The intervals of oscillations in the solutions of the radial Schrödinger differential equation
- Qutaibeh D Katatbeh^{1}Email author,
- Dimitris M Christodoulou^{2} and
- James Graham-Eagle^{2}
https://doi.org/10.1186/s13662-016-0777-7
© Katatbeh et al. 2016
Received: 25 October 2015
Accepted: 27 January 2016
Published: 19 February 2016
Abstract
We have previously formulated a program for deducing the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. In this work, we demonstrate how the oscillation-detection program can be carried out for the radial Schrödinger equation when a Coulomb potential is used to describe the hydrogen atom. The method predicts that the oscillation intervals are finite in radius and their sizes are determined uniquely by the two quantum numbers n and ℓ. Numerical integrations using physical boundary conditions at the origin confirm this oscillatory behavior of the radial Coulomb wavefunctions. Two related differential equations due to Kummer and Whittaker and other attractive electrostatic potentials are also discussed in the same context.
Keywords
MSC
1 Introduction
In the present work, we extend our investigation to the study of oscillatory properties of the radial Schrödinger equation that is of interest to the field of quantum mechanics (see, e.g., Chapter 14 in [6] and Section 67 in [7]). In Section 2, we adopt a classical Coulomb potential of the form \(V(x)\propto-1/x\) that describes the hydrogen atom (as in Section 67 of [7]), and we transform the radial Schrödinger equation into the canonical form (2). Then we use the formulation expressed by equations (5) and (6) to analyze the properties of the resulting coefficient \(q(x)\) in the interval \(x\in(0, +\infty)\). The same \(q(x)\) appears also in the canonical forms of Kummer’s equation [6, 7] and Whittaker’s equation [1, 6]. These closely related equations are discussed in Section 2.2. In Section 3, we carry out numerical integrations of the original equation (1) under physical boundary conditions at \(x=0\) in order to test the predictions of the analytical formulation. Finally, in Section 4, we summarize our results and discuss their relevance to various two-particle quantum mechanical models.
2 Radial Schrödinger equation with Coulomb potential
2.1 The hydrogen atom
- (a)
We know from the results obtained by CGK that some low-order solutions of equation (7) are nonoscillatory and that the lowest-order solutions that may exhibit at least two critical points of the same kind (maxima, or minima, or inflection points) for certain choices of boundary conditions have \(n=3\).
- (b)
For the two largest values of the secondary quantum number, \(\ell= n-2\) and \(\ell_{\max}=n-1\), the wavefunctions have only one node and no node, respectively, in the intervals determined by equation (11). Therefore, although the solutions are distinctly nonmonotonic in these intervals, they do not develop two critical points of the same kind, and they are then nonoscillatory according to the definition given in Section 1.
2.2 Related equations and degeneracy of the canonical form
Equation (14) has a constant coefficient in its \(y'\) term, so a constant damping is folded into the canonical form (2). This is not the case for equations (7) and (13), whose transformations lead to the same canonical form. The fact that \(q(x)\) takes the same form in these three cases implies that differential equations of the form (1) with markedly different structures in their coefficients can be cast to the same canonical form with virtually identical \(q(x)\) coefficients by transformations of the dependent variable y. This is a kind of degeneracy in the canonical transformation of the general form (1). Because of this degeneracy, when we seek the oscillation intervals of a given differential equation in the form of equation (1) or equation (2), we are obliged to undertake the steps of the program described in CGK, irrespective of the symmetries that the given equation may exhibit (e.g., constant damping or being already in canonical form). In particular, the transformation of the independent variable x described in Section 2.3 of CGK must be carried out before a criterion for oscillatory solutions can be established. This is because the transformation of x is not degenerate: two canonical equations of the form (2) with different coefficients \(q(x)\) do not produce the same final canonical form with coefficient \(\hat{q}(x) = k^{2}[(x-c_{1})^{2} q(x) - 1/4]\), where k and \(c_{1}\) are arbitrary constants (see equations [16] and [17] in CGK).
