Quantum distributions for the plane rotator

Physica A 519 (2019) 313–318

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Physica A journal homepage: www.elsevier.com/locate/physa

Quantum distributions for the plane rotator Marius Grigorescu CP 15-645, Bucharest 014700, Romania

highlights • Quantization by using partial Fourier transform as Hermitian operator. • Model of transition from complex wave functions to thermal noise. • Example of non-thermal quantum entropy for localized rotational states.

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Article history: Received 30 August 2018 Received in revised form 10 December 2018 Available online 24 December 2018

a b s t r a c t Quantum phase-space distributions (Wigner functions) for the plane rotator are defined using wave functions expressed in both angle and angular momentum representations, with emphasis on the quantum superposition between the Fourier dual variable and the canonically conjugate coordinate. The standard quantization condition for angular momentum appears as necessary for consistency. It is shown that at finite temperature the time dependence of the quantum wave functions may provide classical sound waves. Non-thermal quantum entropy is associated with localization along the orbit. © 2018 Published by Elsevier B.V.

1. Introduction The action–angle coordinates {(Ji , ϕi ), i = 1, N /Ji ∈ R, ϕi ∈ [−π, π]} on the phase-space M arise in the description of the integrable Hamiltonian systems with periodic orbits [1]. In these variables1 the Hamilton function depends only on J ≡ {Ji , i = 1, N }, and a submanifold ΣJ of constant J is a torus parameterized by {ϕi , i = 1, N }. In the old quantum mechanics Ji takes only a discrete set of values, Ji = ni h ¯ , ni ∈ Z, 2π h¯ = h = 6.626 × 10−34 J s, such that the corresponding Lagrangian submanifolds Σnh¯ generate a partition of M in cells bn of volume hN . Although providing a regular pattern of phase-space ‘‘formatting’’, these cells are not ordered along a complete set of local coordinates, and in the limit h → 0 become singular submanifolds, rather than points. Probability distributions of particles on M may arise from thermal fluctuations, or from an intrinsic ‘‘quantum structure’’, resembling the partition in cells bn of finite volume. On M = T ∗ RN this structure is usually associated with the Wigner transform [2] fψ ∈ F (M) of the quantum ‘‘wave function’’ ψ ∈ L2 (RN ), defined in Cartesian coordinates. For integrable systems the quantum distributions fψn provided by the eigenstates ψn of the Hamiltonian operator show an increased localization probability on Σnh¯ ⊂ M [3,4], but despite constant effort, a direct definition of fψ in terms of the action–angle variables is faced with difficulties. Various aspects of the problem are presented in [5–9]. In this work quantum distributions for the action–angle variables are discussed on the representative example of the plane rotator (M = T ∗ S 1 ). The treatment is similar to the one applied before to the rigid rotator [10], but instead of discretization here the emphasis is on the quantum superposition between the canonically conjugate coordinate and the 1 In the standard approach J are considered as coordinates and ϕ as momenta. For systems with symmetry J are provided by the momentum mapping i i i and ϕi are group coordinates. https://doi.org/10.1016/j.physa.2018.12.021 0378-4371/© 2018 Published by Elsevier B.V.

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(dual) variable introduced by partial Fourier transform. The basic elements of the formalism are presented in Section 2, followed in Section 3 by applications to the Wigner functions of the plane rotator. Finite temperature effects, beyond the single particle coherence time, are discussed in Section 4. Concluding remarks are presented in Section 5. 2. The partial Fourier transform as Hermitian operator Let f (x, y) be a real integrable function of x, y ∈ R and f˜ (x, k) the partial Fourier transform of f only with respect to y, f˜ (x, k) =

∫

dy eiky f (x, y) .

(1)

Because f˜ (x, k)∗ = f˜ (x, −k), we may consider f˜ (x, k) as a matrix element of a Hermitian operator fˆ on L2 (R), having x and k as indices not along the rows and columns, but along the diagonals [11]. In the case of M = T ∗ R parameterized by the canonical variables (q, p), the Fourier transform in momentum f˜ (q, k) of f ∈ F (M) (the set of smooth functions on M), provides a matrix element fˆab ≡ h−1 f˜ ((a + b)/2, (a − b)/h ¯ ) (the ‘‘Weyl quantization’’ of f ) with the row and column indices a = q + h¯ k/2, b=q−h ¯ k/2 defined using h¯ as a conversion factor from k to q-scale. Thus, if f1 , f2 ∈ F (M) then [11] (f1 , f2 ) =

