Quantum Mechanics of Kepler Orbits 1 Introduction - Hikari

Advanced Studies in Theoretical Physics Vol. 8, 2014, no. 21, 889 - 938 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2014.48114

Quantum Mechanics of Kepler Orbits Alexander Rauh and J¨ urgen Parisi Department of Physics, University of Oldenburg, D-26111 Oldenburg, Germany c 2014 Alexander Rauh and J¨ Copyright urgen Parisi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract It is shown that the Kepler orbits both elliptic, hyperbolic, and parabolic ones, are solutions of the Schrödinger equation of the hydrogen atom, and can be inferred from a single wave function in the classical limit. The latter corresponds to large quantum numbers of the angular momentum. The mean initial values of position and velocity are implemented in the state. The orbits evolve by a curve parameter which in the elliptic case corresponds to the eccentric anomaly. The mean values, including mean square deviations, are first calculated for the hyperbolic case and, then, continued into the elliptic region by extending the curve parameter to purely imaginary values. Time dependence is introduced by the assumption that it enters through the curve parameter only. However, as it is shown, this assumption holds in the classical limit only and is violated to next leading order. The wave function is derived by a proper projection of a four-dimensional harmonic oscillator state onto the ordinary three-dimensional space, it appears in a compact form without the necessity of an additional constraint. The initial wave function is connected with a nearly minimum uncertainty product.

Subject Classification: 03.65.Sq; 02.70.Wz; 31.15.-p Keywords: Kustaanheimo-Stiefel transformation; Classical limit; quantum fluctuations; celestial mechanics

1

Introduction

We propose a quantum state ψ in three-dimensional configuration space and calculate the mean values of position and velocity with the result that, in

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Alexander Rauh and J¨ urgen Parisi

the classical limit, the Kepler orbits emerge: elliptic, hyperbolic, or parabolic ones, depending on the initial parameters implemented in ψ. With regard to the Hamiltonian of the hydrogen atom, most previous work was focused on the subspace with discrete energy spectrum which, in the classical limit, leads to elliptic orbits. In the elliptic domain, the wave function has been constructed basically by means of three methods and their combination: (i) The principle of minimum uncertainty coherent states [17, 16, 18, 9, 6]; (ii) The use of the Kustaanheimo-Stiefel (KS) transformation [13, 7, 5, 23, 12, 4, 10, 2]; (iii) The application of Lie groups [19, 15, 3, 6, 2]. The quantum mechanics in the hyperbolic region was elaborated in [20]. In the present work, we do not rely on spectral representations. In the classical limit, we derive from a single wave function the Kepler orbits except for rectilinear ones. Our method is close to using constrained four-dimensional harmonic oscillator states (COS). However, we do not rely on a constraint. The state ψ depends analytically on the configuration vector r ∈ R3 and five parameters as follows: ψ = ψw (r; r0 , v0 ; κ, ν) , where r0 denotes the mean initial distance of an apsis point P0 from the force center, and v0 the length of the mean initial velocity at P0 , which is normal to the apses line; w is a real curve parameter for hyperbolic orbits; in the elliptic case, w is continued to purely imaginary values; κ is an order of magnitude parameter, essentially the angular momentum in units of ~, with κ → ∞ providing us with the classical limit, and ν > 0 is an arbitrary constant. With increasing distance from the initial point, where w = 0, κ has to be replaced by the dynamical parameter K = κh(w), where in the hyperbolic case h(w) monotonically decreases with increasing w from the value h(0) = 1. The initial data (r0 , v0 ) are mean values to leading order in κ. Keeping the initial distance r0 fixed, let us discuss the dependence on v0 in the classical limit. If v0 is large, then the mean energy E is positive and corresponds to a hyperbolic orbit with eccentricity e > 1; the initial point P0 is the peri-center, and one has the relation r0 = (e − 1)a with a denoting the semi-major axis. When v0 decreases, the critical energy E = 0 is reached which corresponds to a parabolic orbit where e = 1. Decreasing v0 further, one comes into the elliptic region with E < 0 and 0 ≤ e < 1; P0 is still the peri-center with r0 = (1 − e)a, until the circular orbit is reached at the speed v0 = vc where e = 0. For speeds below vc , the initial point P0 becomes the apo-center with the connection r0 = (1+e)a, and the eccentricity increases from the value e = 0 to the second singular case e = 1, where v0 = 0 which corresponds to a rectilinear orbit with angular momentum L = 0. As compared to [20], the present theory also applies to parabolic orbits, but it does not cover the case when v0 is close to zero.

Quantum mechanics of Kepler orbits

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The magnitude parameter κ, at first, characterizes the width of ψ0 in configuration space. With increasing κ, the probability density |ψ0 |2 becomes more and more peaked with the consequence that the deviations of mean values r from the Kepler orbit are of order 1/κ. The mean values of the velocity turn out being proportional to κ in the classical limit. At time t = 0, or for w = 0, one can relate κ to the initial parameters r0 , v0 ; in Sec VI. it is found that κ, essentially, equals the angular momentum L = mr0 v0 in units of ~, which implies that the parameter κ and, thus, the halve width of |ψ0 |2 is fixed by the orbital angular momentum. Through the explicit w dependence, the mean values with respect to the state ψw produce the full orbits in the classical limit, but still without time behavior. In the elliptic case, the parameter 2w turns out to be the eccentric anomaly. In the COS formalism, w emerged, up to a dimensional factor, as the time parameter of the harmonic oscillators, and was termed pseudo or fictitious time, since it differs from the time of the Kepler problem, see e.g. [4], [10], [12], [3], [7], [20]. It was observed some time ago in [11] that pseudo time should be related to the eccentric anomaly rather than to a physical time variable. Here, we consider w simply as a curve parameter. As it turns out, if one assumes that time dependence enters exclusively through w = w(t), then this can be proved in the classical limit; in general, however, a one to one correspondence between w and t cannot be maintained. The wave function ψw determines both the mean orbit r(w) and the mean velocity v(w). In Sec. VII, we will infer the time dependence of w from the vector relation d/(dt)r(w) = v(w). As it turns out, only in the classical limit can one find the same scalar function w(t) for all vector components; to next higher approximation with respect to κ, the vector equation is inconsistent with the assumption w = w(t). In the elliptic case, the classical limit reproduces Kepler’s equation for W (t) ≡ 2w(t). In Sec. VII. B, we attempt to solve the time dependent Schrödinger equation in the classical limit for the initial state ψ0 . But we are only partially successful, since we have to restrict our consideration to the neighborhood of the maximum of the probability density |ψw |2 rather than taking into account the full configuration space. In Sec. III, the initial wave function ψ0 is derived from an un-constrained 4D oscillator state by using the fact that the KS space is the direct product of the original R3 and a circle space with the latter described by a fourth coordinate 0 ≤ Φ < 2π, see Sec. II. For any constant value Φ0 , a wave function in R3 appears ; however, the function is not single-valued under rotations in 3D configuration space, a property which is reminiscent of the KS transformation being an outgrowth of spinor theory [13]. A proper state in R3 is obtained by integrating with respect to Φ with equal weight in the interval (0, 2π), see Sec. III. The dependence on the curve parameter w is borrowed from [20]. The calculation of mean values, in particular of mean square deviations of the position and velocity vectors, turned to be quite involved. It is hard to

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imagine that the given problem could have been successfully attacked without the computer software for symbolic mathematical manipulations now available. We have used the version number 9 of Mathematica [22]. Instead of working out the matrix elements in 3D, we made a detour via four-dimensional KS space by keeping, at first, the phase integral with respect to Φ un-evaluated. The advantage is, that we obtain four Gaussian integrals and are left with the Φ integration which eventually leads to modified Bessel functions. As a check of the calculations, we verified the positivity of mean square deviations. This, of course, must be true by definition. However, positivity is not evident in the end formulas which are the result of several intermediate simplification steps. In any case, positivity is verified in the three-dimensional space of the parameters (w, e, ν) which appear explicitly in the results; the parameter ν > 0 shows up as a disposable constant. The mean values are calculated at first in the hyperbolic region, and then analytically continued to elliptic orbits. In the latter case, the positivity proof of the mean square deviations in subsections 1, 2, and 3 of Appendix E shows that the method of analytic continuation is consistent. As a further check, we verify for the hyperbolic case that the uncertainty products obey the quantum mechanical lower bound in the threedimensional parameter space, for details see Appendix E.4. The uncertainty √ products of the initial state ψ0 are close to the minimum, up to a factor 1 + ν 2 . The value ν = 0 is a singular point of the theory; a good choice would be ν = 1. With increasing curve parameter w, there is increasing quantum diffusion which, however, in macroscopic examples like artificial satellites is irrelevant within realistic time spans, see Sec. VIII. The plan of the paper is as follows. In Sec. II, we sketch the KS transformation which serves to establish in Sec. III the wave function. In Sec. IV, we develop two integration methods, one is based on a mean value theorem which is suitable to determine mean values to leading order in the dynamical magnitude parameter K; for higher order correction, integration via detour in KS space is more efficient and outlined in Sec. IV. B. At the end of Sec. IV, the reader will find a list of the main auxiliary functions used in Sec. V, where we present the mean values of orbit vector, velocity, their mean square deviations, uncertainty products at zero time, and of the Hamiltonian. In Sec. VI, the magnitude parameter κ is fixed from the initial data and the classical energy conservation law; furthermore, the classical limit is related to the orbital angular momentum. In Sec. VII, we discuss the time dependence of the wave function. In Sec. VIII, we present numerical examples. In Sec. IX, the results for hyperbolic orbits are analytically continued into the elliptic domain. The conclusions of the paper are in Sec. X followed by five appendices which contain the proof of the mean value lemma, details of the mean value calculations, and several checks.

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2

Coordinate systems

Let us describe the cartesian position components in terms of polar coordinates x = r sin(θ) cos(ϕ), r > 0, 0 < θ < π,

y = r sin(θ) sin(ϕ), 0 ≤ ϕ < 2π.

z = r cos(θ), (1)

By adding the further coordinate 0 ≤ Φ < 2π, we extend to four dimensions and write the KS transformation (r, θ, ϕ, Φ) → u ∈ R4 as in [20] √ √ u1 = √r cos(θ/2) cos(ϕ − Φ); u2 = √r cos(θ/2) sin(ϕ − Φ); (2) u4 = r sin(θ/2) sin(Φ). u3 = r sin(θ/2) cos(Φ); The transformation (2) has been known for a long time, see e.g. [3] and references therein. Equivalent forms, with Φ replaced by Φ0 = ϕ − 2Φ, were used in [4], [12]. According to [20], the volume elements of the different coordinate systems are related as follows du1 du2 du3 du4 = 1/(8r)dxdydzdΦ = (1/8)r sin(θ)drdθdϕdΦ.

(3)

We will use the connection (3) from the right to the left hand side. If the variables (r, ϕ, θ) are fixed, then two different vectors u belonging to the phase Φ0 = 0 and an arbitrary KS phase Φ, respectively, are connected by the matrix T (Φ) u(Φ) = T (Φ)u(0) (4) with



 cos(Φ), sin(Φ), 0, 0  − sin(Φ), cos(Φ),  0, 0 . T(Φ) =   0, 0, cos(Φ), − sin(Φ)  0, 0, sin(Φ), cos(Φ)

(5)

In terms of u, the cartesian components read x = 2(u1 u3 − u2 u4 ),

y = 2(u1 u4 + u2 u3 ),

z = u21 + u22 − u23 − u24 ,

(6)

which can be immediately verified by means of (2). The connection between differential operators is more delicate. In our calculations, we need to transform the velocity operator into u space. We start with the cartesian representation (vx , vy , vz ) = ~/(i m) (∂x , ∂y , ∂z ) ,

(7)

go over to polar coordinates, and eventually to the variables u1 , . . . u4 . In terms of polar coordinates, the cartesian gradient components of f = f (r, θ, ϕ) read ∂x f = sin(θ) cos(ϕ)∂r f + (1/r) cos(θ) cos(ϕ)∂θ f − sin(ϕ)/ (r sin(θ)) ∂ϕ f, ∂y f = sin(θ) sin(ϕ)∂r f + (1/r) cos(θ) sin(ϕ)∂θ f + cos(ϕ)/ (r sin(θ)) ∂ϕ f, ∂z f = cos(θ)∂r f − (1/r) sin(θ)∂θ f. (8)

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For a check of (8), one may specify consecutively the function f by the polar coordinates of the position components to produce the unit matrix (∂x , ∂y , ∂z ) ⊗ (x, y, z) = 1.

(9)

In order to transform the right hand sides of (8) into u space, we use the 4D vector p:=(r, θ, ϕ, Φ). Denoting the right hand sides of (8) by δx , δy and δz , we obtain with aid of (2) Dx ≡ 1/(2u2 ) [u3 ∂u1 − u4 ∂u2 + u1 ∂u3 − u2 ∂u4 ] = δx + 1/(2r) cot(θ/2) sin(ϕ) ∂Φ , Dy ≡ 1/(2u2 ) [u4 ∂u1 + u3 ∂u2 + u2 ∂u3 + u1 ∂u4 ] = δy − 1/(2r) cot(θ/2) cos(ϕ) ∂Φ , Dz ≡ 1/(2u2 ) [u1 ∂u1 + u2 ∂u2 − u3 ∂u3 − u4 ∂u4 ] = δz . (10) One proves (10) from the left to the right hand side by deriving from (2) the 4 × 4 matrix M : du = M dp together with its inverse which gives dp=M −1 du and the transpose N = (M −1 )T , so one finds, for instance, u3 ∂u1 = u3 (r, θ, ϕ, Φ)

4 X

N1k ∂pk .

(11)

k=1

Since our wave function will not depend on the KS phase Φ, we can rigorously replace the differential operators ∂x , ∂y , ∂z by the u forms Di with (vx , vy , vz ) = ~/(i m) (Dx , Dy , Dz ).

(12)

We remark that in previous works, e.g. in [2], [3],[7], [5], where the wave function lived on 4D configuration space, a constraint was required to eliminate the differential ∂Φ . In [7] and [20], the constraint was dealt with approximately only.

3

Construction of wave function in hyperbolic domain

For the initial state ψ0 , at time t = 0, we start from the coherent state given in [20] which is defined in the KS space u ∈ R4 : Ψ0 = C exp a · u − Γ0 u2 /2 , Γ0 > 0. (13) (In the notation of [20], Γ0 = γ0 ΓΩ ). The parameter vector a was fixed in [20] in such a way that, in the classical limit, the mean initial position and velocity vectors come out as r(t = 0) = {r0 , 0, 0},

v(t = 0)i = {0, v0 , 0},

r0 , v0 > 0.

