zero due to

because and , which leads to that satisfies the first order expansion in (A.6).

The author wish to thank the referee for his helpful comments and suggestions.

## References

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- [4] R. L. Greene and C. Aldrich, Phys. Rev. A 14 (1976) 2363.
- [5] C. Lee, Phys. Lett. A 267 (2000) 101.