Normalized eigenfunctions of parametrically factored Schrödinger equations

According to Infeld and Hull [1], the factorization method for Sturm-Liouville differential equations owes its existence primarily to Schrödinger, who at the beginning of 1940’s discussed the method [2] and also factored in four different ways the hypergeometric differential equation [3]. However, they also mention Weyl [4] and Dirac [5] as precursors, who, a decade earlier, considered the spherical harmonics with spin and the creation and annihilation operators, respectively, in a similar framework. Dirac himself gave credit to Fock for the latter factoring operators. After the comprehensive paper of Infeld and Hull from 1951, there were only very few works, e.g., Crum [6], related to the factorization methods that have been published along the next three decades. However, during the 1980’s, with the advent of supersymmetric quantum mechanics, the factorizations in their supersymmetric version with the usage of particular Riccati solutions have become the preferred tool for generating isospectral potentials with different degrees of isospectrality. On the other hand, although the factorizations based on general Riccati solutions for the fundamental cases of quantum harmonic oscillator and the Hydrogen atom have been also addressed in the same period in the pioneering papers of Mielnik [7] and Fernández C. [8], respectively, they really did not enter the mainstream research of supersymmetric quantum mechanics, although one can see chapter 6 in [9] and references therein and in the review paper [10]. In fact, more attention attracted their connection with the version of the Gel’fand-Levitan inverse method developed by Abraham and Moses [11], as mentioned in [7, 8] and also noticed by Nieto [12]. In a parallel line of research, after Matveev [13, 14] revived the Darboux transformations [15], Andrianov et al. [16, 17] showed their equivalence with the supersymmetric factorization method and therefore one may also consider the parametric factorizations as a particular kind of parametric Darboux transformations.

The main goal of this short paper, which is dedicated to Professor D.J. Fernández on the occasion of his forty years of remarkable scientific activity, is to revisit the pioneering papers of 1984, pinpointing some details, such as the issue of normalization constants of the parametrically deformed wavefunctions, which has not been mentioned in the original papers.

The rest of the paper is structured as follows. In Section 2, the parametric factorization of a Schrödinger like equation is presented in general terms. Section 3 is devoted to the application to the quantum harmonic oscillator following [7], with a subsection on the normalization constant of the deformed oscillator eigenfunctions. The application to the radial equation of the Hydrogen atom following [8], with a similar subsection on the normalization constants, is presented in Section 4. The Conclusions section ends up the paper.

Let us consider the one-dimensional Schrödinger equation

where the Schrödinger potential $f(x)$, is an arbitrary $C^{2}$ function, and also primes will be used in the following for the derivatives of some functions. We set $E=0$ for the ground state energy, thus turning the equation into a homogeneous differential equation for the ground state wavefunction $F_{0}$

Equation (2) can be factored in the form

where $\Phi=-F_{0}^{\prime}/F_{0}$ is the negative logarithmic derivative of a solution $F_{0}$ of (2), a.k.a. the seed solution of the Darboux transformation (DT). Comparing (2) and (3), one obtains the Riccati equation,

If one knows a Riccati solution $\Phi$ of (4), the solution of (2) can be obtained from $F_{0}={\rm exp}(-\int^{x}\Phi)$.

The non-parametric Darboux-transformed equation of (3), also known as the supersymmetric partner equation of (3), is obtained by reverting the factoring

The generic interesting fact of the reverted factorizations is that they are not unique, but parametric. This is because in the factorization brackets one can use the general Riccati solution in the form of the Bernoulli ansatz $\Phi_{g}=\Phi+1/u$, where the function $u$ satisfies the first-order differential equation

and not just a particular solution $\Phi$.

Then, it is easy to show that

Furthermore, the left hand side of the latter equation can be written as

Therefore, $\bar{F}_{0}=\hat{F}_{0}$, and it does not matter if one uses the particular or general Riccati solution.