∫

dqdp f1 (q, p)f2 (q, p) = h

∫

∫

db fˆ1ab fˆ2ba ≡ hTr(fˆ1 fˆ2 ) .

da

(2)

M

The change of integration variables from (q, p) to (a, b) is completely formal and it does not change the physics (classical or quantum) of the observables f1 , f2 . However, it distinguishes between a pure quantum distribution fψ ∈ F (M) and other observables by reducing ρˆ ψ = hˆfψ to a projection operator, h(ˆfψ )ab = ψa ψb∗ , ψ ∈ L2 (R), ∥ψ∥ = 1. In this case the expectation value of A ∈ F (M) with respect to fψ is

ˆ = ⟨ψ|Aˆ |ψ⟩ . ⟨A⟩fψ = (fψ , A) = hTr(ˆfψ A)

(3)

Similar results can be obtained using the ‘‘momentum representation’’, defined by the Fourier transform in coordinate, f˜ ′ (k′ , p) =

∫

′

dq eik q f (q, p) ,

(4)

such that fˆb′′ a′ ≡ h−1 f˜ ′ ((a′ − b′ )/h ¯ , (a′ + b′ )/2), with a′ = p + h¯ k′ /2, b′ = p − h¯ k′ /2. It can be shown that if hˆf is separable as ∗ ′ ˆ ˆ hfab = ψa ψb , then hf is also separable, hˆf′b′ a′ = ψb′ ′ (ψa′ ′ )∗ , with

∫

1

ψp′ = √

2π h ¯

dq e−ipq/h¯ ψq .

(5)

This result ensures that both marginal distributions are positive definite,

∫

Fψcs (q) ≡

dp fψ (q, p) = ψq ψq∗ , Fψms (p) ≡

∫

dq fψ (q, p) = ψp′ ψp′∗ ,

(6)

and that we may consider ψq and ψp′ as components of the same ‘‘state vector’’ |ψ⟩ in dual bases, |q⟩ ( ≡ |h ¯ k⟩), and |p⟩, formally related by Fourier transform,

|p⟩ = √

1

2π h ¯

∫

dq eipq/h¯ |q⟩.

(7)

The factor h ¯ is intrinsically related to the triplets (q, h¯ k; p), (q; h¯ k′ , p), and if we replace p by the velocity v = p/M0 then it should be changed to h ¯ v = h¯ /M0 . Thus, one of these factors, h¯ or h¯ v necessarily depends on mass (M0 ), and in the standard ˆ ˆfψ → fψ of the defining representation presumes quantum mechanics this is h ¯ v . It is interesting to note that a change A → A, a certain ‘‘encoding’’, as the ordering of the matrix indices a, b does not always follow the one of the variables q, k. Thus, for variations δ a > 0, δ b > 0 we get also δ q > 0, but δ k > 0 only if δ a > δ b. A related aspect is the sensitivity of fψ to the ˆ u [11], where Π ˆ u is the local inversion symmetry of ψ , as for a localized distribution fu = δ (q − u)δ (p) we get ρˆ u ≡ hˆfu = 2Π ˆ u ψ (q) = ψ (2u − q). inversion operator, Π Because Fψcs (q) ≥ 0 and Fψms (p) ≥ 0, the negative values of fψ (q, p) indicate that in a quantum distribution the canonical coordinates (q, p) are not independent, but correlated by the implicit dependence of fψ on the Fourier dual variables, k or k′ . A measure of these correlations is given by the function Cψ (q, p) = fψ (q, p) − Fψcs (q)Fψms (p). For a pure distribution fψ ≡ ρψ /h the quantum (von Neumann) entropy defined by Sq (ρˆ ) = −Tr(ρˆ ln ρˆ ) vanishes ˆ ρψ ) in general (Sq (ρˆ ψ ) = 0, ρˆ ψ = |ψ⟩⟨ψ|), while the classical (Boltzmann–Gibbs) entropy S(ρψ ) = −(fψ , ln ρψ ) = −Tr(ρˆ ψ ln is complex. The evolution of a distribution function or operator in a thermal environment is described by the Fokker–Planck equations presented in Appendix A.