(14)

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The components of a resulted as follows ak = µ k + i ν k ,

k = 1, . . . 4,

(15)

where, with the abbreviations C0 = cos(Φ0 ) and S0 = sin(Φ0 ), µ1 = ρ 0 C 0 , µ2 = ρ0 S0 , µ3 = ρ0 C0 , µ4 = −ρ0 S0 , ν1 = −ν ρ0 S0 , ν2 = ν ρ0 C0 , ν3 = ν ρ0 S0 , ν4 = ν ρ0 C0

(16)

with ρ20 = Γ20 r0 /2,

ν > 0.

(17)

The phase Φ0 is arbitrary, and ν is a constant disposable number. In [20], the parameters Γ0 and ν were specifically defined in terms an energy eigenvalue of the Schrödinger equation and the eccentricity of the mean Kepler orbit. In the following, we do not rely on that history and keep Γ0 and ν disposable as long as possible. After inserting the parameter vector a, as specified by (15) and (16), into the expression (13) for Ψ0 , and with the aid of (2), we can write √ Ψ0 = C exp[−Γ0 r/2] exp [G0 ] , G0 = ρ0 r [d1 cos(Φ) + d2 sin(Φ)] , (18) where d1 = sin(θ/2) + cos(θ/2) [cos(ϕ) + i ν sin(ϕ)] , d2 = i ν sin(θ/2) + cos(θ/2) [−i ν cos(ϕ) + sin(ϕ)] .

(19)

In (18), we have shifted the origin of Φ from 0 to −Φ0 . It should be noticed that, for a fixed KS phase Φ, the wave function Ψ0 is not invariant under a full rotation in 3D space, one has to go around twice similar to the transformation of a spinor. Formerly, this is due to the occurrence of half of the polar angle, θ/2. In order to ensure the correct transformation behavior, we form the following linear combination with respect to the KS phase Φ: Z 2π ψ0 = dΦ Ψ0 (Φ). (20) 0

The wave function ψ0 is now a function of sin(θ). As a matter of fact, the Φ integration is elementary and can be written in terms of the zero order modified Bessel function I0 as follows p √ (21) ψ0 = 2π C exp[−Γ0 r/2] I0 (U0 ), U0 = ρ0 d r = κ d r/(2r0 ) with d2 = d21 + d22 = 1 − ν 2 + sin(θ) (1 + ν 2 ) cos(ϕ) + 2i ν sin(ϕ) .

(22)

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The classical limit will be controlled by the dimensionless parameter κ 1 with κ = r0 Γ0 . (23) By (21), U0 is proportional to κ, so in the classical limit, with |U0 | 1, the Bessel function can be approximated by the asymptotic formula [8] 1 exp[U0 ] 2 + O(1/U0 ) , (24) 1+ I0 (U0 ) → √ 8U0 2πU0 which gives rise to the asymptotic form of the zero-time wave function r 2π ψ0 → ψas = C exp[−Γ0 r/2 + U0 ] [1 + O(1/U0 )] , |U0 | 1. U0

(25)

The parameter evolution is borrowed from [20]. Instead of the pseudo time σ, we here introduce the curve parameter w = ω σ. According to [20], the parameter dependence enters through the vector a → a(w) and the scalar parameter Γ0 → Γ(w) with consequences for G0 → G(w) and U0 → U (w) as follows a(w) = a(0)/f ∗ (w), G(w) = G0 /f ∗ (w), U (w) = U0 /f ∗ (w), Γ(w) = ΓR + i ΓI = −i Γ0 tanh (w + i C1 ) ,

(26)

where f (w) is defined in (60a) below and tanh(C1 ) = γ0 which implies that ΓR = Γ0 h(w),

ΓI = −Γ0 h(w)(1 + γ02 )/(2γ0 ) sinh(2w)

(27)

with h(w) defined in (60b). The integration constant γ0 , which in [20] was set equal to 1, is kept disposable. The above parameter dependence, with w being real, refers to hyperbolic orbits. The parameter dependent wave function now reads Ψw (r, θ, ϕ, Φ) = C(w) exp(−Γ(w)r/2] exp[G(w)], (28) which, since f (0) = h(0) = 1, reproduces ψ0 for w = 0. Analogously to (20) and (21), the KS phase is averaged out with the result Z 2π ψw = dΦ Ψw = 2πC(w) exp[−Γ(w)r/2] I0 (U ). (29) 0

The magnitude parameter now becomes w dependent as κ → K = r0 ΓR (w) = κh(w),

(30)

which implies that K decreases with increasing w > 0, since h(0) = 1 and h(w) < 1. In the classical limit, if K 1, the asymptotic form of the wave function reads r 2π exp [−Γ(w)r/2 + U ] [1 + O(1/K)] . (31) ψw → ψas = C(w) U The state ψw lives in three-dimensional configuration space and is normalizable for all parameters w.

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4

Integration Methods

4.1

Mean value theorem

From the maximum of the probability density, P = |ψas |2 , we infer that the mean values of the orbit vector r = (x, y, z), to leading order in K, come out as follows √ x = a [e − cosh(2w)] , y = a e2 − 1 sinh(2w), z = 0, e > 1. (32) where a and e denote the semimajor axis and the eccentricity, respectively. Clearly, (32) is the parametric representation of a hyperbola with parameter w ∈ R. In order to derive the maximum of P , we write the asymptotic probability density in the following form P = |C(w)|2 √

2π exp [G2 ] , UU∗

(33)

where hp i2 G2 = −ΓR r + U + U ∗ = −K r/r0 − D) + KD2 , √ D = 2/4 [d(θ, ϕ)f (w) + d∗ (θ, ϕ)f ∗ (w)]

(34)

with d defined in (22) and U in (26) and (21). Since D is independent of r, G2 is maximal at rm = r0 D2 (θ, ϕ), (35) with the consequence that G2 (rm ) = KD2 (θ, ϕ) ≥ 0.

(36)

It is easily verified that the only extremum of D(θ, ϕ) with respect to θ ∈ (0, π) is at θm = π/2, which means that the extremum lies in the z = 0 plane. Then, from ∂ϕ D(θm , ϕ) = 0 we find the further condition tan(ϕm /2) = γ0 ν tanh(w).

(37)

As is discussed below and in Appendix A, the extremum of G2 is actually a maximum, which for K 1 is narrowly peaked. For w ≥ 0, condition (37) is equivalent to p p cos(ϕm /2) = cosh(w) (e − 1)Zw , sin(ϕm /2) = sinh(w) (e + 1)Zw (38) with Zw = [e cosh(2w) − 1]−1 . Evaluating rm (θ, ϕ) at the maximum (θ = π/2, ϕ = ϕm ), we obtain from (35) and (37) rm (w) = r0 cosh2 (w) + γ02 ν 2 sinh2 (w) . (39)

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Now, we dispose of the parameter γ0 such, that rm (w) is the hyperbolic distance from the force center, i.e. rm = a [e cosh(2w) − 1] ,

r0 = (e − 1)a,

(40)

which is true, if γ0 = (1/ν)

p (e + 1)/(e − 1),

e > 1.

(41)

With the aid of θm = π/2, (40), and (38), one easily derives the following vector components of the orbit at the maximum of the probability density for e > 1: xm ≡ rm sin(θm ) cos(ϕm ) = a [e − cosh(2w)] , √ ym ≡ rm sin(θm ) sin(ϕm ) = a e2 − 1 sinh(2w), zm ≡ rm cos(θm ) = 0,

(42)

which describe a hyperbolic orbit. In Appendix A, we prove the following mean value lemma for real functions F: Lemma : hψas |F (r)ψas i = F (rm , θm , ϕm )[1 + O(1/K)]. (43) As a consequence, the vector components at the maximum are actually mean values to leading order in K: (xm , ym , zm ) = r[1 + O(1/K)].

(44)

The proof of the Lemma in Appendix A is based on the Taylor expansion of G2 around the maximum with √ (45) r = rm + δr, θ = θm + δθ, ϕ = ϕm + δϕ, = 1/ K. We obtain to leading order in , for details see Appendix A, G2 = E0 /2 − Er δr2 + Eϕ δϕ2 + Eθ δθ2 + O(), E0 = [Zw (e − 1)]−1 ,

Er = Zw (e − 1)/(4r02 ),

Eϕ = [4Zw (e − 1)]−1 ,

−1

Eθ

(ν 2 + 1) [2(e − 1)Zw2 ] = . (e − 1) (1 + cosh(2w)) + (e + 1)ν 2 (cosh(2w) − 1)

(46)

Clearly, the above E coefficients are positively definite for e > 1 and real w. They remain positive after the analytic continuation to elliptic orbits, w → i w0 , with the eccentricity confined to the interval 0 ≤ e < 1 and with w0 being a real number. The expansion method implies, that the mean deviations δr, δθ, δϕ are of √ order = 1/ K.

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4.2

Integration via KS space

The integration in √3D configuration space becomes rather awkward when higher orders in 1/ K are needed as it is the case for mean square deviations. In what follows, we adopt a more efficient method by means of a detour in KS space. To this end, one postpones the averaging with respect to the KS phase Φ, and writes the mean value of an observable O as Z 2π Z 2π 0 dΦ dΦ hΨw (Φ0 )|OΨw (Φ)i, (47) hψw |Oψw i = 0

0

where Ψw (Φ) = C exp a(w) · u(Φ) − Γ(w)u2 /2

(48)

with a(w) and Γ(w) defined in (26) and C denoting the normalization constant; we used the property r = u2 and the fact that u2 ≡ u · u does not depend on Φ. In the scalar product a · u(Φ), the Φ dependence is shifted to the parameter vector a by means of the transposed matrix T with the result Ψw (Φ) = C exp a(w, Φ) · u − Γ(w)u2 /2 , Ψ∗w (Φ0 ) = C ∗ exp a∗ (w, Φ0 ) · u − Γ∗ (w)u2 /2 , u = u(0), (49) where the star denotes the complex conjugation and a(w, Φ) = T (Φ) a(w).

(50)

The explicit form of a(w, Φ) is, with Φ0 = 0 and ρ0 defined in (17), a(w, Φ) = [ρ0 /f ∗ (w)] {cos(Φ) − i ν sin(Φ), i ν cos(Φ) + sin(Φ), cos(Φ) + i ν sin(Φ), i ν cos(Φ) − sin(Φ)} .

4.3

(51)

Normalization

The normalization constant C is calculated by setting in (47) the observable O ≡ 1 . The corresponding scalar product is written, at first, in the metric of polar coordinates and, then, transformed by means of (3): hψw |ψw i = C

2

2π 0

Z

dΦ Z0 2π

0

Z

dr dθ dϕ dΦ r2 sin(θ) exp A · u − ΓR u2 , ∞

dΦ du1 du2 du3 du4 (8u2 ) exp A · u − ΓR u2 , 0 −∞ Z 2π Z ∞ 2 ∂ 0 = −8C dΦ du1 du2 du3 du4 exp A · u − ΓR u2 , ∂ΓR 0 −∞ ∗ A = a (w, Φ0 ) + a(w, Φ), ΓR = Γ0 h(w). (52) = C

2

Z

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The u integrations are Gaussian and immediately give rise to ∂ π2 1 = −8|C| ∂ΓR Γ2R 2

Z

2π

dΦ0 exp [A · A/ (4ΓR )] .

(53)

0

It is straightforward to show that A · A/(4ΓR ) = [k0 + k1 cos (Φ0 − Φ)] /ΓR ,

(54)

where k0 and k1 are defined in (60d) below. The Φ0 integral in (53) does not depend on Φ and can be expressed by the zero order modified Bessel function I0 [8], k0 1 k1 ∂ exp I0 1 = −16π C 2 ∂ΓR ΓR Γ ΓR R k0 1 k1 k1 3 2 = 16π C exp [2ΓR + k0 ] I0 + k1 I1 4 ΓR ΓR ΓR ΓR 3 r = 16π 3 C 2 exp [Kκ0 ] 03 {[2 + Kκ0 ] I0 (Kκ1 ) + Kκ1 I1 (Kκ1 )} , (55) K 3

2

where in the last equation we set ΓR = K/r0 . In macroscopic cases, the argument Z := Kκ1 1, so we use the asymptotic approximation of the Bessel functions [8] as exp[Z] i1 (ν) i2 (ν) −3 Iν (Z) → √ + + O(Z ) , 1+ Z Z2 2πZ

ν = 0, 1, 2, . . . , (56)

where, for ν = 0, 1, 2 which is needed further below, i1 (0) = 1/8, i2 (0) = 9/128, i1 (1) = −3/8, i2 (1) = −15/128, (57) i1 (2) = −15/8, i2 (2) = 105/128, i1 (3) = −35/8, i2 (3) = 945/128. The normalization condition now reads, including the first order correction with respect to 1/K, √ 1 = C 2 8 2 r03 κ21 (κ0 + κ1 ) [π/(κ1 K)]5/2 exp [K (κ0 + κ1 )] (1 + δn/K), (58) δn = (κ0 + 13κ1 ) [8κ1 (κ0 + κ1 )]−1 , K = κ h(w), (59) where κ0 and κ1 are defined in (60f) below. At the end of Appendix A it is shown, that the zero order result above, with the δn term neglected, agrees with result (A13) of the 3D integration.


4.4

901

List of formulas for the hyperbolic case

For e > 1, we give a list of functions and symbols used above and further below. f (w) = cosh(w) − i γ0 sinh(w). (60a) −1 ∗ h(w) ≡ (f (w)f (w)) . (60b) p γ0 = (e + 1)/(e − 1)/ν. (60c) 2 2 (60d) k0 = r0 ΓR κ0 , k1 = r0 ΓR κ1 . 2 2 2 K1 = (e − 1) cosh (w) + (1 + e)ν sinh (w), K2 = (e − 1)ν 2 cosh2 (w) + (1 + e) sinh2 (w). (60e) 2 2 2 ∗ κ0 = (1/4)(1 − ν ) f (w) + (f (w)) , κ1 = (1/2)(1 + ν 2 )/h(w), κ0 + κ1 = [e cosh(2w) − 1]/(e − 1), κ2 = 2κ0 (1 + ν 2 )/(1 − ν 2 ), κ3 = 2κ1 (1 − ν 2 )/(1 + ν 2 ), κ4 = (1/(4Γ0 )) (1 + ν 2 ) (ΓI + i ΓR )f 2 (w) + (ΓI − i ΓR ) (f ∗ (w))2 , 1 + γ02 2 κ5 = (1/(2Γ0 ))ΓI (1 − ν 2 )f (w)f ∗ (w) = (ν − 1) sinh(2w), 4γ0 r 2 e+1 2 ∗ sinh(2w). (60f) κ6 = i ν f (w) − (f (w)) = 2 e−1 IA (Γ) = exp [(A · A)/(4Γ)] = exp [(k0 + k1 cos(Φ0 − Φ))/Γ] . (60g) K = r0 ΓR = κh(w), κ = r0 Γ0 . (60h) 2 2 −1 g = ΓI /ΓR = −(1 + γ0 )(4νγ0 ) κ6 . (60i) −1/2 Zw = [e cosh(2w) − 1]−1 , Ze = e2 − 1 . (60j)

5

Mean values and Kepler orbits

In the following, we list the main results: In subsection A the mean components xi , i = 1, 2, 3, of the hyperbolic orbit vector r together with the mean square deviations (∆xi )2 = x2i − xi 2 ; in B the mean velocity components vi and their mean square deviations (∆vi )2 = vi2 − v 2i ; in C the uncertainty products at w = 0; in D the mean Hamiltonian. At the curve parameter w = 0, where r(0) = (r0 , 0, 0) and v(0) = (0, v0 , 0), the uncertainty products turn to be close to or equal to the quantum mechanical minimum, see subsection D. The mean values are expressed in terms of the functions defined in (60). The results are asymptotic approximations for K 1. The eccentricity e is larger 1, and the semi-major axis a obeys the relation a = r0 (e − 1). For derivations, we refer to Appendix B and C in the case of position and velocity mean values, respectively.

