Comment on: “Some aspects of the interaction of a magnetic quadrupole moment with an electric field in a rotating frame” [J. Math. Phys. 58, 102103 (2017)]

Fernández, Francisco M.

doi:10.1063/5.0152523

We discuss results obtained recently for a quantum-mechanical model given by a neutral particle with a magnetic quadrupole moment in a radial electric field and a scalar potential proportional to the radial variable in cylindrical coordinates that also includes the noninertial effects of a rotating reference frame. We show that the conjectured allowed values of the cyclotron frequency are a mere artifact of the truncation of the power series used to solve the radial eigenvalue equation. Our analysis proves that the analytical expression for the eigenvalues is incorrect.

In a paper published recently in this journal, Fonseca and Bakke¹ discussed a neutral particle with a magnetic quadrupole moment in a radial electric field and a scalar potential proportional to the radial variable in cylindrical coordinates. They also considered the noninertial effects of a rotating reference frame. The Schrödinger equation for this quantum-mechanical model is separable in cylindrical coordinates and the authors solved the radial eigenvalue equation by means of the Frobenius (power-series) method. Upon forcing the truncation of the series they derived exact analytical polynomial expressions for the radial eigenfunctions as well as an exact analytical formula for the energy levels. They concluded that there are some permitted cyclotron frequencies determined by the angular velocity of the rotating frame, the parameter associated to the scalar potential and the quantum numbers. In this Comment we test the validity of those results and conclusions.

The starting point of our discussion is the eigenvalue equation

\begin{matrix} F^{''} (r) + \frac{1}{r} F^{'} (r) - \frac{l^{2}}{r^{2}} F (r) - r^{2} F (r) - ν r F (r) + W F (r) = 0, \\ W = \frac{4}{α} (E + \frac{1}{2} ϑ l + l ϖ), α^{2} = ϑ^{2} + 4 ϖ ϑ, \\ ν = \frac{2^{5 / 2} a}{\sqrt{m α^{3}}}, \end{matrix}

(1)

where m is the mass of the particle, E the energy, l = 0, ±1, ±2, … the rotational quantum number, a a constant in the scalar potential, ϑ the cyclotron frequency and ϖ the angular velocity of the rotating frame.¹ We can draw some straightforward conclusions from this equation. Since the behavior of F(r) at r → 0 and r → ∞ is determined by the terms l²/r² and r², respectively, we conclude that there are square integrable solutions for all values of ν. Therefore, there is no restriction on the values of the cyclotron frequency ϑ, contrary to what the authors stated.

There are square integrable solutions

\int_{0}^{\infty} {|F (r)|}^{2} r d r < \infty,

(2)

for particular values of W = W_i,l(ν), i = 0, 1, …, that are continuous functions of ν. Besides, from the Hellmann–Feynman theorem (HFT)^2,3

\frac{\partial W}{\partial ν} = ⟨r⟩ > 0,

(3)

we conclude that each W_i,l(ν) is an increasing function of ν.

Fonseca and Bakke¹ focused on polynomial solutions to Eq. that we discuss in what follows. If we look for a solution of the form

F (r) = r^{s} \exp (- \frac{r^{2}}{2} - \frac{ν r}{2}) \sum_{j = 0}^{\infty} c_{j} r^{j}, s = | l |,

(4)

we conclude that the expansion coefficients should satisfy the three-term recurrence relation

\begin{align} c_{j + 2} & = A_{j} c_{j + 1} + B_{j} c_{j}, j = - 1, 0, 1, \dots, c_{- 1} = 0, \\ A_{j} & = \frac{ν (2 j + 2 s + 3)}{2 (j + 2) [j + 2 (s + 1)]}, \\ B_{j} & = - \frac{4 W - 8 j + ν^{2} - 8 (s + 1)}{4 (j + 2) [j + 2 (s + 1)]} . \end{align}

(5)

In order to obtain a polynomial solution of order n, n = 0, 1, …, we require that c_n ≠ 0, c_n+1 = 0 and c_n+2 = 0 that leads to B_n = 0. From the last equality we obtain

W = W_{l}^{(n)} = 2 (n + s + 1) - \frac{ν^{2}}{4},

(6)

which leads to

B_{j} = B_{j, n} = \frac{2 (j - n)}{(j + 2) [j + 2 (s + 1)]} .

(7)

We immediately realize that something is amiss because the eigenvalues

W_{l}^{(n)}

do not satisfy the HFT .