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3. Characteristic distributions for the plane rotator A distribution function f(ϕ, J) of angle (ϕ ) and orbital angular momentum (J ≡ Lz ) on M = T ∗ S 1 ≃ S 1 × R may describe ensembles encountered both at mesoscopic [12] and macroscopic [13] scale, of particles moving on a circle. It can be therefore regarded as a constrained distribution on T ∗ R2 (Appendix B), but to take into account the specific geometry of S 1 as configuration space we should start along the lines of Section 2, with the partial Fourier transform

˜f(ϕ, k) =

∫

dJ eikJ f(ϕ, J) .

(8)

To proceed towards the one-particle quantum distributions one should note that if we let k ∈ R and ϕ ∈ [−π, π] then α = ϕ + h¯ k/2, β = ϕ − h¯ k/2 are not well defined as indices for a matrix element ˆfαβ = h−1˜f((α + β )/2, (α − β )/h¯ ) of a Hermitian operator ˆf on the quantum Hilbert space H = L2 (S 1 ). Following the example of the rigid rotator [10], quantum distributions fψ can be defined though using a separable expression ˜fψ (ϕ, k) ≡ ψα ψβ∗ , if the range of γ = h ¯ k is restricted to the first ‘‘Brillouin zone’’, γ ∈ [−π, π]. In this case one obtains π

∫

1

fψ (ϕ, J) =

2π h ¯

dγ e−iγ J /h¯ ψ (ϕ +

−π

γ 2

) ψ ∗ (ϕ −

γ 2

) ,

(9)

in agreement with Vψ (θ, p) of [6]. The overlap between two such functions is (fψ 1 , fψ 2 ) =

∞

∫

∫

−∞

where ⟨ψ1 |ψ2 ⟩ ≡ by (9) are Fψcs (ϕ ) =

π

dJ

∮

dϕ fψ 1 fψ 2 = hTr(fˆψ 1 fˆψ 2 ) =

−π

|⟨ψ1 |ψ2 ⟩|2 , 2π h ¯

(10)

dϕψ1∗ ψ2 is the scalar product between ψ1 and ψ2 as elements of H. The marginal distributions provided

∞

∫

−∞

dJ fψ (ϕ, J) = ψϕ ψϕ∗ ,

(11)

positive definite, and Fψms (J) =

∮

dϕ fψ (ϕ, J) =

1

⟨ψ|Pˆ J |ψ⟩ ,

(12)

ˆ dγ eiγ (J −J)/h¯ , Jˆ = −ih ¯ ∂ϕ ,

(13)

h ¯

where Pˆ J =

∮

1 2π

√

ˆ (Jˆψn = nh¯ ψn ), then ⟨ψn |Pˆ J |ψn′ ⟩ = δnn′ j0 (π (n−J /h¯ )), (∂x ≡ ∂/∂ x). If ψn (ϕ ) = einϕ / 2π , n ∈ Z, is an ‘‘integral’’ eigenstate2 of J, ms j0 (x) = sin(x)/x. This shows that Fψ (J) is not positive definite if J /h ¯ ∈ R, but if J /h¯ ∈ Z then Pˆ nh¯ , n ∈ Z becomes a projection

operator on ψn , Pˆ nh¯ = |ψn ⟩⟨ψn |, and Fψms (nh ¯ ) = |⟨ψ|ψn ⟩|2 /h¯ ≥ 0. Moreover, if J /h¯ ∈ Z and ψ is a function of good parity, ψ (ϕ + π ) = ±ψ (ϕ ), then the integral (9) becomes intrinsic on S 1 , namely invariant to a change of parameter γ → γ + 2π . The set {ρψn ∈ F (M)/ρψn (ϕ, J) = ⟨ψn |Pˆ J |ψn ⟩, n ∈ Z} may be called of characteristic distributions associated to a quantum ˆ because by the properties partition of M based on the integral eigenvalues of J, h−1 (ρψn , J) = nh ¯ , (ρψn , 1) = h , (ρψm , ρψn ) = hδmn ,

∑

ρψn (ϕ, J) = 1 ,

(14)

n∈Z

resemble the characteristic functions χn : M ↦ → {0, 1} of the subsets bn ⊂ M = ∪n∈Z bn , bn = {(ϕ, J)/−π ≤ ϕ ≤ π, n − 1/2 < J /h¯ ≤ n + 1/2}, χn |bk = δnk . To obtain quantum distributions in the angular momentum representation the approach is similar, but also faced with difficulties. Because ϕ has a finite range a function f ∈ F (M) can be expanded in a Fourier series, f (ϕ, J) =

1 2π h ¯

∑

e−imϕ f˜ ′ (m, J /h ¯) ,

(15)

m∈Z

but with J /h ¯ ∈ R and m ∈ Z the variables a = J /h¯ + m/2 and b = J /h¯ − m/2 are unsuitable as matrix indices. However, if J /h ¯ ∈ Z and fψ (ϕ, J) has the form (9) with ψ of good parity,3 then a, b ∈ Z too, and ˜f′ψ ′ (m, J /h¯ ) = ψb′ (ψa′ )∗ with 1

ψn′ = √

2π

∮

dϕ e−inϕ ψϕ .