The relevant result is obtained only when one reverts back the factorization brackets in the intent to return to the initial equation. Then, one obtains

Substituting the Bernoulli form of $\Phi_{g}$ in (9) leads to

The latter equation is actually a one-parameter family of equations having the same Darboux-transformed partner, the running parameter of the family being the integration constant that occurs in Bernoulli’s function $1/u$ obtained by the integration of (6)

It is easy to show that equation (10) differs from the initial equation (1) by

which is the additive Darboux deformation of $f(x)$ introduced by the parametric DT. Moreover, the solutions $\widetilde{{\cal F}}_{0}$ of the parametric Darboux-transformed equations are related to the undeformed solutions $F_{0}$ as follows

Before ending this section, we mention that including a scalar scaling $c$ in front of $D^{2}$ can be manipulated by democratically distributing $\sqrt{c}$ in front of each of the factoring brackets $(\pm D+\Phi)$. The only change is the scaling by $c$ of the eigenspectrum of the given Schrödinger problem. Therefore, in the following we will use the usual quantum-mechanical scaling $1/\sqrt{2}$ in front of the factoring operators.

For this case, the factorization brackets are similar to Fock’s creation and annihilation operators $a=\frac{1}{\sqrt{2}}\left(\frac{d~{}}{dx}+x\right)$ and its adjoint conjugate $a^{\dagger}$, which factorize the Hamiltonian $H_{qho}=-\frac{1}{2}\frac{d^{2}~{}}{dx^{2}}+\frac{1}{2}x^{2}$ as

are replaced by the new operators $b=\frac{1}{\sqrt{2}}\left(\frac{d~{}}{dx}+\beta(x)\right)$, and its adjoint conjugate $b^{\dagger}$, requiring that

In order to satisfy this, $\beta(x)$ is found to satisfy the Riccati equation

and the reversed factorization $b^{\dagger}b$ provides the one-parameter isospectral Hamiltonian with the one-parameter contribution to the potential

where the real parameter $\gamma$ is required to be $|\gamma|>\frac{1}{2}\sqrt{\pi}$ if we want to avoid singular potentials.

From (17) and (13) with $F_{0}=e^{-\frac{x^{2}}{2}}$ and $\widetilde{{\cal F}}_{0}=\widetilde{\Psi}_{0}(x;\gamma)$, one obtains

respectively. In Fig. 2, one can observe plots of singular and regular one-parameter deformed counterparts of the harmonic oscillator potential and ground state for $\gamma=\mp 0.5$, respectively.

The radial Schrödinger equation for the Hydrogen atom may be written as the following Sturm-Liouville eigenvalue problem

where $H_{\ell}$ is the radial Hamiltonian operator and the used units are $m=\hbar=e=4\pi\varepsilon_{0}=1$, together with $Z$=1.

With this convention, the energy eigenvalues and the eigenfunctions can be written in the form

where $C_{n,\ell}$ are the normalization constants given by

$n\in\mathbb{N}$, $n>\ell$, and $L^{2\ell+1}_{n-\ell-1}(2r/n)$ are derivatives of order $2\ell+1$ of the Laguerre polynomials of degree $n-\ell-1$, a.k.a. associated Laguerre polynomials. The polynomial degree $n_{r}=n-\ell-1$ provides the number of nodes of the eigenfunctions in the radial direction. In particular, we are interested in ’ground-state’ wavefunctions for which $n=\ell+1$.

The ladder operators of Infeld and Hull provide the following factorizations of the radial H operator (23)

where the Infeld-Hull ladder operators act on the radial functions as follows:

where $c_{n\ell}=\frac{\sqrt{(n-\ell)(n+\ell)}}{n\mathop{\ell}\nolimits}$.

Using Mielnik’s factorization approach, Fernández C. introduced the following modified Infeld-Hull raising and lowering operators [8]

where

obtained as the general solution of the Riccati equation

The parametric radial Hydrogen potentials and ground state wavefunctions (for the eigenvalue $-1/\ell^{2}$) are given by

where

to avoid singularities.

We have revisited the pioneering papers of Mielnik and Fernández that opened the research of parametric families of isospectral potentials in supersymmetric quantum mechanics. The uncovered issue of normalization of the parametric deformed ground state wavefunctions is discussed in some detail. It is found that there exist pairs of deformation parameters (one positive, the other negative) for which the deformed wavefunctions have the same normalization constant as the undeformed wavefunctions.

Acknowledgment

The second author wishes to thank the organizers of the workshop honoring Professor David J. Fernández C. for the enjoyable days of the event. Thanks also to Professors David Fernández and Asim Gangopadhyaya for some clarifying remarks.