902

5.1


Mean values of position variables

We write down the mean values of the cartesian components, including first order corrections with respect to K, as x = hxi0 [1 + ξ/K + O 1/K 2 ], y = hyi0 [1 + η/K + O 1/K 2 ], z = 0, (61) where, in terms of the κi defined in (60f), hxi0 = (1/2)r0 (κ2 + κ3 ) = a[e − cosh(2w)], √ hyi0 = (1/2)r0 κ6 = a e2 − 1 sinh(2w), 4κ1 κ2 − κ0 κ3 + 3κ1 κ3 , ξ = 2κ1 (κ0 + κ1 )(κ2 + κ3 ) η = 2/(κ0 + κ1 ).

(62)

We remark that the result z = 0 comes out without approximation. The mean square deviations result as, for the definitions of K1 and K2 see (60e), 2r02 e cosh(2w) − 1 K2 [1 + O(1/K)] , κ (e − 1)2 ν 2 = (∆x)2 , 2r02 K1 K2 = [1 + O(1/K)] . κ (e − 1)2 ν 2 (1 + ν 2 )

(∆x)2 = (∆y)2 (∆z)2

5.2

(63)

Mean values of velocity variables

The mean velocity components come out as vx = hvx i0 1 + ξv /K + O(1/K 2 ) , vy = hvy i0 1 + ηv /K + O(1/K 2 ) , vz = 0,

(64)

with r

e − 1 ~κν sinh(2w) , e + 1 2mr0 e cosh(2w) − 1 ~κν (e − 1) cosh(2w) , = = 2mr0 e cosh(2w) − 1 3κ1 (κ4 + κ5 ) + κ5 (κ0 + κ1 ) = − , 2κ1 (κ0 + κ1 )(κ4 + κ5 ) = −3[2(κ0 + κ1 )]−1 .

hvx i0 = − hvy i0 ξv ηv

(65)

903


From the initial condition hvy i0 = v0 at w = 0, one can substitute for ~κν the term 2mr0 v0 so that, to leading order in κ, the velocity components are the classical expressions without bearing on quantum mechanics through ~. For comparison, the velocity vector may be also derived from the eccentricity (Runge-Lenz) vector by making use of r = rm according to (40) and from the mean position components given in (62). For the mean square deviations of the velocity components, we find 2

(∆vx )

2

= κ F0 (e − 1)

Zw3

3 X

c2j cosh(2jw) [1 + O(1/K)] ,

j=0

(∆vy )

2

=

κ F0 Zw3 (e

− 1)/(e + 1)

3 X

d2j cosh(2jw) [1 + O(1/K)] ,

j=0

(∆vz )2 = κ F0

1)Zw2

(e − [f0 + f4 cosh(4w)] [1 + O(1/K)] , (1 + ν 2 )(e + 1)

(66)

where F0 c0 d0 d4 f0

5.3

= = = = =

~2 /(16m2 r02 ), −2(1 + ν 2 ), c2 = e − 1 + (e + 1)ν 2 , c4 = 0, c6 = e + 1 + (e − 1)ν 2 , 2 1 + e + ν 2 (1 − e) , d2 = ν 2 − 1 − 3e(1 + ν 2 ) + 2e3 (1 + ν 2 ), −2e(1 + e − ν 2 + eν 2 ), d6 = 1 + e + (e − 1)ν 2 , e2 (1 + ν 2 )2 − 1 − 6ν 2 − ν 4 , f4 = e2 (1 + ν 2 )2 − (ν 2 − 1)2 . (67)

Uncertainty products at initial time

The uncertainty products (∆xi )2 (∆vi )2 = ~2 /(4m2 ) Pi ,

i = x, y, z,

(68)

have to obey the quantum mechanical inequality Pi ≥ 1. The initial state, corresponding to w = 0, is connected with a nearly minimum uncertainty which depends on the disposable constant ν > 0. One easily derives from (63) and (66) Px (w = 0) = Py (w = 0) = 1 + ν 2 , Pz (w = 0) = 1. (69) As a check of the analytical results, we will prove in Appendix E.4 that Pi ≥ 1 in the three-dimensional parameter space (w, e, ν).

5.4

Mean Hamiltonian

The mean energy reads E = H = (m/2)(vx2 + vy2 ) − α1/r,

(70)

904


where α is the interaction constant. To leading order, see Appendix C, 2 2 = v vx,y x,y [1 + O(1/K)] ,

(71)

and by the mean value Lemma (43), 1/r = (1/rm ) [1 + O(1/K)] ,

(72)

where rm is stated in (40). With the aid of (65) and (40), we find C1 + C2 cosh(2w) e−1 [1 + O(1/K)] , 2 8(e + 1)mr0 e cosh(2w) − 1 = ~2 κ2 ν 2 − 8(1 + e)αmr0 , C2 = e~2 κ2 ν 2 .

E = C1

6

e > 1,

(73)

Disposing of open parameters and classical limit

In this section, we will fix the half width parameter Γ0 from the initial data and from energy conservation. We also relate the magnitude parameter κ to the angular momentum and determine the eccentricity e in terms of the initial magnitudes r0 , v0 . The still open parameters are Γ0 , which has the dimension of a reciprocal length and characterizes the half width of the wave function ψ0 and, furthermore, the number ν > 0. The parameter Γ0 appears through the combination κν = r0 Γ0 ν ≡ νK/h(w).

(74)

From the initial condition on vy , we can fix κν with the aid of (65): v0 ≡ vy (w = 0) = ~κν/(2mr0 ) [1 + O(1/K)] ,

(75)

which gives rise to κν = 2(mr0 v0 )/~ [1 + O(1/K)] .

(76)

In the result (65) for the mean velocities, the factors ~κν can be eliminated in terms of v0 by means of (76). As a consequence, the arbitrary constant ν does not enter the mean values of the Kepler orbit; however, it does affect the mean square deviations, as can be seen from (63) and (66). As a further condition on κν, we require that the mean energy E does not depend on the curve parameter w to leading order: E = α/(2a) [1 + O(1/K)] ,

a = r0 /(e − 1), e > 1.

(77)

905


As is easily verified, condition (77) together with the expression (73) for E is fulfilled, if p (78) κν = (2/~) (e + 1)mr0 α [1 + O(1/K)] . As a consequence of (78), we can write the coupling parameter α of the potential, α/r, as follows α = ~2 κ2 ν 2 [4(e + 1)mr0 ]−1 [1 + O(1/K)] ,

(79)

which tells that the energy E is of leading order κ2 . With (76) and (78), we have two expressions for the product κν. For given initial data r0 and v0 , we now get the following equation for the eccentricity e: e + 1 = mr0 v02 /α [1 + O(1/K)] .

(80)

From the first order corrections in (76) and (77), the eccentricity e can be determined, in principle, to next higher order. For simplicity, we consider the symbol e being correspondingly re-normalized without further specification. In order to see the connection between the formula (80) for e and a standard formulation of celestial mechanics, one detects the angular momentum in (80), L = mr0 × v0 = m(r0 , 0, 0) × (0, v0 , 0) = mr0 v0 ˆ z,

(81)

so we can write e + 1 = Lv0 /α [1 + O(1/K)] .

(82)

On the other hand, a typical textbook result reads, see e.g. Eq. (15.4) of [14], p (83) e = 1 + 2EL2 /(mα2 ). After squaring (83), substituting α(e−1)/(2r0 ) for E, and dividing by (e−1) > 0, we recover our result (80). We remark that the textbook formula (83) is consistent with the sign switch in the connection r0 = |1 ∓ e|a at the initial speed v0 = vc of a circle orbit. To see this, one has to insert into (83) L = mr0 v0 together with the energy E = −α/(2r0 )(1 − e) and E = −α/(2r0 )(e + 1) for v0 > vc and v0 < vc , respectively; see also Sec. IX. We have also checked the sign switch by means of numerical integration of the classical equations of motion at constant r0 and for varying v0 . As to the dynamical magnitude parameter K = κh(w), the function h(w) ≤ 1 with h(0) = 1, see (60b). If w 1, then h(w) → 4 exp[−2w]/(1 + γ02 ) gets exponentially small. We have assumed that K 1 which, of course, cannot be maintained for arbitrary large curve parameters w. Let us assume that h(w) is not too far away from the value 1 and that the still open parameter ν is of the order 1. Then, large K is equivalent to large κ and by (76) amounts to κ ≈ mr0 v0 /~ 1,

or L ~.

(84)

906


In words, the classical limit amounts to the orbital angular momentum L being large as compared to ~ or, equivalently, to the quantum number l 1 with L = ~l(l + 1). For large distances r from the force center, the initial probability density |ψ0 |2 , see (21), decreases with the exponent (−Γ0 r) = (−r/∆κ ), where the halve width ∆κ = r0 /(r0 Γ0 ) ≡ r0 /κ. We now see that the half width essentially is fixed by the angular momentum L and decreases proportional to 1/L.

7 7.1

Time dependence From mean values

We introduce the time t from the classical property that the mean velocity is the time derivative of the mean position vector. The mean values are taken to leading order in K. In view of the classical behavior, we assume that the t dependence is through the curve parameter w = w(t). Both the x and the y component should lead to the same scalar function w(t), as they actually do. From the relations dhxi0 = hvx i0 [1 + O(1/K)] , dt

dhyi0 = hvy i0 [1 + O(1/K)] , dt

(85)

and from the mean values defined in (62) and (65), one derives the same differential equation for W ≡ 2w: dW N = , dt e cosh(W ) − 1

N=

~κν (e − 1)3/2 √ , 2mr02 e+1

e > 1.

(86)

Integration leads to the standard result for hyperbolic orbits N (t − t0 ) = W0 − W − e [sinh(W0 ) − sinh(W )] .

(87)

It remains to be shown that N agrees with the standard expression of celestial mechanics, p N = α/(ma3 ), (88) see, for instance, Eqs. (4.101) and (4.97) in [21]. To obtain (88), we use (78) to eliminate the factor ~κν in the expression of N in (86). Then, substitution of r0 by r0 = (e − 1)a leads to (88). To next higher order, if the correction terms of (62) and (65) are included, different functions W = W (t) emerge for the x and y component, respectively. This can be rather easily seen by setting the disposable constant ν = 1, which significantly simplifies the coefficients κi as a function of w. As a consequence, the assumption that time dependence exclusively enters via the curve parameter w, can be only maintained in the classical limit K → ∞.

907


7.2

Solving the Schr¨ odinger equation

In the following, we show that the time dependence of w, as derived above, is consistent with the time dependent Schrödinger equation under certain conditions, which are: (i) keeping only terms to leading order in K, (ii) taking the distance √ variable r close to the maximum of probability density with r = rm + δr/ K; the angle variables θ and ϕ will not be restricted. As more technical assumptions, we take h(w) of order 1 which allows us to equivalently use the magnitude parameter κ or K; furthermore, we set the constant ν = 1, which will significantly simplify intermediate formulas. Partial derivatives will be indicated by a suffix. The task is to solve the Schrödinger equation i ~ ∂t ψ = Hψ

(89)

for the initial state at t = 0 ψas = 2πCR (w)

p 2π/U0 exp [i c(w) − Γ0 r/2 + U0 ] ,

(90)

where, as compared to (25), we wrote the normalization constant C(w) = CR (w) exp[i c(w)] in terms of modulus CR > 0 and real phase c(w); the latter will serve as a disposable function. For the time dependent state, we adopt the asymptotic form (31) (neglecting 1/K corrections) p (91) ψ = CR (w) 2π/U exp [i c(w) − Γ(w)r/2 + U ] with the assumption that time dependence enters as w = w(t),

w(0) = 0.

(92)

It should be noticed that f (0) = f ∗ (0) = 1, which implies that U (t = 0) = U0 , and as a consequence, the time dependent state (91) reduces to the initial state (90). For the left hand side of (89) we proceed as follows: ψt = wt ψw , ψw = [i c0 + CR0 /CR + (f ∗ )0 /(2f ∗ ) − U (f ∗ )0 /f ∗ − (1/2)r Γ0 ] ψ,(93) where prime denotes the partial derivative with respect to w. For ν = 1, we find from the normalization condition (58) to leading order in K CR0 /CR = −(2/3) [f 0 /f + (f ∗ )0 /f ∗ ] .

(94)

In order to project out of (93) the leading order with respect to κ, we take into account that both Γ and U are proportional to Γ0 = κ/r0 with U = U0 /f ∗ , see (21). Since the quotient f 0 /f is of order 1, we obtain ψw /ψ = [i c0 − U (f ∗ )0 /f ∗ − (1/2)r Γ0 ] [1 + O(1/κ)] .

(95)

908


As to the right hand side of the Schrödinger equation, we find to leading order in κ: ∆ψ/ψ = Ur2 + (1/r2 ) Uθ2 + Uϕ2 / sin2 (θ) − Γ(w)Ur + (1/4)Γ2 (w) [1 + O(1/κ)] . (96) From (21) and (22), we infer the partial derivatives of U with the result (∆ψ)/ψ = −U/(2r) Γ(w) + (1/4)Γ2 (w) [1 + O(1/κ)] ,

(97)

where we used the fact that Ur2 + (1/r2 ) Uθ2 + Uϕ2 / sin2 (θ) = 0.