(16)

2 For the harmonic oscillator Hamiltonian H on T ∗ R the action variable J = H /ω is positive, (√2J , ϕ ) are polar coordinates on T ∗ R, (dp ∧ dq = dJ ∧ dϕ ), and the eigenstates of Jˆ are real. 3 This means that m is an even integer. The odd values of m enlarge the domain {a ∈ Z} of ψ ′ by new points, a = n + 1/2, n ∈ Z. a

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In this representation Fψcs′ (ϕ ) = h ¯

∑

∗ fψ ′ (ϕ, nh ¯ ) = ψϕ ψϕ ,

(17)

n∈Z

∫

(considering

Fψms′ (J) =

dJ = h ¯

∮

∑

), n=J /h ¯

and

dϕ fψ ′ (ϕ, J) = ψJ′/h¯ (ψJ′/h¯ )∗ .

Moreover, for f1 , f2 ∈ F (M) we get (f1 , f2 ) = hTr ′ (fˆ1′ fˆ2′ ), where Tr ′ Aˆ ′ ≡ ′ J =h = h¯ aδba , and ¯ (a + b)/2 . In particular, 1ˆ ′ba = δba , Jˆba ′ =− ϕˆ ba

i a−b

(18)

∑

b∈Z

′ Aˆ ′bb , and fˆba = f˜ ′ (m, J /h¯ )/h, m = a − b,

(−1)a−b (1 − δab ) .

(19)

The angle operator ϕˆ ′ coincides with (φˆ −π )p from [14], and corresponds to the series expansion

ϕ=−

∑ (−1)m m

m̸ =0

sin mϕ .

(20)

One should consider though ϕ only as a local coordinate, because at the points of discontinuity ϕ = ±π this series contains ± sin mπ = 0, while instead of π , as is limn→∞ (π − ϕ/n), the limit lim 2

n ∑ sin(mϕ/n)

(21)

m

n→∞

m=1

yields 1.08949π (the ‘‘Gibbs phenomenon’’). 4. Coherence properties and temperature effects Similarly to the case of the free particle on the R-axis [15], also for the free plane rotator the quantum distribution fψ (ϕ, J) is coherent, in the sense that if fψ is a solution of the classical Liouville equation, J

∂t fψ + ∂ϕ fψ = 0 ,

(22)

I

then ψ is a solution of the Schrödinger equation, ih ¯ ∂t ψ = Hˆ ψ , Hˆ = Jˆ2 /2I, by I denoting the moment of inertia. At the temperature T > 0 we may consider the thermal average fT over pure distributions fψ of the form (15), fT (ϕ, J) =

1

∑

2π h ¯

e−imϕ ˜f′T (m, J /h ¯) ,

(23)

m∈Z

s s ∗ where ˜f′T (m, J /h ¯) = ¯ + m/2, b = J /h¯ − m/2, and ws,T is the thermal distribution function, s∈S ws,T ψb (ψa ) , a = J /h −Es /kB T ), over a set S of one-particle states s with energy Es and average angular momentum Js . (e.g. ws,T ∼ e At thermal equilibrium, during a small single-particle coherence time τ [16], a quantum wave function ψ s (J /h ¯ + µ, t), µ = ±m/2, will become

∑

ψ s (J /h¯ + µ, t + τ ) = e−iτ (Es −µh¯ Js /I)/h¯ ψ s (J /h¯ + µ, t) ,

(24)

such that ˜f′T changes into

˜f′T (m, J /h¯ , t + τ ) =

∑

ws,T e−iτ mJs /I ψbs (ψas )∗ |t .

