# Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule

## Abstract

**:**

_{θ}of the θ-oscillations, and the change of the angular variable φ during one half-period of the θ-motion—all in the forms of one-fold integrals in the general case and illustrated it pictorially. We also produce the corresponding explicit analytical expressions for relatively small values of the projection p

_{φ}of the angular momentum on the axis of the electric dipole. We also derive a general condition for this conditionally-periodic motion to become periodic (the trajectory of the electron would become a closed curve) and then provide examples of the values of p

_{φ}for this to happen. Besides, for the particular case of p

_{φ}= 0 we produce an explicit analytical result for the dependence of the time t on θ. For the opposite particular case, where p

_{φ}is equal to its maximum possible value (consistent with the bound motion), we derive an explicit analytical result for the period of the revolution of the electron along the parallel of latitude.

## 1. Introduction

_{min}< ξ < ξ

_{max}, η

_{min}< η < η

_{max}). The motion occurs inside a torus created by the revolution (about the dipole axis) of the two limiting ellipses, corresponding to ξ = ξ

_{min}and ξ = ξ

_{max}, and of the two limiting parabolas, corresponding to η = η

_{min}and η = η

_{max}.

_{φ}of the angular momentum on the axis of the electric dipole.

_{φ}= 0 we produce an explicit analytical result for the dependence of the time t on θ. For the opposite particular case, where p

_{φ}is equal to its maximum possible value (consistent with the bound motion), we derive an explicit analytical result for the period of the revolution of the electron along the parallel of latitude.

_{θ}of the θ-oscillations, and the change of the angular variable φ during one half-period of the θ-motion—all in the form of one-fold integrals in the general case and illustrate it pictorially. We also produce the corresponding explicit analytical expressions for relatively small values of the projection p

_{φ}of the angular momentum on the axis of the electric dipole.

## 2. Classical Non-Circular Orbits

^{2}+ r

^{2}(dθ/dt)

^{2}+ r

^{2}sin

^{2}θ (dφ/dt)

^{2}]/2 − eDcosθ/r

^{2},

_{z}is conserved, as manifested by the fact that the energy, while depending on dφ/dt, does not depend on φ. Fox [6] denoted M

_{z}as p

_{φ}because it is the generalized momentum corresponding to the dynamical variable φ:

_{φ}= mr

^{2}sin

^{2}θ (dφ/dt) = const.

^{2}= [2eD/(mr

^{4})](− x

^{3}+ x − K),

_{φ}

^{2}/(2meD).

_{φ}/(mr

^{2}sin

^{2}θ) = p

_{φ}/[mr

^{2}(1 − x

^{2})].

^{3}+ x − K

^{3/2}at x = − 1/3

^{1/2}and a maximum equal to −K + 2/3

^{3/2}at x = 1/3

^{1/2}. Obviously, for the range of the bound motion to exist, the maximum should be positive, leading to the following requirement (Fox [6]):

^{3/2}= K

_{max}

^{3/2}p

_{φ}

^{2}/(4me).

_{φ}in the right side of Equation (8) can be zero. The same is true for classical bound states for any finite size of the dipole, as shown in paper [3]. In contrast, the quantum bound states exist only if the dipole moment exceeds some critical value D

_{min}= 0.6393148771999813 (in atomic units). This value of D

_{min}with the accuracy of the first 16 digits was calculated in paper [2] of 2007 (paper [2] contains references to most of the previous calculations of D

_{min}). However, the existence of the critical dipole moment D

_{min}and its first three digits were calculated as early as in 1947 by Fermi and Teller [1].

_{max}= 2/3

^{3/2}. It is seen that in this range of K, the maximum value of y is positive, so that there is indeed a range of x that allows classical bound motion.

_{2}and x

_{3}(x

_{3}> x

_{2}. They are the real roots of the following cubic equation:

^{3}− x + K = 0.