(98)

Thus, we obtain to leading order in κ ~2 ∆ψ α ~2 U 1 2 ~2 κ2 Hψ =− − =− − Γ(w) + Γ (w) − , (99) ψ 2m ψ r 2m 2r 4 4(e + 1)mr0 r where, for the potential term (−α/r), we used the relation (79) with ν = 1. The Schrödinger equation now amounts to the condition S ≡ SR + i SI = i ~ wt [i c0 − U (f ∗ )0 /f ∗ − (1/2)r Γ0 ] − [Hψ]/ψ. (100) We have the two real unknown functions wt and c0 to solve the two equations SR = 0 and RI = 0. To split S into real and imaginary parts, we introduce the following abbreviations U = UR + i UI ,

Γ = ΓR + i ΓI ,

Γ0 = Γ0R + i Γ0I ,

(f ∗ )0 /f ∗ = FR + i FI (101)

and obtain SI = −~/(4mr) [~A + 2mrBwt ] , A = UI ΓR + ΓI UR − rΓI ΓR , B = 2 (FR UR − FI UI ) + rΓ0R ; (102) SR = ~ [8(1 + e)mrr0 ]−1 [C wt + D wt c0 + E] , (103) 0 C = 4(1 + e)mrr0 [rΓI + 2UI FR + 2UR FI ] , D = −8(1 + e)mrr0 , E = ~ (1 + e)r0 2UI ΓI − 2UR ΓR + r Γ2R − Γ2I + 2κ2 . From (102), we derive ~ A ~κ wt = − = 2mr B 4mr0 r

r

e−1 . e+1

(104)

909


Eq. (104) is based on the following auxiliary formulas where we use the abbreviations (60j) for Zw and Ze : h√ i √ UR = cU e − 1 cos(ϕ/2) cosh(w) + e + 1 sin(ϕ/2) sinh(w) , i h√ √ e − 1 sin(ϕ/2) cosh(w) − e + 1 cos(ϕ/2) sinh(w) , UI = cU p p cU = κ Zw r/r0 (e − 1) sin(θ); (105) √ ΓR = (κ/r0 ) Zw (e − 1), ΓI = −(κ/r0 ) Zw e e − 1 sinh(2w); (106) 0 2 ΓR = −2(κ/r0 ) Zw e(e − 1) sinh(2w), Γ0I = −2(κ/r0 ) Zw2 Ze e(e − 1) [e − cosh(2w)] ; (107) FR = e Zw sinh(2w), FI = Zw /Ze . (108) To evaluate the quotient A/B, we write A = A0 + A1 cos(ϕ/2) + A2 sin(ϕ/2) and B = B0 + B1 cos(ϕ/2) + B2 sin(ϕ/2). Then, it turns out that A0 /B0 = A1 /B1 = A2 /B2 = −κ/(2r0 ) Ze (e − 1),

(109)

which implies that A/B = A0 /B0 . The dependence on the angles θ and ϕ has dropped out completely. With wt known, we find c0 (w) from SR = 0. With the aid of the auxiliary formulas (105) to (108), one can prove that in SR the coefficients of UR and UI vanish. Thus, after inserting the result (104) into (103), we obtain h i p c0 = 1/(2κ)Ze −(1 + e)Γ2I rr0 + 2κ2 + (1 + e)r r0 Γ2R + (e − 1)/(e + 1)Γ0I κ . (110) Further evaluation with the aid of (106) and (107) leads to c0 = (κ/2) Ze [(1 − e)(r/r0 ) + 2] .

(111)

The results (104) for wt and (111) for c0 are now approximated by confining r to the neighborhood of the probability maximum by setting √ (112) r = rm + δr, = 1/ K. With the aid of (40), we arrive at h √ i 1 N1 wt = ( 1 + O(1/ K) , 2 e cosh[2w) − 1

N1 =

~κ (e − 1)3/2 √ , 2mr02 e+1

which is the result (86) for ν = 1 and W = 2w. Furthermore, h √ i c0 (w) = −(κ/2) Ze (e cosh(2w) − 3) 1 + O(1/ K) ,

(113)

(114)

which after integration, with the initial value c(0) = 0 and to leading order, gives the result c(w) = κZe [(3/2)w − (e/4) sinh(2w)] . (115)

910

8


Numerical examples

Let us consider an artificial satellite with mass m = 103 kg and initial data r0 = 4 × 104 km and v0 = 5000 m/s. One finds from (76), with ν = 1, the parameter κ ≈ 1048 which, for the probability density |ψ0 |2 , gives rise to the half width ∆κ ≡ r0 /κ ≈ 10−41 m and to the angular momentum quantum number l ≈ 1024 .

8.1

Escape velocity

In the gravitational case with α = GmM , where M = 5.979 × 1024 kg is the earth mass and G = 6.673 × 10−11 m3 kg−1 s−2 the gravitational constant, the eccentricity, as calculated from (80), is given by e = r0 v02 /(GM ) − 1 = 1.5064. By (80) and at the earth surface with r0 = RE = 6.37812 × 103 km, the parabolic limit with e = 1 comes out at the velocity v0 = 11.185 km/s, which is the well known escape velocity from the earth surface.

8.2

Quantum diffusion

For the given example, we use (88) to calculate the time constant T = 1/N ≈ 10 hour. We introduce the dimensionless time ζ as p (116) ζ = t/T, T = ma3 /α. The hyperbolic ”Kepler equation” (87), with t0 = 0 and W0 = 0, is then approximately solved by W ≈ log[(2/e)ζ],

ζ 1.

(117)

Let us estimate the mean square deviation (∆x)2 = (∆y)2 for large time numbers ζ. We use the properties sinh[W ] → cosh(W ) → (1/2) exp[W ] → ζ/e,

K2 → ζ.

(118)

e > 1, ζ 1.

(119)

From (63) we derive for ν = 1: (∆x)2 →

r0 ~ ζ 2 [1 + O(1/ζ)] , mv0 (e − 1)2

Asking for a mean fluctuation |∆x| of one meter, we obtain for the above example ζ1 ≈ 1016 which approximately corresponds to 2 × 1013 year. We have to check, whether the dynamic magnitude parameter K = κh(w) is still large for ζ1 . With ν = 1, one finds from (60b) and (60a) that h(w) → (e − 1)/ζ1 > 10−18 and, thus, K = κh(w) > 1030 1, which evidently is within the classical approximation.

911


8.3

Parabolic limit

For the parabolic limit e → 1, we have to scale W = order with respect to (e − 1) → 0, (87) amounts to ζp = W 0 + (1/6)(W 0 )3 ,

ζp = t/Tp ,

√

e − 1 W 0 . To lowest

3/2

Tp = r0

p m/α

(120)

which for large ζp is approximated by W 0 ≈ (6 ζp )1/3 ,

ζp 1.

(121)

In the case of the satellite example above (v0 has to replaced by the initial speed p 2α/(mr0 ) for a parabolic orbit), one finds Tp ≈ 3.5 hour. Furthermore, with the aid of (63) and after taking the limit e → 1, a mean square deviation of 1 m would be observable after about 1021 year.

9

Analytic continuation to elliptic orbits

In the following, we continue the curve parameter W ≡ 2w into the complex plane by the map √ (122) W = e − 1 W 0 , W 0 ∈ R, e ≥ 0. The continuation proceeds from the hyperbolic to the elliptic side. We assume that the initial (apsidal) point P0 is fixed together with its distance r0 from the force center, which is origin of our cartesian coordinate system. The initial velocity is perpendicular to the apsidal line. As was mentioned in Sec. I, the connection between r0 and the semi-major axis a are r0 = a|e − 1| if v0 ≥ vc , where vc is the initial speed for a circular orbit, and r0 = a(e + 1) if v0 < vc . By (80), the speed vc , where e = 0, is given by vc2 = α/(mr0 ).

9.1

(123)

Elliptic orbits, Kepler’s equation, energy conservation

We will confine ourselves, at first, to initial speeds with v0 ≥ vc , where P0 coincides with the peri-center. We start from (32) and use a = r0 /(e − 1) to write, for e > 1 and W 0 ∈ R, √ √ r0 √ 2 r0 e − cosh e − 1 W 0 , y = e − 1 sinh e − 1 W 0 . x= e−1 e−1 (124) The above equations, differing by a re-scaled curve parameter, describe the same hyperbola as (32), of course. Now, let us assume the eccentricity in the

912


interval 0 ≤ e < 1, keeping W 0 real. √Then, since cosh(iW ) = cos(W ) and sinh(iW ) = i sin(W ), where now W = 1 − e W 0 and W 0 real, we obtain: √ x = a [cos (W ) − e] , y = a 1 − e2 sin (W ) , a = r0 /(1 − e), (125) which is the parameter representation of an ellipse. In the limit e → 1, one finds the same parabola both from the hyperbolic and the elliptic side, namely √ x = r0 (1 − W 02 /2), y = r0 2 W 0 , or x = r0 1 − y 2 /(4r02 ) . (126) The above continuation also produces Kepler’s equation for elliptic orbits. To see this, we write (88) in the form s r α e − 1 α(e − 1) 1 = , e > 1. (127) N≡ a ma r0 mr0 Going to the elliptic region with 0 ≤ e < 1, we obtain s r 1 − e α(1 − e) α 0 0 , a = r0 /(1 − e). N → −i N , N = = r0 mr0 ma3

(128)

In (87), the imaginary unit cancels out and, setting W0 = 0 at t0 = 0, we obtain Kepler’s equation N 0 t = W − e sin(W ),

0 ≤ e < 1.

(129)

For the mean distance rm and the mean velocity components hvx,y i0 which are given in (40) and (65), respectively, we find the analytically continued expressions, to leading order in K, rm = a [1 − e cos(W )] , r 1−e sin(W ) hvx i0 = −v0 , 1 + e 1 − e cos(W ) a = r0 /(1 − e),

0 ≤ e < 1,

cos(W ) hvy i0 = v0 (1 − e) , 1 − e cos(W ) √ (130) vc ≤ v0 ≤ 2 vc .

From (130), we check the energy conservation to leading order in K: (131) E = (m/2) hvx i20 + hvy i20 − α/rm = −α/(2a). Setting a = r0 /(1 − e), we obtain E + α(1 − e)/(2r0 ) = F1 F2 , F1 = (1 − e)Zw (1 + e cos(W )) [8mr02 (1 + e)]−1 , F2 = 4(1 + e)mr0 α − (~κν)2 . (132)

913


The above balance vanishes identically in W , if F2 = 0 which is equivalent to (80) with (76) and gives the same result for the eccentricity: e = mr0 v02 /α − 1.

(133)

Obviously, in order√that 0 ≤ e < 1, the initial speed must be confined to the interval vc ≤ v0 < 2 vc . What happens if v0 < vc ? Then, the eccentricity goes through zero, and we obtain the continued components from (125) and (130) by changing, at fixed r0 , e into −e which leads to √ x = a(cos(W ) + e), y = a 1 − e2 sin(W ), rm = a [1 + e cos(W )] , r sin(W ) cos(W ) 1+e , hvy i0 = v0 (1 + e) hvx i0 = −v0 1 − e 1 + e cos(W ) 1 + e cos(W ) a = r0 /(1 + e), 0 ≤ e < 1, v0 < vc . (134) As a check, the energy conservation is fulfilled with E+

α(1 + e) (1 + e) (e cos(W ) − 1) , = G1 G2 , G1 = 2r0 8mr02 (e − 1) (e cos(W ) + 1) G2 = 4(e − 1)mr0 α + (~κν)2 .

(135)

The factor G2 vanishes, if e = 1 − (~κν)2 /(4mr0 α) = 1 − mr0 v02 /α,

v0 ≤ vc .

(136)

At zero speed, where e = 1, the classical orbit is rectilinear with the angular momentum L = 0.

9.2

Analytic continuation of the mean square deviations

We substitute w → i w or, equivalently, W → i W . As compared to the definition of mean square deviations by means of expectation values, which cannot be negative by definition, the derivation by analytic continuation should be checked for positivity in the eccentricity interval 0 ≤ e < 1. In the following, we examine the mean square deviations of the position components; for the velocity components we refer to Appendix E. From (63), one infers for 0 ≤ e < 1: 2r02 e cos(2w) − 1 (K2 )w→i w [1 + O(1/K)] , κ (e − 1)2 ν 2 = (∆x)2 , 2r02 (K1 K2 )w→i w → [1 + O(1/K)] . κ (e − 1)2 ν 2 (1 + ν 2 )

(∆x)2 → (∆y)2 (∆z)2

(137)

914


In the interval 0 ≤ e < 1, the factor e cos(2w) − 1 in the expression of (∆x)2 is negative. It remains to be shown that (K2 )w→i w < 0, which immediately follows from the definition in (60e), provided 0 ≤ e < 1. Also (K1 )w→i w < 0 is easily verified, so that the mean square deviations of all three components x, y, z are larger than zero in the elliptic region. From (66) we derive the following analytic continuations of the mean square deviations of the velocity components, if 0 ≤ e < 1, 2

(∆vx )

2

(∆vy )

3 X (1 − e)2 c2j cos(2jw) [1 + O(1/K)] , → −κF0 [1 − e cos(2w)]3 j=0 3 X 1−e → κF0 d2j cos(2jw) [1 + O(1/K)] , (e + 1)[1 − e cos(2w)]3 j=0

(∆vz )2 → −κF0

1−e f0 + f4 cos(4w) [1 + O(1/K)] . (1 + ν 2 )(e + 1) [1 − e cos(2w)]2

(138)

In Appendix E, we prove that the above mean square deviations, which were obtained by analytic continuations, are positively definite in the three-dimensional parameter space (w ∈ R, 0 ≤ e < 1, ν > 0).

10

Conclusions

We have demonstrated that the Schrödinger equation of the hydrogen atom, in the classical limit, includes as solutions the Kepler orbits. The elliptic, hyperbolic, and parabolic orbits are obtained from a single wave function in the limit of large quantum numbers of the angular momentum; exceptions are rectilinear and nearby orbits with small angular momentum. The paper unifies and supplements existing results which were based on different methods for elliptic and hyperbolic orbits, respectively. The properties of elliptic orbits were obtained by analytical continuation of the results of the hyperbolic case. The orbits evolve depending on a curve parameter w which, in the elliptic case, corresponds to half of the eccentric anomaly. At zero time, the initial wave function √ is characterized by a nearly minimum uncertainty product, up to the factor 1 + ν 2 , where ν > 0 is a disposable constant. Time is introduced by the assumption that it enters only through the curve parameter as w = w(t). This assumption is justified in the classical limit. The orbits are subject to quantum diffusion which, however, is hardly observable for macroscopic examples like artificial satellites within realistic time spans. Appendix A: Proof of mean value lemma (43) The following proof is based on the series expansion of G2 according to (34), (35), and (45). Since D does not depend on r, by substituting r = rm + δr

915


with 2 = 1/K, we derive from (34): G2 = KD2 (θ, ϕ) − Er δr2 + O(δ),

Er = 1/(4r0 rm ).

(A1)

From (22), (34), and (60a), we obtain, by using θm = π/2, D(θm , ϕ) = cos(ϕ/2) cosh(w) + γ0 ν sin(ϕ/2) sinh(w).