(25)

s∈S

Presuming that in the sum above we can approximate ws,T (Js2 − ⟨J 2 ⟩T ) ≈ 0, with ⟨J 2 ⟩T =

∂t2˜f′T (m, J /h¯ , t) = ∂τ2˜f′T (m, J /h¯ , t + τ )|τ =0 ≈ −m2 ΩT2˜f′T (m, J /h¯ , t) ,

∑

s

ws,T Js2 , we get (26)

where = ⟨J ⟩T /I . In this approximation we find the transition expected in [15], from the complex wave functions ψ s to real classical waves (thermal noise), because for a time t ≫ τ , fT of (23) is a solution of the classical wave equation ∂t2 fT = ΩT2 ∂ϕ2 fT . The result is independent of h¯ and should hold also in a classical ensemble during a macroscopic (sound) perturbation, with the condition of constructive interference along the circle providing a discrete spectrum of ‘‘angular wavelengths’’.

ΩT2

2

2

M. Grigorescu / Physica A 519 (2019) 313–318

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5. Concluding remarks The partial Fourier transform of an observable A ∈ F (T ∗ R), only in momentum or in coordinate, can be regarded as matrix element Aˆ ab , of a Hermitian operator Aˆ on L2 (R), after a change of variables to the linear combinations a, b, between the Fourier dual (×h ¯ ) and the canonically conjugate coordinate (Section 2). A pure quantum quasiprobability distribution (Wigner function) f = ρ/h ∈ F (T ∗ R) ensures that the non-linear factor ρˆ (1 − ρˆ ) from the dissipative term of Eq. (32) vanishes, and ρˆ = ρˆ 2 is a projection operator. In the case of the plane rotator A ∈ F (T ∗ S 1 ), and the indices a, b should be defined in terms of variables assuming values in real sets with different topological properties. Therefore, a consistent definition of the quantum distributions f(ϕ, J) for angle (ϕ ) and orbital angular momentum (J) introduces constraints such as the integrality condition J /h ¯ ∈ Z (Section 3). According to (14), the Planck’s constant h can also be regarded as volume of a ‘‘quantum cell’’, with the characteristic distribution ρψ = hfψ ∈ F (T ∗ S 1 ). This finite volume, which is a heuristic element in the definition of thermal entropy (produced by random forces), generates the quasi-entropy due to the spreading of ρψ over classical states, S = −(fψ , ln ρψ ), (or S2 = 1 − (fψ , ρψ ), [17,18]), and the entropy Sq of the mixed state (35). It is interesting to note that the Titius–Bode law for the planetary system suggests a constraint resembling a form of ‘‘entropy quantization’’, such as log2 (

J JG

)3 = n , n = 0, 1, 2, . . .

(27)

where JG = M0 cRG with RG = 2γG Mo /c 2 denoting the Schwarzschild radius of the central body (the Sun for the planets or Jupiter for its satellites, Appendix C). At the atomic scale a constraint of this type is unlikely, but for the high circular Rydberg levels under active investigation [19,20], a Poisson distribution (35) could indicate a stage of localization along the orbit. Appendix A. Classical and quantum Fokker–Planck equations A distribution function f(q, p) on T ∗ R3 , for an ensemble with the single-particle Hamiltonian H = p2 /2M + V (q), at the temperature T , evolves according to the Fokker–Planck equation 1 γ p · ∇ f − ∇ V · ∇p f = ∇p · (p + MkB T ∇p )f , M M where γ denotes the friction coefficient. By Fourier transform in momentum (28) becomes

∂t f +

∂t ˜f −

i M

∇k · ∇ ˜f + k · (i∇ V +

γ M

∇k )˜f = −γ kB T k2˜f .

(28)

(29)

Introducing the new variables a = q + h ¯ k/2, b = q − h¯ k/2, and the notation ˆfab ≡ ˜f(q, k)/h3 , we get ∂t ˆfab = (∂t ˆf)ab , k = (a − b)/h ¯ , kˆfab = [qˆ , ˆf]ab /h¯ , k2ˆfab = [qˆ ·, [qˆ , ˆf]]ab /h¯ 2 , ∇k = h¯ (∇a − ∇b )/2, ∇kˆfab = h¯ {∇, ˆf}′ab /2, ∇k · ∇ ˆfab = h¯ [∆, ˆf]ab /2, where {, }′ denotes the anticommutator. With these expressions and the approximation (at small k, necessary only if V (q) is not a linear or quadratic function of q), h ¯ k · ∇ V ˆfab ≈ (Va − Vb )ˆfab = [V , ˆf]ab , (29) becomes, as indicated in [15], the quantum Fokker–Planck equation, ih ¯ ∂t ˆf = [Hˆ , ˆf] +

γ 2M

[qˆ , ·{pˆ , ˆf}′ ] −

iγ k B T h ¯

[qˆ , ·[qˆ , ˆf]] .