3 Numerical solutions
Additional numerical integrations of equation (7) confirm that (a) for values of \(n\geq3\) and \(\ell\leq n-3\), the solutions exhibit increasingly more oscillations with increasing n for a fixed value of ℓ; and (b) for a fixed value of n, such oscillations become fewer with increasing ℓ and eventually disappear for \(\ell= n-2\) and for \(\ell_{\max}=n-1\).
- (a)
For \(n=1\), then \(\ell= 0\), and the wavefunction is nodeless.
- (b)
For \(n=2\), then \(\ell= 0\) or 1, and the wavefunctions have one node and no node, respectively.
- (c)
For \(\ell= n-2\), then \(\mathcal{N}=1\), and these wavefunctions have one node.
- (d)
For \(\ell_{\max} = n-1\), then \(\mathcal{N}=0\), and these wavefunctions are all nodeless.
4 Summary and discussion
4.1 Summary
In Section 2, we have presented an analysis of the oscillatory properties of the solutions of the radial Schrödinger equation (7) with a Coulomb potential of the form (9) that describes the hydrogen atom [6, 7]. This equation has a regular singular point at \(x=0\), and its canonical form has the same structure and properties as the canonical forms of the closely related Kummer [6, 7] and Whittaker [1, 6] equations (Section 2.2). Our analysis of these equations makes use of a program that was described in [5] by CGK for investigating the x-intervals in which the solutions exhibit an oscillatory character.
The main result of our investigation is that the radial Schrödinger wavefunctions of the classical Coulomb potential show oscillations only in a finite interval the extent of which is determined analytically and given by equation (12) in terms of the two quantum numbers n and ℓ. This result could not have been obtained by classical oscillation theory [2–4] because the Coulomb wavefunctions do not exhibit infinitely many nodes. Numerical integrations of equation (7) using physical boundary conditions (\(y(0)=0\) and \(y'(0)=1\)) are described in Section 3, and they confirm the predicted oscillatory intervals in the wavefunctions for various choices of \(n\geq3\) and \(\ell\leq n-3\) (two examples are shown in Figures 1 and 2).
On the other hand, there exist solutions in which one complete oscillation (a full ‘cycle’) does not develop in the intervals specified by equation (12). Even in such cases, however, the wavefunctions are distinctly nonmonotonic in the predicted intervals. Complete oscillations do not occur for low-order solutions (\(n\leq2\)) and for values of \(\ell\geq n-2\) (a related example with \(n=10\) and \(\ell= 8\) is shown in Figure 3). The wavefunctions in these cases exhibit at most one node in \(x\in(0, +\infty)\), which is not enough for two extrema of the same kind to develop (see Section 3 for details).
4.2 Discussion
In quantum mechanics, the radial Schrödinger equation (7) is often studied for a variety of potentials other than the classical Coulomb potential. These include the Kratzer and Morse diatomic potentials [7], the two-particle Yukawa [7] and Hellmann [10] potentials, the cutoff Coulomb potential [11], and various power-law attractive potentials [12]. In all of these cases, the coefficient \(q(x)\) of the canonical Schrödinger form is given by equation (8), where \(V(x)\) is the particular type of the adopted potential, and the criterion for the appearance of oscillations in the wavefunctions is given by equation (5).
On the other end of the spectrum, the wavefunctions of highly excited states with large values of n and low-to-moderate values of ℓ will certainly exhibit oscillations since, for \(n\gg1\) and \(\ell\ll \sqrt{n}\), the condition for oscillations (16) reduces asymptotically to \(V(x) < 0\), which is automatically satisfied for attractive electrostatic potentials. In the same case, however, the wavefunctions are expected to become nonoscillatory for \(\ell\geq n-2\) because for the two largest values of ℓ, the solutions of the radial Schrödinger equation (7) are not expected to have enough radial nodes (see equation (15) in Section 3).
Declarations
Acknowledgements
During this research project, DMC and JG-E were supported by the University of Massachusetts Lowell, whereas QDK was on a sabbatical visit and was fully supported by the Jordan University of Science and Technology.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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