(A2)

The explicit forms of E0 = D2 (θm , ϕm ) and Er are stated in (46). Next, we expand with respect to θ and ϕ up to second order in . To this end, we make use of the following relations: √ 2 [cos(ϕ/2) + i ν sin(ϕ/2)] , d(θm , ϕ) = ∂θ d(θm , ϕ) = ∂θ ∂ϕ d(θ, ϕ)θ=θm = 0, √ ∂ϕ d(θm , ϕ) = (1/ 2 [i ν cos(ϕ/2) − sin(ϕ/2)] , ∂θ ∂θ d(θm , ϕ) = (1/8) (1 + ν 2 ) cos(ϕ) + 2i ν sin(ϕ) / [∂ϕ ∂ϕ d(θm , ϕ)] , √ ∂ϕ ∂ϕ d(θm , ϕ) = −( 2/4) [cos(ϕ/2) + i ν sin(ϕ/2)] . (A3) Furthermore, by (A2), ∂ϕ D(θm , ϕ)ϕ=ϕm = 0,

(A4)

which is a consequence of (37). From (A1) and the first equation of (46), the expansion coefficients of G2 (rm , θ, ϕ) are determined at (θm , ϕm ) as E0 = D2 , Eθ = −DDθθ , Eϕ = −DDϕϕ ,

(A5)

which, with the aid of (A3) and the conditions (37) and (41), are evaluated to give the results stated in (46). Now, let us explicitly write down the mean value as Z ∞ Z π Z 2π 2π 2 2 F (r) = C dr r dθ sin(θ) dϕ F (r) √ exp[G2 ]. (A6) UU∗ 0 0 0 In view of the definition of U defined by (26) and (21), we introduce the amplitude function F1 , √ √ F1 (r) = β r2 sin(θ) F (r) [r d f (w)d∗ f ∗ (w)]−1/2 , β = 2 2π r0 /K, (A7) and transform the integration variables (r, θ, ϕ) → (δr, δθ, δϕ) according to (45). Then, for large K, the integration limits can be replaced by (−∞, ∞) which amounts to neglect exponentially small terms of order exp[−K]. The exponent G2 of the integrand has the form given in (46) in terms of δr, δθ, δϕ. We expand F1 as F1 (r) = F1 (rm + δr, θm + δθ, ϕ + δϕ) = P F1 (rm , θm , ϕm ), P = 1 + f1 + f2 2 + . . .

(A8)

916


and obtain with the aid of (46) and to leading order in K Z ∞ C2 F (r) = (dδr) (dδθ) (dδϕ) exp [G2 ] F1 (rm , θm , ϕm ) K 3/2 −∞ −1/2 = C 2 π 3/2 K 3 Er Eθ Eϕ F1 (rm , θm , ϕm ) exp[KE0 ],

(A9)

where we used the fact that the linear term of P drops out by parity; the term 2 f2 of P is consumed in the error term of relativ order 1/K. We use (A9) and the definition (A7) of F1 to define the normalization constant C 2 by the following condition −1/2 1 = β C 2 (rm π)3/2 K 3 dm f f ∗ d∗m Er Eθ Eϕ exp[KE0 ],

(A10)

where dm = d(θm , ϕm ). We, thus, arrive at Lemma (43). In order to show the equivalence of (A10) with the normalization condition (58), we further evaluate (A10) with the aid of (38), (40), (41), (46), and the definition (22) of d. We obtain (ν 2 + 1) (e cosh(2w) − 1)2 , 64(e − 1)K1 r02 −1 dm f (w)d∗m f ∗ (w) = 2Zw K1 K2 (e − 1)ν 2 Er Eθ Eϕ =

(A11) (A12)

with K1 and K2 defined in (60e) and Zw in (60j). After inserting (A11) and (A12) into (A10), we obtain the following normalization condition to zero order with respect to 1/K

e cosh(2w) − 1 1 = C exp K e−1 2

16π 3/2 r03 ν (e cosh(2w) − 1) . √ √ √ e − 1 K 5/2 K2 1 + ν 2

(A13)

On the other hand, we have to insert in (58) the explicit expressions of κ0 and κ1 , as defined in (60f), in order to reproduce the 3D method result (A13). Appendix B: Mean values of position variables The mean values will be calculated in 4D space to leading and next higher approximation with respect to K. The integration method is outlined in Sec. IV. B and exemplified in Sec. IV. C. The u dependence of the position and velocity components xi and vi is given in (6) and (12) with (10), respectively. B.1 Mean x component


Z

2π 0

Z

917

∞

(8u2 )du1 ...du4 2(u1 u3 − u2 u4 ) exp A · u − Γu2 −∞ 0 Z ∞ Z 2π ∂2 ∂2 2 ∂ 0 = −16C du1 ...du4 dΦ − exp A · u − Γu2 ∂Γ 0 ∂A1 ∂A3 ∂A2 ∂A4 −∞ Z 2π ∂2 1 ∂2 0 2 2 ∂ − dΦ exp [A · A/(4Γ)] = −16π C ∂Γ 0 ∂A1 ∂A3 ∂A2 ∂A4 Γ2 Z 2π 2 2 ∂ 1 = −4π C dΦ0 [A1 A3 − A2 A4 ] exp [A · A/(4Γ)] , (B1) ∂Γ Γ4 0

x = C

2

dΦ

where the vector A is defined in (51) and (52). We obtain A1 A3 − A2 A4 = Γ2R r0 [κ2 + κ3 cos(Φ0 − Φ)]

(B2)

with κ2 and κ3 defined in (60f). The Φ0 integral in (B1) does not depend on Φ and can be expressed by modified Bessel functions as follows ∂ Γ2 r0 exp [k0 /Γ] R4 [κ2 I0 (k1 /Γ) + κ3 I1 (k1 /Γ)] ∂Γ Γ = C 2 4π 3 r04 K −3 exp[Kκ0 ] {[2(4 + Kκ0 )κ2 + Kκ1 κ3 ] I0 (Kκ1 ) + 2 [Kκ1 κ2 + 4κ3 + Kκ0 κ3 ] I1 (Kκ1 ) + Kκ1 κ3 I2 (Kκ1 )} , (B3)

x = −8C 2 π 3

where, after differentiation with respect to Γ, we replaced Γ by ΓR and then ΓR by K/r0 . We approximate the Bessel functions by the asymptotic form (56) with (57) to obtain the two leading orders with respect to K √ x = 4 2C 2 r04 [π/(Kκ1 )]5/2 exp [K(κ0 + κ1 )] κ21 (κ0 + κ1 )(κ2 + κ3 ) [1 + δξ/K] = (1/2)r0 (κ2 + κ3 ) [1 + ξ/K] , (B4) where, for the latter equation, we made use of the normalization condition (58) and (59) which gives rise to δξ = −

κ0 κ2 + 29κ1 κ2 − 3κ0 κ3 + 25κ1 κ3 . 8κ1 (κ0 + κ1 )(κ2 + κ3 )

(B5)

With (B4), (B5) and by the definition ξ = δξ − δn, we arrive at the result given in (61) and (62). B.2 Mean y component In analogy to (B2), the amplitude function comes out as A2 A3 + A1 A4 = r0 Γ2R κ6 ,

(B6)

918


where κ6 is defined in (60f). The amplitude factor (B6) does not depend on the phase φ = Φ − Φ0 . Analogously to (B4) and (B5), we obtain, after using Γ0 h(w) = K/r0 , √ y = 4 2 C 2 r04 [π/(Kκ1 )]5/2 κ21 (κ0 + κ1 )κ6 exp [K(κ0 + κ1 )] [1 + δη/K] , δη = [κ0 + 19κ1 ] / [8κ1 (κ0 + κ1 )] . (B7) We exploit the normalization condition (58) and (59) to obtain y = (1/2)r0 κ6 1 + η/K + O(1/K 2 ) , η = (δη − δn) = 2/(κ0 + κ1 ),

(B8)

as stated in (61) and (62). B.3 Mean z component In analogy to (B2), the amplitude function reads A21 + A22 − A23 − A24 = 4i r0 Γ2R 2ν/(1 + ν 2 ) κ1 sin(φ).

(B9)

The function κ1 is defined in (60f) and does not depend on the phase φ = Φ−Φ0 . Thus, the phase integral vanishes identically implying z = 0: Z 2π Z k0 + k1 cos(φ) k0 + k1 cos(φ) d ΓR 2π dφ sin(φ) exp dφ exp =− = 0. ΓR k1 0 dφ ΓR 0 (B10)

B.4 Mean square deviation of x Analogously to (B1), we have to evaluate Z 2π Z ∞ 2 0 x2 = C dΦ (8u2 )du1 ...du4 4(u1 u3 − u2 u4 )2 exp A · u − Γu2 0

−∞ 2π

2 ∂2 ∂2 du1 ...du4 dΦ − exp A · u − Γu2 ∂A1 ∂A3 ∂A2 ∂A4 0 −∞ 2 Z 2π ∂2 ∂2 1 2 2 ∂ 0 dΦ − exp [A · A/(4Γ)] = −32π C ∂Γ 0 ∂A1 ∂A3 ∂A2 ∂A4 Γ2 Z 2π 1 2 2 ∂ 0 2 2 2 2 = −32π C dΦ −2A A A A + A A + 2Γ + A A + 2Γ + 1 2 3 4 1 3 2 4 ∂Γ 16Γ6 0 (B11) 2Γ A23 + A24 + 4Γ exp [A · A/(4Γ)] . ∂ = −32C ∂Γ 2

Z

0

Z

∞

After differentiation with respect to Γ, we set Γ = ΓR and substitute for A1 , . . . A4 the explicit expressions which are defined through (51) and (52).


919

With the abbreviation φ = Φ0 − Φ we write x2 = C 2 π 2 r05 /(8K 5 ) exp[Kκ0 ] × Z 2π dφ [c0 + c1 cos(φ) + c2 cos(2φ) + c3 cos(3φ)] exp[Kκ1 cos(φ)], 0 c0 = 512 + K 3 (1 − ν 2 )(1 + ν 2 )2 f 6 + (f ∗ )6 + 192K(1 − ν 2 ) f 2 + (f ∗ )2 + 32K 2 (1 + ν 2 + ν 4 ) f 4 + (f ∗ )4 + 128K 2 (1 + ν 4 ) (f f ∗ )2 + K 3 (9 + ν 2 − ν 4 − 9ν 6 ) f 2 (f ∗ )4 + f 4 (f ∗ )2 , c1 = 2K(1 + ν 2 )f f ∗ 192 + 64K(1 − ν 2 ) f 2 + (f ∗ )2 + K 2 (3 − 2ν 2 + 3ν 4 )× 4 f + (f ∗ )4 + K 2 (9 − 10ν 2 + 9ν 4 ) (f f ∗ )2 , c2 = −2K 2 (f f ∗ )2 −32(1 − ν 2 + ν 4 ) + K(−3 + ν 2 − ν 4 + 3ν 6 ) f 2 + (f ∗ )2 , c3 = 2K 3 (1 − ν 2 )2 (1 + ν 2 ) (f f ∗ )3 . The φ integration leads to modified Bessel functions: Z 2π dφ cos(nφ) exp[Kκ1 cos(φ)] = 2πIn (Kκ1 ), n = 0, 1, . . . .

(B12)

(B13)

0

In view of K 1, we make use of the zero and first order asymptotic approximations (56) and (57) to obtain x2 = F (9 + 16Kκ1 + 128K 2 κ21 ) c0 + (−15 − 48Kκ1 + 128K 2 κ21 ) c1 + (105 − 240Kκ1 + 128K 2 κ21 ) c2 + (945 − 560Kκ1 + 128K 2 κ21 ) c3 , i−1 h √ 5 2 2 5/2 [Kκ1 ]−5/2 exp[K(κ0 + κ1 )]. (B14) F = C (r0 π) 512 2K We now insert the coefficients c0 to c3 from (B12), and, in order to take into account normalization, we divide the right hand side of (B14) by the right hand side of (58). The result is given by the two leading orders in K as x2 = hx2 i0 + (1/K) hx2 i1 + O(1/K 2 ); (B15) 2 −1 2 2 2 ∗ 2 2 ∗ hx i0 = (−r0 ) [64(κ0 + κ1 )] (ν − 1) f + (f ) − 2(1 + ν )f f × 2 (1 + ν 2 ) f 2 + (f ∗ )2 + 2(1 − ν 2 )f f ∗ ; (B16) 2 2 4 ∗ 2 hx i1 = F1 128κ1 (κ0 + κ1 )(3 − ν + 3ν ) (f f ) + 256κ1 (κ0 + κ1 )(1 − ν 4 ) f (f ∗ )3 + f 3 f ∗ + 64κ1 (κ0 + κ1 )(1 + ν 2 + ν 4 )× 4 f + (f ∗ )4 + (ν 2 − 1) 8κ0 (3 + 2ν 2 + 3ν 4 ) + κ1 (69 + 58ν 2 + 69ν 4 ) 4 ∗ 2 f (f ) + f 2 (f ∗ )4 + 3κ1 (ν 2 − 1)(1 + ν 2 )2 f 6 + (f ∗ )6 − 2(κ0 + 4κ1 )(3 + ν 2 + ν 4 + 3ν 6 ) f 5 f ∗ + f (f ∗ )5 − 4(1 + ν 2 ) 8κ1 (3 − 4ν 2 + 3ν 4 ) + κ0 (9 − 14ν 2 + 9ν 4 ) (f f ∗ )3 , −1 F1 = r02 128κ1 (κ0 + κ1 )2 . (B17)

920


We simplify (B16) by expressing f 2 + (f ∗ )2 and f f ∗ in terms of κ0 and κ1 , respectively, see (60f)and (60b); furthermore, we exploit the connections κ2 = const. κ0 and κ3 = const. κ1 specified in (60e) to write hx2 i0 = (r02 /4)(κ2 + κ3 )2 ,

(B18)

which tells that hx2 i0 = (hxi0 )2 , see (62), and that the mean square deviation (∆x)2 is of order 1/K: (∆x)2 = 1/K(∆x)21 + O(1/K 2 ),