Since c_n+1 is a polynomial function of ν of degree n + 1, the second condition c_n+1 = 0 leads to n + 1 particular values of ν, ν_n,i.l, i = 1, 2, …, n + 1. From these roots Fonseca and Bakke¹ concluded that “only specific values of the cyclotron frequency ϑ are permitted.” However, they did not attempt to investigate the actual meaning of these values of the model parameter ν. For convenience, here we organize them in decreasing order ν_n,i,l > ν_n,i+1,l. We appreciate that for each value of n the resulting eigenvalues ,

W_{l}^{(n, i)} = 2 (n + s + 1) - \frac{ν_{n, i, l}^{2}}{4},

(8)

are located on an inverted parabola. When n + 1 is odd there is a root ν = 0 that leads to the exact eigenvalues of the harmonic oscillator.

The truncation of the series in Eq. leads to particular polynomial solutions of the form

F_{l}^{(n, i)} (r) = r^{| l |} \exp (- \frac{r^{2}}{2} - \frac{ν_{n, i, l} r}{2}) \sum_{j = 0}^{n} c_{j, n, l} r^{j} .

(9)

They are square integrable but there are other solutions that satisfy the condition that do not have polynomial factors. Since Fonseca and Bakke overlooked the latter, they drew the wrong conclusion mentioned above.

This kind of problems is commonly called quasi-exactly solvable or conditionally solvable because one obtains eigenvalues W only for particular values of ν. They have been studied by several authors (see, for example, the review by Turbiner⁴ and the references therein). Fonseca and Bakke¹ seemed to be unaware of this fact and appeared to believe that the only quadratically integrable solutions to Eq. (1) are the polynomial ones (9). For this reason they concluded, wrongly, that there are allowed values of the cyclotron frequency ϑ. The true fact is that the allowed energies E_n,l are continuous functions of this model parameter.

There is no doubt that $W_{l}^{(n, i)}$ and $F_{l}^{(n, i)} (r)$ are eigenvalues and eigenfunctions, respectively, of the differential equation (1). A question now arises about the connection between the eigenvalues $W_{l}^{(n, i)}$ of such polynomial solutions and the actual eigenvalues W_j,l(ν) mentioned above. Taking into account the HFT (3) and the convenient ordering of the roots ν_n,i,l chosen above, we conclude that $W_{l}^{(n, i)} (ν_{n, i, l}) = W_{i - 1, l} (ν_{n, i, l})$ ⁠; in other words, $W_{l}^{(n, i)} (ν_{n, i, l})$ is a particular point on the continuous curve $W_{i - 1, l} (ν)$ ⁠.

Figure 1 shows some selected points $W_{0}^{(n, i)} (ν_{n, i, 0})$ ⁠, n ≤ 22, i ≤ min(n + 1, 3), connected by continuous lines that draw the curves W_j,0(ν), j = 0, 1, 2. The inverted parabola $W_{0}^{(10)} (ν) = 22 - ν^{2} / 4$ connects some of the solutions $W_{0}^{(10, i)}$ given by Eq. (8). The intersections of the vertical dashed line with the curves W_j,0(ν) are the actual eigenvalues of the quantum model with a given value of the parameter ν. Such a vertical line passes through, at most, one value of $W_{0}^{(n, i)} (ν_{n, i, 0})$ as shown in Fig. 1.

FIG. 1.

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Eigenvalues $W_{0}^{(n, i)}$ from the truncation method and actual eigenvalues W_n,0(ν) of the differential equation (1).

In order to obtain W_j−,l(ν) for ν_n,j,l < ν < ν_n+1,j,l, we simply resort to any suitable interpolation method. For example, from least squares we obtain

\begin{matrix} W_{0, 0} (ν) & = & 2 + 0.852 300 284 4 ν - 0.029 750 465 92 ν^{2} + 0.000 870 657 743 9 ν^{3}, \\ W_{1, 0} (ν) & = & 6 + 1.547 791 990 ν - 0.042 027 302 46 ν^{2} + 0.001 218 822 726 ν^{3}, \\ W_{2, 0} (ν) & = & 10 + 2.010 156 364 ν - 0.045 621 569 39 ν^{2} + 0.001 269 456 909 ν^{3}, \end{matrix}

(10)

which are sufficiently accurate in the range of ν values shown in Fig. 1.

Summarizing: The eigenvalues W_i,l(ν) of the differential equation (1) are continuous functions of the model parameter ν. Consequently, the energy eigenvalues E_n,l are continuous functions of the cyclotron frequency ϑ that can take any physically acceptable value. The truncation of the series in Eq. (4) only yields eigenvalues $W_{l}^{(n . i)}$ for particular values ν_n,i,l. From this particular values of ν Fonseca and Bakke¹ concluded, wrongly, that there are allowed or permitted values of the cyclotron frequency ϑ. The eigenvalues $E_{n, l}$ in Eq. (17) of the paper by Fonseca and Bakke are meaningless because the model parameters change with the quantum numbers n and l. More precisely, $E_{n, l}$ and $E_{n^{'}, l^{'}}$ are energy eigenvalues of different physical problems.