_{2}(K) = (−1)

^{2/3}2

^{1/3}/[(729K

^{2}− 108)

^{1/2}− 27K]

^{1/3}− (−1)

^{1/3}[(729K

^{2}− 108)

^{1/2}− 27K]

^{1/3}/(2

^{1/3}3).

_{3}(K) = 2

^{1/3}/[(729K2 − 108)

^{1/2}− 27K]

^{1/3}− [(729K2 − 108)

^{1/2}− 27K]

^{1/3}/(2

^{1/3}3).

_{2}(K) and x

_{3}(K), where x

_{2}(K) and x

_{3}(K) are the lower part and the upper part of the double-valued curve, respectively. The lower and upper parts intersect at K = K

_{max}= 2/3

^{3/2}, where x

_{2}(2/3

^{3/2}) and x

_{3}(2/3

^{3/2}) = 1/3

^{1/2}.

_{2}(K) and x

_{3}(K) are the lower part and the upper part of the double-valued curve, respectively.

^{4})]

^{1/2}.

^{3}+ x − K)

^{1/2}.

_{max}= 2/3

^{3/2}, there is no θ-motion: the electron follows a circular path along the parallel of latitude corresponding to cosθ = 1/3

^{1/2}. The latter equation yields θ = 0.9553 rad = 54.74 degrees. (We note that McDonald [8] considered this circular motion for a positive charge, in which case cosθ = −1/3

^{1/2}, so that θ = 2.1863 rad = 125.26 degrees.) From Equation (5) it follows that the electron rotates with the constant angular velocity

_{φ}/(2mr

^{2}),

^{2}/(3p

_{φ}).

_{φ}= (2KmeD)

^{1/2}, so that for K = K

_{max}= 2/3

^{3/2}we have p

_{φ}= 2(meD)

^{1/2}/3

^{3/4}, so that Equations (14) and (15) can be rewritten as follows:

^{1/4}[eD/(mr

^{4}]

^{1/2}, T = (2π/3

^{1/4}) [mr

^{4}/(eD)]

^{1/2}.

_{2}(K) ≈ K, x

_{3}(K) ≈ 1 − K/2.

_{φ}= 0, there is no φ-motion. The electron oscillates along a semicircle located in a meridional plane in the upper hemisphere. (This is analogous to the corresponding result by Jones [7], later reproduced by McDonald [8], for a positive charge: the only difference is that for the positive charge, the semicircular orbit is in the lower hemisphere.) Let us study this special case in more detail before proceeding to the general case.

**F**[arcsin(–x)

^{1/2}, −1]},

**F**(α, q) is the incomplete elliptic integral of the first kind. Despite the formal appearance of the imaginary unit i in Equation (18), the right side of Equation (18) is actually real for the range of x from 0 to 1 where the motion occurs. In particular, for x << 1, we obtain from Equation (18) the following:

_{0}= 4τ(1) = 10.488.

^{4}/(2eD)]

^{1/2}.

_{max}= 2/3

^{3/2}). Based on Equation (13), the dependence of the scaled time τ on x (i.e., the dependence of τ on cosθ) in the general case:

_{2}(K) to x

_{3}(K) and back to x

_{2}(K)—for K = 0.1 (solid line), K = 0.2 (dashed line), and K = 0.3 (dotted line).

_{2}(K)] + Kg(x) − Kg[x

_{2}(K)]}.

**F**[arcsin(–x)

^{1/2}, −1],

^{2}− 2 − x

^{2}(1 − x

^{2})

^{1/2}

_{2}F

_{1}(3/4, 1/2, 7/4, x

^{2})]/[2(x − x

^{3})

^{1/2}]

_{2}F

_{1}(a, b, c, z) is the hypergeometric function.

_{θ}of the θ-motion. It is calculated by the following formula (in units of mr

^{4}/(2eD)):

_{θ}on K is shown in Figure 6.