(∆x)21 = hx2 i1 − 2(hxi0 )2 ξ,

(B19)

where ξ is defined in (62). With the aid of (B17) and (62), we find (∆x)21 = F1 {32(κ0 + κ1 )(κ2 + κ3 )(κ0 κ3 − 4κ1 κ2 − 3κ1 κ3 )+ (B20) 7 ∗ 2 6 8 ∗ 7 2 4 (5 + 6ν − 6ν − 5ν ) f f + f (f ) + 2(15 + 20ν − 6ν + 20ν 6 + 15ν 8 ) f 6 (f ∗ )2 + f 2 (f ∗ )6 + (75 + 106ν 2 − 106ν 6 − 75ν 8 ) 5 ∗ 3 f (f ) + f 3 (f ∗ )5 + 4(25 + 36ν 2 + 6ν 4 + 36ν 6 + 25ν 8 )f 4 (f ∗ )4 . To simplify (B20), we first substitute κ0 , . . . κ3 in terms of f and f ∗ according to (60f) and obtain 3 (∆x)2 = −2 (F1 /K) (1 + ν 2 )f f ∗ (ν 2 − 1) f 2 + (f ∗ )2 − 2(1 + ν 2 )f f ∗ . (B21) After substituting the function f according to (60a) and γ0 by (41), we find to leading order in K −1 (∆x)2 = (F1 /K) K2 128(1 + ν 2 ) (e − 1)4 ν 2 e cosh(2w) − 1)3 . (B22) We substitute κ0 , κ1 , and K = κh(w) with the result −1 (F1 /K) = (e − 1)2 r02 Zw2 64κ(1 + ν 2 ) ,

(B23)

With (B22) and (B23), we arrive at the mean square deviation of x as stated in (63). B.5 Mean square deviation of y We have to evaluate Z ∞ Z 2π 0 2 2 dΦ (8u2 )du1 ...du4 4(u1 u4 + u2 u3 )2 exp A · u − Γu2 y = C 0

−∞ 2π

2 ∂2 ∂2 dΦ du1 ...du4 + exp A · u − Γu2 ∂A1 ∂A4 ∂A2 ∂A3 0 −∞ 2 Z 2π ∂2 ∂2 1 2 2 ∂ 0 = −32π C dΦ + exp [A · A/(4Γ)] ∂Γ 0 ∂A1 ∂A4 ∂A2 ∂A3 Γ2 Z 2π 1 2 2 ∂ dΦ0 (2A1 A2 A3 A4 + A22 (A23 + 2Γ) + A21 (A24 + 2Γ)+ = −32π C 6 ∂Γ 16Γ 0 2 2Γ(A3 + A24 + 4Γ) exp [A · A/(4Γ)] . (B24) ∂ = −32C ∂Γ 2

Z

0

Z

∞

921


After differentiation with respect to Γ, we set Γ = ΓR and substitute for A1 , . . . A4 the explicit expressions (51) and (52). With the abbreviation φ = Φ0 − Φ, we write Z 2π 2 2 5 5 dφ [c0 + c1 cos(φ) + c2 cos(2φ)] exp[K (κ0 + κ1 cos(φ))], y 2 = C π r0 /(2K ) 0 c0 = 128 + 48K(1 − ν 2 ) f 2 + (f ∗ )2 + 8K 2 (1 + 6ν 2 + ν 4 ) (f f ∗ )2 + 2K 2 (1 − 14ν 2 + ν 4 ) f 4 + (f ∗ )4 + K 3 ν 2 (1 − ν 2 ) f 4 (f ∗ )2 + f 2 (f ∗ )4 + K 3 ν 2 (ν 2 − 1) f 6 + (f ∗ )6 , c1 = −2K(1 + ν 2 )f f ∗ −48 + K 2 ν 2 f 4 + (f ∗ )4 − 2K 2 ν 2 (f f ∗ )2 + (B25) 4K(ν 2 − 1) f 2 + (f ∗ )2 , c2 = 4K 2 (1 + ν 2 )2 (f f ∗ )2 . We integrate with respect to φ and use the asymptotic approximation of the Bessel functions including the first order correction y 2 → F 9 + 16Kκ1 + 128K 2 κ21 c0 + −15 − 48Kκ1 + 128K 2 κ21 c1 + 105 − 240Kκ1 + 128K 2 κ21 c2 , (B26) i−1 h √ exp[K(κ0 + κ1 )] [π/(Kκ1 )]5/2 . (B27) F = r05 C 2 128 2K 5 We now insert the coefficients c0 to c2 from (B25) and, for normalization, divide the right hand side of (B27) by the right hand side of (58). The result is arranged by orders of K with the result y 2 = hy 2 i0 + (1/K) hy 2 i1 + O(1/K 2 ), 2 hy 2 i0 = r02 ν 2 f 2 − (f ∗ ]2 [16(κ0 + κ1 )]−1 (ν 2 − 1) f 2 + (f ∗ )2 − 2(1 + ν 2 )f f ∗ , hy 2 i1 = F1 3κ1 ν 2 (1 − ν 2 ) f 6 + (f ∗ )6 + 2(κ0 + 4κ1 )ν 2 (1 + ν 2 ) f 5 f ∗ + f (f ∗ )5 + 3κ1 ν 2 (ν 2 − 1) f 4 (f ∗ )2 + f 2 (f ∗ )4 − 4(κ0 + 4κ1 )ν 2 (1 + ν 2 ) (f f ∗ )3 + 4κ1 (κ0 + κ1 )(1 − 14ν 2 + ν 4 ) f 4 + (f ∗ )4 + 8κ1 (κ0 + κ1 )(3 + 14ν 2 + 3ν 4 ) (f f ∗ )2 + 16κ1 (κ0 + κ1 )(1 − ν 4 )× −1 f 3 f ∗ + f (f ∗ )3 , F1 = r02 32κ1 (κ0 + κ1 )2 . As it turns out, the zero order term amounts to hy 2 i0 = hyi20 . As a consequence, the mean square deviation (∆y)2 is of order 1/K with (∆y)2 = 1/K(∆y)21 + O(1/K 2 ),

(∆y)21 = hy 2 i1 − 2(hyi0 )2 η,

(B29)

where η is defined in (62). With the aid of (B28) and (62), we find (∆y)21 = F1 3ν 2 (1 − ν 2 ) f 6 + (f ∗ )6 + (2(f 5 f ∗ + f (f ∗ )5 )(κ0 + 4κ1 )ν 2 (1+ ν 2 ))/κ1 + 3(f 4 (f ∗ )2 + f 2 (f ∗ )4 )ν 2 (ν 2 − 1) − 4(f f ∗ )3 (κ0 + 4κ1 )ν 2 × (1 + ν 2 )/κ1 + 4(f 4 + (f ∗ )4 )(κ0 + κ1 )(1 − 14ν 2 + ν 4 ) + 16(f 3 f ∗ + f (f ∗ )3 )(κ0 + κ1 )(1 − ν 4 ) + 8(f f ∗ )2 (κ0 + κ1 )(3 + 14ν 2 + 3ν 4 )− 32(κ0 + κ1 )κ26 . (B30)

(B28)

922


After substituting κ0 , κ1 , and κ6 from (60f), we obtain (∆y)21 = (1/2)r02 (1 − ν 2 )f 2 + 2(1 + ν 2 )f f ∗ + (1 − ν 2 )(f ∗ )2 .

(B31)

With the aid of the relation κ(e − 1)ν 2 /K2 = K, the mean square deviation (∆y)2 = (1/K)(∆y)21 = (∆x)2 as stated in (63). B.6 Mean square deviation of z

z2

We have to evaluate Z ∞ Z 2π 2 0 2 dΦ (8u2 )du1 ...du4 u21 + u22 − u23 − u24 exp A · u − Γu2 = C 0

−∞ 2π

2 2 ∂ ∂2 ∂2 ∂2 ∂ 1 0 = −8π C dΦ + − − exp [A · A/(4Γ)] 2 2 2 2 ∂Γ 0 ∂A1 ∂A2 ∂A3 ∂A4 Γ2 = (π 2 /8)C 2 Q exp [A · A/(4Γ)] , (B32) 2

2

Z

where Q = (1/Γ8R ) A61 + A62 + A63 + 3A43 A24 + 3A23 A44 + A64 − A42 (A23 + A24 − 32ΓR )+ 32A43 ΓR + 64A23 A24 ΓR + 32A44 ΓR + 192A23 Γ2R + 192A24 Γ2R + 512Γ3R + A41 × (3A22 − A23 − A24 + 32ΓR ) + A21 3A42 − A43 − A44 − 2A22 (A23 + A24 − 32ΓR ) − 32A24 ΓR + 192Γ2R − 2A23 (A24 + 16ΓR ) − A22 A43 + A44 + 32A24 ΓR − 192Γ2R + 2A23 (A24 + 16ΓR ) . (B33) We skip some intermediate steps which are analogous to the calculation of the mean square deviations above. The leading order hz 2 i0 = hzi20 = 0 comes out as expected. Thus, the mean square deviation of z equals the following first order correction: (∆z)2 = (1/K)hz 2 i1 (1 + O(1/K)) (B34) with hz 2 i1 = Fz f f ∗ (ν 2 − 1) f 2 + (f ∗ )2 − 2(1 + ν 2 )f f ∗ × 4 (ν − 1) f 2 + (f ∗ )2 − 2(1 + ν 4 )f f ∗ , Fz = r02 [16Kκ1 (κ0 + κ1 )]−1 . Evaluation produces the result as stated in (63). Appendix C: Mean values of velocity variables C.1 Mean x component of velocity

(B35)

923


We calculate the mean value of the velocity operator as defined by (12) and (10). Z Z 2π 0 2 dΦ drdθdϕ r2 sin(θ) exp a∗ (Φ0 ) · u − Γ∗ u2 /2 (1/(2r)) × Dx = C 0 Z 2π [u3 ∂u1 − u4 ∂u2 + u1 ∂u3 − u2 ∂u4 ] dΦ exp a(Φ) · u − Γu2 /2 . (C1) 0

By partial integration, we apply the differential operator to the left hand exponent, which produces a minus sign. The factor (1/r) is combined with the metric factor. According to (3), we transform the metric as dΦ0 (r/2)drdθ sin(θ)dϕ → 4 du1 du2 du3 du4 ,

(C2)

and write ∞ du1 du2 du3 du4 exp a(Φ0 ) · u − Γu2 /2 × Dx = −4C dΦ −∞ 0 [u3 ∂u1 − u4 ∂u2 + u1 ∂u3 − u2 ∂u4 ] exp a∗ (Φ) · u − Γ∗ u2 /2 Z 2π Z ∞ 2 = −4C dΦ du1 du2 du3 du4 [u3 (a∗1 − Γ∗ u1 ) − u4 (a∗2 − Γ∗ u2 ) + 0 −∞ ∗ ∗ u1 (a3 − Γ u3 ) − u2 (a∗4 − Γ∗ u4 )] exp A · u − ΓR u2 , (C3) 2

Z

2π

Z

where the vector A is defined in (52). In the next step, we produce the components ui by derivatives with respect to Ai to be followed by the Gaussian u integrations. With the notation Γ = ΓR + i ΓI , we obtain Z C 2 π 2 2π Dx = −4 2 dΦ Dx (a∗ , A) exp [A · A/(4ΓR )] , ΓR 0 x ∗ 2 −1 D (a , A) = (2ΓR ) {i ΓI (A1 A3 − A2 A4 ) + ΓR (a∗1 A3 − A1 A3 + a∗3 A1 + A2 A4 − a∗2 A4 − a∗4 A2 )} . (C4) We make use of the explicit functions a∗ and A, as given in (51) and (52), to obtain Dx (a∗ , A) = i κ (κ4 + κ5 cos(φ)) ,

φ = Φ0 − Φ,

κ = r0 Γ0

(C5)

with κ4 and κ5 defined in (60f). The phase integral, thus, reads Z 4π 2 C 2 κ 2π Dx = −i dφ [κ4 + κ5 cos(φ)] exp [(k0 + k1 cos(φ))/ΓR ] Γ2R 0 8π 3 C 2 κ = −i exp[κ0 K] [κ4 I0 (κ1 K) + κ5 I1 (κ1 K)] , (C6) Γ2R 5/2 π C 2κ = −i √ 2 exp [(κ0 + κ1 )K] (9 + 16κ1 K + 128(κ1 K)2 )κ4 + 16 2ΓR κ1 K (−15 − 48κ1 K + 128(κ1 K)2 )κ5 [1 + O(1/K)] , (C7)

924


where in the last equation we applied the asymptotic formulas (56) with (57); furthermore, we use (60d), the normalization condition (58), and the definition (41) of γ0 , in order to get the intermediate expression i κ (κ4 + κ5 ) Dx = − 1 + (δvx − δn)/K + O(1/K 2 ) , 2r0 (κ0 + κ1 ) −1 . δvx = 1 + e − ν 2 + ν 4 − eν 4 8κ1 ν 2

(C8)

Using δn of (59), the auxiliary functions defined in (60f), and vx = ~/(i m) Dx , we produce the results as stated in (64) and (65). C.2 Mean y component of velocity In (C4) and (C5), we substitute Dy for Dx with Dy (a∗ , A) = (2Γ2R )−1 {i (A2 A3 + A1 A4 )ΓI + [a∗2 A3 + A2 (a∗3 − A3 ) − A1 A4 + a∗1 A4 + A1 a∗4 ] ΓR } = −(1/2)r0 ν (ΓI + i ΓR )f 2 − (ΓI − i ΓR )(f ∗ )2 . (C9) With the aid of the definitions given in (60) and (41), we obtain Dy = −i κν cosh(2w),

(C10)

which does not depend on the phase φ = Φ−Φ0 . After integration with respect to φ, application of the asymptotic approximation of I0 (κ1 K) and taking into account the normalization (58), we obtain the intermediate result Dy = hDy i0 1 + (δvy − δn)/K + O(1/K 2 ) , Kν 4κ0 1 + γ02 2 hDy i0 = i + 2 2 κ6 , 4r0 (κ0 + κ1 ) 1 − ν 2 4ν γ0 δvy = 1/(8κ1 ),

(C11)

which, in view of vy = ~/(i m) Dy and with the aid of the formulas of (60), leads to the results stated (64) and (65). C.3 Mean z component of velocity In (C4) and (C5), we substitute Dz for Dx with Dz (a∗ , A) = (4Γ2R )−1 i A21 + A22 − A23 − A24 ΓI + −A21 + 2A1 a∗1 − A22 + 2A2 a∗2 + A23 − 2A3 a∗3 + A24 − 2A4 a∗4 ΓR = −r0 ΓI νf f ∗ sin(φ). (C12)

925


Due to the amplitude factor sin(φ), the phase integral with respect to φ vanishes identically with the consequence that vz = 0. C.4 Mean square deviation of x component of velocity We use the differential operators (12) and (10) and shift one vx to the bra vector by means of partial integration to write (note that r = u2 ) vx2

Z Z ~2 2 2π 0 dΦ dΦ r2 dr sin(θ)dθdϕ (C13) = C 2 m 0 1 [u3 ∂u1 − u4 ∂u2 + u1 ∂u3 − u2 ∂u4 ] exp a∗ (Φ0 ) · u − Γ∗ u2 /2 × 2 2u 1 [u3 ∂u1 − u4 ∂u2 + u1 ∂u3 − u2 ∂u4 ] exp a(Φ) · u − Γu2 /2 . 2 2u