(30)

This equation is similar to the one proposed in [21], but still it contains a dissipative term of classical form. In fact, when ˆ = pˆ 2 /2M the quantum equilibrium distributions H

ρˆ ± (α ) =

1

, ±1 β = 1/kB T , ρˆ − (α ) = −ρˆ + (α + iπ ), are stationary solutions of the nonlinear equation eβ Hˆ −α

ih ˆ + ¯ ∂t ρˆ = [Hˆ , ρ]

γ 2M

[qˆ , ·{pˆ , ρˆ (1 ∓ ρˆ )}′ ] −

iγ kB T h ¯

[qˆ , ·[qˆ , ρ]] ˆ ,

(31)

(32)

in which ∓γ {pˆ , ρˆ 2 }′ /2M could be assigned to a density-dependent correction to the friction force. Appendix B. Rotational coherent states Let us consider a particle of mass M, in uniform rotation with the angular frequency ω > 0, around the Z -axis, on a circle of radius R in the XY plane. Thus, if u = (ux , uy ) and v = (vx , vy ) are the position and momentum vectors, then u = (R cos ϕ, R sin ϕ ), v = (−P sin ϕ, P cos ϕ ), with P = M ωR and ϕ = ϕ0 + ωt. It can be shown that a Gaussian distribution on T ∗ R2 , centered on u and v, of the form fu,v (q, p) =

1

π 2 h¯ 2

2 /a−(p−v)2 /b

e−(q−u)

, a = h¯ 2 /b = h¯ /M ω

(33)

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can be obtained by a standard Wigner transform of the rotational coherent state (‘‘symmetry breaking vacuum’’), ˆ†

|ζ ⟩ = eζ bu −ζ bu |0⟩ , (34) √ √ √ † † † −iϕ 2 ˆ ˆ ˆ ˆ ˆ where ζ = J /h ¯ e , J = M ωR , and bu = (bx + iby )/ 2, with bq = M ω/2h¯ (qˆ + ipˆ q /M ω), bq |0⟩ = 0, q = x, y. Moreover, the average of fu,v over ϕ ∈ [−π, π] at constant J is the Wigner transform of the density operator ρˆ w =

∗ˆ

∞ ∑

1

wn |n⟩⟨n| , |n⟩ = √ (bˆ †u )n |0⟩ , n! n=0

(35)

4 expressed by a Poisson (non-thermal) distribution wn = e−J /h¯ (J /h ¯ )n /n! of quantum entropy √ Sq = − n wn ln wn , over the † † † † † eigenstates |n⟩ of the angular momentum operator Lˆ z = h ¯ (bˆ u bˆ u − bˆ d bˆ d ), bˆ d = (bˆ x − ibˆ y )/ 2. It is worth noting that when the states ξn (x, y) ≡ ⟨x, y|n⟩ describe a charged particle in the uniform magnetic field Bz along the Z -axis, confined in the XY plane by a symmetric double or triple well potential [22] V (z) = V (−z), tuned at the resonance for the wave packets ηL (z), ηR (z) of width Wz , localized at zL,R = ±Dz /2, then quantum coherence oscillations across Dz ≫ Wz between the states Ψn,L = ξn ηL and Ψn,R = ξn ηR can appear.

∑

Appendix C. Constrained orbits for Kepler’s problem The condition (27) and the third law of J. Kepler, (in Harmonices Mundi, Linz, 1619), yield for the n’th circular orbit the radius rn = RG 21+2n/3 , n = 0, 1, 2, . . .. In the case of Jupiter RG = 2.82 m, and as indicated in Table 1, with n = 39, 40, 41, 42 we get values (rn ) close to the ones observed (robs ) for its largest satellites: Io, Europa, Ganimede, Callisto. Table 1 Comparison between the observed orbital radius (robs ) of the Jupiter satellites and the calculated value (rn ). Satellite/n

Io/39

Eu/40

Ga/41

Ca/42

robs (103 km) rn (103 km) robs /rn

421.6 378.5 1.11

670.8 600.8 1.12

1070 953.7 1.12

1882 1514 1.24

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4 A Gaussian distribution on R with the same mean and variance as {w } has the entropy S = [1 + ln(2π J /h)]/2. ¯ n J