_{θ}(K) ≈ 2 {f (1 − K/2) − f (K) + Kg(1 − K/2) − Kg(K)},

_{φ}dt/[mr

^{2}(1 − x

^{2})].

_{2}(K) to x

_{3}(K)) for K = 0.1 (solid line), K = 0.2 (dashed line), and K = 0.3 (dotted line). It is seen that as the parameter K increases, the curve φ(x) becomes steeper and the change of φ over one half-period of the θ-motion slightly increases.

^{1/2}[j(x) − j(K)]

^{2})]

^{1/2}{1 − (x − 1/x)

^{1/2}

**F**[arccsc(x

^{1/2}), −1]}

**F**(α, q) is the incomplete elliptic integral of the first kind. In Equation (30) we used the fact that x

_{2}(K) ≈ K for K << 1.

_{max}= 2/3

^{3/2}, we have Δφ ≈ π/2

^{1/2}, so that

^{1/2}.

_{θ}(K) of the θ-oscillation (given by Equation (26) and presented in Figure 6 in units of mr

^{4}/(2eD)), the angle φ advances by Δφ(K) given by Equation (32). In general, Δφ(K) is not equal to nπ/m, where n and m are relatively small integers, so that the combined motion is conditionally-periodic: the trajectory generally is not a closed curve.

## 3. Conclusions

_{φ}

^{2}/(2meD) << 1, i.e., for relatively small values of the projection p

_{φ}of the angular momentum on the axis of the electric dipole.

_{max}= 2/3

^{3/2}and the electron follows a circular path along the parallel of latitude corresponding to θ = 0.9553 rad = 54.74 degrees, we obtained an explicit analytical result for the period of the revolution.

_{θ}of the θ-oscillations in the form of a one-fold integral in the general case and illustrated it pictorially. We also derived the corresponding explicit analytical expressions for the case of K << 1.

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Plot of both positive roots x

_{2}and x

_{3}of the cubic Equation (8) versus K = p

_{φ}

^{2}/(2meD).

**Figure 3.**Dependence of the scaled time τ (defined in Equation (12)) on x = cosθ during one half-period of the electron oscillation along a semicircular path through the north pole of the upper hemisphere (K = 0).

**Figure 4.**Evolution of the scaled time τ during one period of the θ-motion for K = 0.1 (solid line), K = 0.2 (dashed line), and K = 0.3 (dotted line).

**Figure 5.**Comparison of the approximate analytical result for τ(x, K) from Equation (23) for K = 0.01 (solid line) with the corresponding exact result obtained by the numerical integration in Equation (22) (dashed line).

**Figure 6.**Dependence of the scaled period T

_{θ}of the θ-motion on the parameter K = p

_{φ}

^{2}/(2meD). The period T

_{θ}is in units of mr

^{4}/(2eD).

**Figure 7.**Dependence of φ on x = cosθ during one half-period of the θ-motion for K = 0.1 (solid line), K = 0.2 (dashed line), and K = 0.3 (dotted line).

**Figure 8.**Comparison of the approximate analytical result for φ(K, x) from Equation (30) for K = 0.02 (solid line) with the corresponding exact result from Equation (29) (dashed line).

**Figure 9.**Intersections of the curve Δφ(K)/π (thick line) with the tree horizontal lines corresponding to 2/3 (the top thin line), 3/5 (the middle thin line), and 4/7 (the bottom thin line). For the values of K, corresponding to these intersections, the motion, which is generally conditionally-periodic, becomes truly periodic.

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**MDPI and ACS Style**

Oks, E.
Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule. *Symmetry* **2020**, *12*, 1275.
https://doi.org/10.3390/sym12081275

**AMA Style**

Oks E.
Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule. *Symmetry*. 2020; 12(8):1275.
https://doi.org/10.3390/sym12081275

**Chicago/Turabian Style**

Oks, Eugene.
2020. "Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule" *Symmetry* 12, no. 8: 1275.
https://doi.org/10.3390/sym12081275