As it turns out, the integrand depends on the difference Φ0 − Φ and, after the integration with respect to Φ0 , does not anymore depend on Φ. We then use (3) and, after the differentiation with respect to ui , we get Z

2π

vx2 = F

0

Z

dΦ 0 Z ∞ Z = F dγ ΓR

1 2 exp A · u − Γ u R u2 2π dΦ0 du1 du2 du3 du4 Qx exp A · u − γu2 , (C14) du1 du2 du3 du4 Qx

0

where A is defined in (52) and Qx = qx∗ (Φ0 )qx (Φ), qx (Φ) = u3 [a1 (Φ) − (ΓR + i ΓI ) u1 ] − u4 [a2 (Φ) − (ΓR + i ΓI ) u2 ] + u1 [a3 (Φ) − (ΓR + i ΓI ) u2 ] − u2 [a4 (Φ) − (ΓR + i ΓI ) u4 ] ; F = 2~2 C 2 /m2 . (C15) In the second equation of (C14), we produced the factor 1/u2 by means of integration with respect to the variable γ. The amplitude factor Qx is multiplied out, and each monomial in u generated by partial differentiation ui → ∂/∂Ai . After carrying out the u integrations, the derivatives ∂/∂Ai act on the function IA which is defined in (60g): Z ∞ du1 du2 du3 du4 exp A · u − γu2 = (π 2 /γ 2 ) IA . (C16) −∞

The mean value is now expressed as follows Z ∞ Z 2π 2 2 ˜ x IA . vx = π F dγ dφ Q ΓR

0

(C17)

926


˜ x by powers of ΓR /γ and by cos(nφ) terms: We order Q ˜ x = r2 Q 0

6 X 2 X

x qmn

m=3 n=0

ΓR γ

m cos(n φ),

(C18)

where, x x = 0, = κ1 /K, q32 q31 1 3 + 2ν 2 + 3ν 4 2 2 2 2 4g ν − gK(ν − 1) κ6 + ν 4 − 4K κ0 + K κ1 , 2K 2 ν (1 + ν 2 )2 x 2(κ0 − 1/K) κ1 , q42 = κ23 /8, 32κ0 (1 + g 2 )ν 2 − K κ26 (ν 2 − 1)2 + 8gκ0 κ6 ν(ν 2 + 1)2 (ν 2 − 1)−1 + 2 −2 io −1 2 2 2 4 16κ0 ν 3 + 2ν + 3ν ν −1 16Kν 2 , κ1 2 x 4g ν + 4ν(1 − 2Kκ0 ) + gKκ6 (1 − ν 2 ) , q52 = −κ23 /4, 2Kν x x (1 + g 2 )(2κ22 + κ23 )/8, q61 = 2(1 + g 2 )κ0 κ1 , q62 = (1 + g 2 )κ23 /8. (C19)

x = 0, q30 x q40 = x q41 = x q50 =

x q51 = x q60 =

For the functions κi and g, see (60f) and (60i), respectively. The φ integrals are stated in (B13). There remains the γ integration which, after the variable transformation γ → s = ΓR /γ, enters as follows vx2

3

= 2π F r0 K

6 X 2 X

x qmn Wmn ,

(C20)

m=3 n=0

Z

1

Wmn =

ds sm−2 exp [Kκ0 s] In ((Kκ1 s) ,

0 x where we used r0 ΓR = K, k0 /ΓR = Kκ0 and k1 /ΓR = Kκ1 . The functions qmn and Wmn , as well as K, are dimensionless. Since the normalization constant C 2 has dimension 1/length3 , one verifies immediately that r0 F has the dimension velocity2 . For large K, we derive in Appendix D the asymptotic approximation exp [K(κ0 + κ1 )] 1 i1 (n)(κ0 + κ1 ) + (5/2 − m)κ1 2 Wmn = 3/2 √ 1+ + O 1/K K κ1 (κ0 + κ1 ) K 2πκ1 (κ0 + κ1 ) (C21) with m = 3, . . . , 6, n = 0, 1, 2, and i1 (n) defined in (57). In order to eliminate the normalization constant C 2 , we divide F by (58). After ordering by powers of K and using K = h(w)κ, we write (C20) as (C22) vx2 = κ2 V0x + V1x /K + O(1/K 2 ) , κ2 V0x = hvx i20 ,

2

κ

V1x /K

2

=~

κ (e−1)Zw3

3 X j=0

−1 cx2j cosh(2jw) 32m2 r02 (e + 1)(1 + ν 2 ) , (C23)

927


where the c coefficients read: cx0 = −4 e + 2ν 2 − eν 4 + e2 (1 + ν 2 )2 , cx2 = 5 + 4eν 2 − 5ν 4 + 2e3 (1 + ν 2 )2 + 3e2 (ν 4 − 1), cx4 = 4(1 + e + 4ν 2 + ν 4 − eν 4 ), cx6 = 2e3 (1 + ν 2 )2 + 5(ν 4 − 1) − 3e2 (ν 4 − 1) − 4e(1 + 3ν 2 + ν 4 ).

(C24)

To leading order including the first order correction, the mean square deviation is given by (∆vx )2 ≡ vx2 − vx 2 = κ2 V1x /K − hvx i0 ξv /K [1 + O(1/K)], (C25) which after evaluation with the aid of (C23), (C24), and (65) leads to the result stated in (66). C.5 Mean square deviation of y component of velocity In analogy to (C20), we write vy2

3

= 2π F r0 K

6 X 2 X

y qmn Wmn

(C26)

m=3 n=0

with y q30 = 0, y q40 y q41 y q50 y q51 y q60

= = = = =

y q31 = κ1 /K,

y q32 = 0,

2 2(1 + g 2 )/K 2 + [−4κ0 + gκ6 (1/ν − ν)] /(2K) + 4κ21 ν 2 / 1 + ν 2 , y −2κ1 /K, q42 = 0, 2 2(1 + g )κ0 /K − κ26 /2 + 2gκ0 κ6 ν/(ν 2 − 1), y 2(1 + g 2 )κ1 /K, q52 = 0, y y (1/4)(1 + g 2 )κ26 , q61 = 0, q62 = 0. (C27)

Instead of (C22) and (C23), we obtain vy2 = κ2 V0y + V1y /K + O(1/K 2 ) ,

κ2 V0y = hvy i20 ,

3 −1 X κ2 V1y /K = ~2 κ(1 − e)Zw3 32m2 r02 (e + 1) cy2j cosh(2jw),

(C28) (C29)

j=0

with cy0 cy2 cy4 cy6

= = = =

2(3e2 − 1) −1 − ν 2 + e(ν 2 − 1) , 7(ν 2 − 1) − 9e2 (ν 2 − 1) − 3e(1 + ν 2 ) + 5e3 (1 + ν 2 ), 2 5e(1 − ν 2 ) + 3e3 (ν 2 − 1) + 3(1 + ν 2 ) − e2 (1 + ν 2 ) , (3e2 − 5)(1 + e − ν 2 + eν 2 ).

(C30)

928


To leading order including the first order correction, the mean square deviation is given by (∆vy )2 ≡ vy2 − vy 2 = κ2 V1y /K − hvy i0 ηv /K [1 + O(1/K)], (C31) which after evaluation with the aid of (C29), (C30), and (65) leads to the result stated in (66). C.6 Mean square deviation of z component of velocity In analogy to (C20), we write vz2

3

= 2π F r0 K

6 X 2 X

z qmn Wmn

(C32)

m=3 n=0

with z q30 = 0, z q40 z q41 z q50 z q51 z q60

= = = = =

z q31 = κ1 /K,

z q32 = 0,

(C33) 2 2(1 + g 2 )/K 2 + [−4κ0 + gκ6 (1/ν − ν)] /(2K) − 2κ21 ν 2 / 1 + ν 2 , z −2κ1 /K, q42 = (2κ21 ν 2 )/(1 + ν 2 )2 , 2 2Kκ21 ν 2 + (1 + g 2 )κ0 (1 + ν 2 )2 / K(1 + ν 2 )2 , z 2(1 + g 2 )κ1 /K, q52 = −(4κ21 ν 2 )/(1 + ν 2 )2 , z z −2(1 + g 2 )κ21 ν 2 /(1 + ν 2 )2 , q61 = 0, q62 = (2(1 + g 2 )κ21 ν 2 )/(1 + ν 2 )2 .

Instead of (C22) and (C23), we obtain hvz2 i = κ2 V0z + V1z /K + O(1/K 2 ) ,

κ2 V0z = hvz i20 = 0.

(C34)

To leading order including the first order correction, the mean square deviation is given by (∆vz )2 ≡ hvz2 i − hvz i2 = κ2 V1z /K[1 + O(1/K)],

(C35)

which leads to the result stated in (66). Appendix D: Integrals with Bessel functions In the following, we derive an asymptotic approximation of Wmn for K 1: Z 1 Wmn = ds sm−2 exp[Kκ0 s]In (Kκ1 s) , m = 3, 4, 5, 6, n = 0, 1, 2. (D1) 0

For large K, the main contribution to the integral comes from s close to 1, where the Bessel function can be approximated by the asymptotic series. However, in the integration interval with small s, the asymptotic approximation is unjustified. We, therefore, split the integral into two parts, A + B, with Z 1/2 Z 1 A= ds . . . , B = ds . . . . (D2) 0

1/2

929


Part A is estimated by an upper bound of the Bessel function, see formula (9.6.18) in [1]. √ 0 ≤ In (x) ≤ π(x/2)n Γ−1 (n + 1/2) exp(x), n ≥ 0, x ≥ 0, (D3) and can, thus, be estimated as √ A ≤ π(1/2)n Γ−1 (n + 1/2)(1/2)n+m−1 exp [K(κ0 + κ1 )/2] .

(D4)

As compared to A, part B has the dominating exponential term with s = 1/2 replaced by s = 1, so A + B = B(1 + A/B),

A/B ∝ exp [−K(κ0 + κ1 )/2] ,

(D5)

which tells that the relative contribution of A is exponentially small for K 1; by (60f), κ0 + κ1 ≡ [e cosh(w) − 1]/(e − 1) ≥ 1, e > 1. (D6)

® LogHKL

-5 -6 -7 -8 -9 -10

LogÈ1-QÈ ®

-11

3.6

3.8

4.0

4.2

4.4

4.6

Figure 1: Numerical check of the accuracy of the asymptotic approximation (D12). Q = Q(K), with 34 < K < 110, is the quotient of the numerically determined integral (D1) divided by the analytic approximation (D12). Parameters are κ1 = 0.8 and κ0 = 0.4. The upper three curves refer to m = 6 and to the Bessel functions In with n = 0 (solid curve), n = 1 (dashed), and n = 2 (dot-dashed), respectively; the lower curves refer to m = 4. In the integrand of B, we approximate the Bessel functions by the asymptotic form (56): Z 1 1 Wmn → √ ds exp [K(κ0 + κ1 )s] × (D7) 2πKκ1 1/2 m−5/2 s + i1 (n)sm−7/2 /(Kκ1 ) + i2 (ν)sm−9/2 /((Kκ1 )2 ) + . . . ,

930


where the i-coefficients are listed in (57) for n = 0, 1, 2. The s integral can be expressed in terms of the incomplete Γ functions, see Section 6.5 of [1]; with m = 3, 4, 5, 6 and the abbreviation K01 = K(κ0 + κ1 ), we obtain Z

1

Km : =

3/2−m

ds exp [K01 s] sm−5/2 = i (−1)m K01

×

1/2

[Γ(m − 3/2, −K01 ) − Γ(m − 3/2, −K01 /2)] .

(D8)

The contribution from the lower integration boundary is exponentially small. Using the asymptotic form, see (6.5.32) of [1], Γ(a, s) ∼ sa−1 exp(−s) 1 + (a − 1)/s + (a − 1)(a − 2)/s2 + . . . , (D9) we obtain m − 5/2 (m − 5/2)(m − 7/2) 1 + ... . exp[K01 ] 1 − + Km ∼ 2 K01 K01 K01 Thus, we get the following asymptotic approximation: 1 i1 (n) i2 (n) Wmn = √ Km + Km−1 + Km−2 + . . . . Kκ1 (Kκ1 )2 2πKκ1

(D10)

(D11)

After the amplitude is ordered by powers of 1/K and neglecting terms of order (1/K 2 ), we arrive at exp [K(κ0 + κ1 )] 1 i1 (n)(κ0 + κ1 ) + (5/2 − m)κ1 . (D12) Wmn = 3/2 √ 1+ K κ1 (κ0 + κ1 ) K 2πκ1 (κ0 + κ1 ) As can be inferred from Fig. 1, for K > 34, the difference Q − 1, with Q = (D1)/(D12), approaches the value zero proportional to 1/K b with b ≈ 2, which is consistent with the O(1/K 2 ) error in (D12). Appendix E: Check of mean square deviations and uncertainty products As a check of the analytical results, we prove in the following that the analytically continued mean square deviations of the velocity components of elliptic orbits are positive, see subsections 1, 2, and 3 to follow; furthermore, in subsection 4, we show for the hyperbolic case that the uncertainty products obey the quantum mechanical lower bound. E.1 Mean square deviation of x component of the velocity

931


By (138), it has to be shown that for 0 ≤ e < 1 Kx :=

3 X

c2j cos(2jw) < 0.

(E1)

j=0

It is convenient to express the functions cos(2jw) in terms of powers of X := cos2 (w) by means of the substitutions cos(2w) = 2X −1, cos(4w) = 1−8X +8X 2 , cos(6w) = 18X −48X 2 +32X 3 −1. (E2) With the aid of the c coefficients defined in (67) and the abbreviation ν 2 = n > 0, we obtain Kx = A0 + A1 n, A0 = −2(1 + e) + 4(4 + 5e)X − 48(1 + e)X 2 + 32(1 + e)X 3 , A1 = −2(1 + e) + 4(5e − 4)X + 48(1 − e)X 2 − 32(1 − e)X 3 , 0 ≤ e < 1, 0 ≤ X ≤ 1, n > 0.

(E3)

Since n can be arbitrarily small and large, both A0 and A1 have to be negatively definite. At the end points of the X interval, we get the values Kx (X = 0) = −2(1+e)(1+n) < 0,

Kx (X = 1) = −2(1−e)(1+n) < 0. (E4)

Let us first consider A0 . In the inner interval with 0 < X < 1, we find a maximum and a minimum at the points p (E5) X± = (1/12)(1 + e)−1 [6(e + 1) ± d0 ] , d0 = 6(2 + 3e + e2 ), which give rise to the extrema values (A0 )± = −(2/9)(1 + e)−1 [9(1 + e) ± 2(1 + e/2) d0 ] .

(E6)

The minimum, corresponding to the ”+” sign, is negatively definite, obviously. It remains to be shown that the maximum is negatively definite, which is the case, since [9(1 + e)]2 − [2(1 + e/2) d0 ]2 = 3(1 − e2 )(11 + 14e + 2e2 ) > 0,

0 ≤ e < 1. (E7)

The fact that the internal maximum is negative, together with the negative end point values of A0 (set n = 0 in (E4)), proves that A0 < 0. As to A1 , the end point values are negative as can be read from the coefficient of n in (E4). Once more, there exist a maximum and a minimum in the interval 0 < X < 1; however, the inner extrema exist only if the eccentricity is confined to the interval 0 ≤ e ≤ 4/5; in the interval 4/5 < e < 1, the extrema lie outside the X interval. The maximum lies at p Xmax = (1/12)(1 − e)−1 [6(1 − e) + d1 ] , d1 = 6(2 − 3e + e2 ), (E8)

932


and, in the interval 0 ≤ e ≤ 4/5, has the value (A1 )max = −(2/9)(1 − e)−1 [a1 − a2 ] , a1 = 9(1 − e), a2 = 2(1 − e/2)d1 . (E9) Both a1 and a2 are positive magnitudes. In order that (A1 )max < 0, we have to check a1 > a2 , or after squaring a21 − a22 = 3(1 − e2 )(11 − 14e + 2e2 ) ≥ 729/625 > 0,

0 ≤ e < 4/5.

(E10)

In the eccentricity interval 4/5 < e < 1, the coefficient A1 monotonically increases with X in the interval 0 ≤ X ≤ 1 with the consequence that A1 ≤ A1 (X = 1) = −2(1 − e) < 0. This completes the proof that Kx < 0. E.2 Mean square deviation of y component of the velocity By (138), it has to be shown that Ky :=

3 X

d2j cos(2jw) > 0.

(E11)

j=0

After expressing the cosine functions according to (E2) and inserting the di coefficients from (67), we obtain Ky = 2(1 + e)A0 + 2(1 − e)A1 n, A0 = 1 − e2 + 2(4 + 3e + e2 )X − 8(3 + e)X 2 + 16X 3 , A1 = (1 + e)2 − 2(4 + 5e + e2 )X + 8(3 + e)X 2 − 16X 3 , 0 ≤ e < 1, 0 ≤ X ≤ 1, n > 0.

(E12)

In (E12), since n > 0 is arbitrary, both A0 and A1 have to be positively definite. The coefficient A0 can be transformed into the following, evidently positive, expression A0 = (1 − e2 )(1 − X) + 16X [X − (3 + e)/4]2 > 0.

(E13)

As to A1 , we immediately get A1 (X = 0) = (1 + e)2 > 0 and A1 (X = 1) = 1 − e2 > 0. In the interval 0 < X < 1, one finds a maximum and a minimum at h p i (E14) X± = (1/12) 6 + 2e ± d2 , d2 = 2(6 − 3e − e2 ). The minimum is at X− with p (A1 )min = (1/27) [a0 − a1 ] , a0 = 27 − 9e(1 + e) − 5e3 , a1 = (6 − 3e − e2 ) d2 . (E15) It is noticed that both a0 and a1 are positive for 0 ≤ e < 1, so we check a20 − a21 = 27(1 − e)(1 + e)2 11 − 5e − 3e2 − e3 > 0, (E16)

933


where the square bracket factor is larger than 2 in the eccentricity interval 0 ≤ e < 1. Thus, it has been shown that (A1 )min > 0 and since at the end points A1 (X = 0) > 0 and A1 (X = 1) > 0, it is proved that A1 > 0 and, thus, Ky > 0. E.3 Mean square deviation of z component of the velocity By (138), it has to be shown that Kz := f0 + f4 cos(4w) < 0.

(E17)

After expressing the cosine functions according to (E2) and inserting the fi coefficients from (67), we obtain Kz B0 B1 B2

= = = =

B0 + B1 n + B2 n2 , −2(1 − e2 )(1 − 2X)2 , 4 −1 + e2 (1 − 2X)2 − 4X(1 − X) , −2(1 − e2 )(1 − 2X)2 .

The coefficients B0 and B2 are negatively definite for 0 ≤ e < 1, and B1 is easily estimated as B1 ≤ 4 −1 + e2 − 4X(1 − X) < 0. (E18) Thus, it is proved that Kz < 0. E.4 Uncertainty products for hyperbolic orbits In the following, we prove that the uncertainty products obey the lower bound condition of quantum mechanics according to (68). In particular, we show that Pi ≥ 1 holds true for all three components i = x, y, z in the parameter domains (w ∈ R, e > 1, n = ν 2 > 0). The proofs to follow are consistent with (69) which tells that if ν > 0 , then Px (w = 0) > 1, Py (w = 0) > 1 and Pz (w = 0) = 1. E.4.1 Uncertainty product of x component We obtain from (63), (66) and (67), using the abbreviations W = 2w and n = ν 2, Px = Nx /Dx , Nx = [−2(1 + n) + (e − 1 + n(1 + e)) cosh(W ) + (1 + e + n(e − 1)) cosh(3W )] × (e − 1)n cosh2 (W/2) + (e + 1) sinh2 (W/2) , Dx = 2n [e cosh(W ) − 1]2 ,

e > 1,

n > 0.

(E19)

934


Since Dx > 0, the condition Px ≥ 1 is equivalent to the condition dx := Nx − Dx ≥ 0. We find for this difference dx dx0 dx1 dx2

= = = =

dx0 + dx1 n + dx2 n2 , (E20) 2 (e + 1) sinh (W/2) [−2 + (e − 1) cosh(W ) + (1 + e) cosh(3W )] , 4(e2 − 1) sinh2 (W/2) [cosh(W/2) + cosh(3W/2)]2 , (e − 1) cosh2 (W/2) [−2 + (1 + e) cosh(W ) + (e − 1) cosh(3W )] .

The coefficient dx1 ≥ 0, obviously. The other coefficients are easily estimated as dx0 ≥ dx0 (W = 0) ≥ 0, dx2 ≥ dx2 (W = 0) > 0. (E21) As a consequence, Px > 1 for e > 1 and n > 0. E.4.2 Uncertainty product of y component From (63), (66) and (67), we obtain Py = Ny /Dy , Ny = N1y N2y N1y = 2 (1 + e + n(1 − e)) + n − 1 − 3e(1 + n) + 2e3 (1 + n) cosh(W ) − 2e (1 + e + (e − 1)n) cosh(2W ) + (1 + e + (e − 1)n) cosh(3W ), y N2 = (e − 1)n cosh2 (W/2) + (e + 1) sinh2 (W/2), Dy = 2(e2 − 1)n [e cosh(W ) − 1]2 , e > 1, n > 0. (E22) Once more, since Dy > 0, we can equivalently examine the condition dy := Ny − Dy ≥ 0, where dy = dy0 + dy1 n + dy2 n2 , (E23) y y d0 = (e + 1)2 sinh2 (W/2) f0 , f0y = 2 + [2e(e − 1) − 1] cosh(W ) − 2e cosh(2W ) + cosh(3W ), dy1 = 4(e2 − 1) cosh(W ) [cosh(W ) − e] sinh2 (W ), dy2 = (e − 1)2 cosh2 (W/2) f2y , f2y = −2 + [2e(e + 1) − 1] cosh(W ) − 2e cosh(2W ) + cosh(3W ). Since one can choose n > 0 arbitrarily small or arbitrarily large, both f0y and f2y must not be negative which, however, is not obvious. The coefficient dy1 , which can be negative, if e > 1 and W close to zero, has to be compensated by the n2 term. Thus, we examine dy = D0 + dy2 [n + dy1 /(2dy2 )]2 ,

D0 = dy0 − (dy1 )2 /(4dy2 ),

(E24)

and prove in the following that (f0y , f2y , D0 ) > 0 which implies that dy > 0.

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To show this, we make use of the identities cosh(W ) : = Y, cosh(2W ) = 2Y 2 − 1, cosh(3W ) = 4Y 3 − 3Y, cosh(4W ) = 8Y 4 − 8Y 2 + 1, (E25) which allows us to write f0y = 2 1 + e + e2 − e − 2 Y − 2eY 2 + 2Y 3 .

(E26)

There are √ three cases: (a.) 1 < e ≤ 4 , ( b.) 4 < e < ec , (c.) e > ec with ec = (3 + 33)/2. In case (a.), there exists one extremum only, a minimum, for Y ≥ 1, in case (b.) there exist a maximum and a minimum, and in case (c.) f0y is monotonically increasing with Y , and f0y ≥ f0y (Y = 1) = 2(e − 1)2 > 0. Since we are interested in a lower bound, in the cases (a.) and (b.), we consider the minimum only for 1 < e < ec , which is at √ (E27) Ymin = (1/6) [2e + s0 ] , s0 = 12 + 6e − 2e2 , 1 ≤ e ≤ ec , and has the value (f0y )min = y0 + y1 s0

(E28)

with y0 = (2/27) 27 + 9e − 9e2 + 5e3 ,

y1 = (2/27) e2 − 3e − 6 .

(E29)

In order to see that this minimum is larger zero, we observe that y0 > 0 and, for 1 ≤ e ≤ ec , consider the difference y02 − y12 s20 = (4/27)(e − 1)2 11 + 16e + e2 1 + (e − 1)2 > 0. (E30) Concluding f0y ≥ min 2(e − 1)2 , (f0y )min > 0,

e > 1.

We proceed similarly with f2y = 2 e − 1 + (e2 + e − 2)Y − 2eY 2 + 2Y 3 ,

Y ≥ 1.

(E31)

(E32)

There exists no extremum in the interval Y ≥ 1 for e > 1, which implies that f2y monotonically increases with Y with the consequence that f2y ≥ f2y (Y = 1) = 2(e2 − 1) > 0,

e > 1.

(E33)

With the positivity of f0y and f2y , we have the properties that dy0 ≥ 0 and dy2 > 0, see (E23). It remains to be shown that D0 ≥ 0. As a matter of fact, we find D0 = N0 /f2y , N0 = 4(e − 1)(e + 1)3 sinh2 (W/2) [cosh(2W ) − e cosh(W )]2 ≥ 0.

(E34) (E35)

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At w = 0, we have D0 = 0; however, in (E24), the cofactor of dy2 is larger 0, if n > 0. Thus, it is proved that Py > 1 for e > 1 and n > 0. E.4.3 Uncertainty product of z component From (63), (66) and (67), we write after setting ν 2 = n Pz = Nz /Dz = N1z N2z N3z /Dz , N1z = −(1 + 6n + n2 ) + e2 (1 + n)2 + e2 (1 + n)2 − (n − 1)2 cosh(2W ), N2z = (e − 1)n cosh2 (W/2) + (1 + e) sinh2 (W/2), N3z = (e − 1) cosh2 (W/2) + (1 + e)n sinh2 (W/2), Dz = 2(e2 − 1)n(1 + n)2 [e cosh(W ) − 1]2 , e > 1,

n > 0.

(E36)

Since Dz > 0, the condition Pz ≥ 1 is equivalent with dz = Nz − Dz ≥ 0 which can be written as 2 dz = (1/2) 4e n + (n − 1)2 Y − e2 (1 + n)2 Y (Y 2 − 1). (E37) Since Y = cosh(W ) ≥ 1, we have dz ≥ 0, which proves that the uncertainty product amounts to Pz ≥ 1. To achieve the above compact form of dz , Mathematica [22] tools were quite helpful: by successively applying the commands Expand, TrigToExp, Expand, P ExpToTrig, the difference dz comes out in the form j Cj cosh(jW ). Then, one uses (E25) followed by the Mathematica command Simplify. As a numerical test with the data e = 5/4, n = 4/5, and W = 1, Mathematica delivers the same value to 20 digits, N [A, 20] = N [B, 20] = 9.7114786183862078599, where A = Nz − Dz according to (E35) and B = dz according to (E36).

References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions, Dover Publications, New York, 1968. [2] M. Boiteux, The three-dimensional hydrogen atom as a restricted fourdimensional harmonic oscillator, Physica, 65(1973), 381. [3] A. C. Chen and M. Kibler, Connection between the hydrogen atom and the four-dimensional oscillator, Phys. Rev. A, 31(1985), 3960. [4] I. H. Duru and H. Kleinert, Solution of the path integral for the H-atom, Phys. Lett. B, 84(1979), 185. [5] T. Fabˇciˇc, J. Main, and G. Wunner, Fictitious-time wave-packet dynamics. I. Nondispersive wave packets in the quantum Coulomb problem, Phys. Rev. A, 79(2009), 043416-1.


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[6] J.C. Gay, D. Delande, and A. Bommier, Atomic quantum states with maximum localization on classical elliptical orbits, Phys. Rev. A, 39(1989), 6587. [7] C. C. Gerry, Coherent states and the Kepler-Coulomb problem, Phys. Rev. A, 33(1986), 6. [8] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965. [9] V. P. Gutschick and M. M. Nieto, Coherent states for general potentials. V. Time evolution, Phys. Rev. D, 22(1980), 403. [10] R. Ho and A. Inomata, Exact-path-integral treatment of the Hydrogen atom, Phys. Rev. Lett., 48(1982), 231. [11] B. R. Johnson, Time variables in propagators and coherent states for the Kepler-Coulomb problem, Phys. Rev. A, 35(1987), 1412. [12] M. Kibler and T. Négadi, Connection between the hydrogen atom and the harmonic oscillator, Phys. Rev. A, 29(1984), 2891. [13] P. Kustaanheimo and E.Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218(1965), 204. [14] L. D. Landau and E. M. Lifschitz, Mechanik, , Akademie-Verlag, Berlin, 1964. [15] I. Mostowski, On the classical limit of the Kepler problem, Lett. Math. Phys., 2(1977), 1. [16] S. Nandi and C. S. Shastry, Classical limit of the two-dimensional and the three-dimensional hydrogen atom, J. Phys. A: Math. Gen., 22(1989), 1005. [17] M. Nauenberg, Quantum wave packets on Kepler elliptic orbits, Phys. Rev. A, 40(1989), 1133. [18] M.M. Nieto, Coherent states for general potentials: IV. Three-dimensional systems, Phys. Rev. D, 22(1980), 391. [19] M. Perelomov, Coherent states for arbitrary Lie group, Commun. Math. Phys., 26(1972), 222. [20] A. Rauh and J. Parisi, Quantum mechanics of hyperbolic orbits in the Kepler problem, Phys. Rev. A, 83(2011), 042101 [21] A. E. Roy, Orbital Motion, Adam Hilger, Bristol, 1988.

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[22] S. Wolfram, Mathematica Version 9.0.0.0,, Wolfram Research, Champaign, IL., 2012. [23] B.W. Xu, G.H. Ding, and F.M. Kong, Nonspreading coherent states for the hydrogen atom, Phys. Rev. A, 62(2000), 022106. Received: August 25, 2014