Quantum trajectories emerging from classical phase space

Physica A 390 (2011) 1926–1930

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Quantum trajectories emerging from classical phase space F. Olivares a , A. Plastino a , Bernard. H. Soffer b,∗ a

IFLP-CCT-Conicet, National University La Plata, 1900 La Plata, Argentina

b

Pacific Palisades, CA, United States

article

abstract

info

Article history: Received 5 December 2010 Available online 15 February 2011

Starting with a classical Hamiltonian function H, we show that, if we select among its phase–space trajectories those that minimize Fisher’s information measure, the resulting ˆ of H. trajectories coincide with the stationary ones of their quantum counterparts. H © 2011 Elsevier B.V. All rights reserved.

Keywords: Classical Hamiltonian Phase–space trajectories Fisher information Quantum orbits

1. Introduction In the last 15 years much effort has been invested in physical applications of Fisher’s information measure (FIM) [1,2], the source of a powerful variational principle, extreme physical information (EPI), that yields most of the canonical Lagrangians of theoretical physics [1,2], and also correctly characterizes an ‘‘arrow of time’’, alternative to that one associated with Boltzmann’s entropy [3,4]. The FIM measure associated with translations of a one-dimensional observable x with corresponding probability density ρ(x) is [5]

∫ Ix =

∂ ln ρ(x) dxρ(x) ∂x 

2

∫ ≡

dxρ(x) (∇ ln ρ(x))2 ,

(1)

or, setting ρ(x) = ψ(x)2 [2],

∫ Ix = 4

dx[∇ψ(x)]2 ,

(2)

which obeys the so-called Cramer–Rao inequality

(1x)2 ≥ Ix−1

(3)

involving the variance of the stochastic variable x [5]

(1x) = ⟨x ⟩ − ⟨x⟩ = 2

2

2

∫

dxρ(x)x − 2

∫

dxρ(x)x

2

.

(4)

From the earliest days of quantum Mechanics (QM), there has been a continuous inquiry into its meaning, earlier focusing attention on spectra, cross sections, and expectation values, and lately on information content and its relation

∗

Corresponding author. Tel.: +1 310 454 0721. E-mail address: [email protected] (B.H. Soffer).

0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.02.008

F. Olivares et al. / Physica A 390 (2011) 1926–1930

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to measurement, the emergence of the classical, and the nature of the quantum. For a lucid summary of recent concerns see Ref. [6], who believes and we concur that many of the remaining gaps in our understanding of quantum physics and its relation to the classical domain, such as the definition of systems, or the still mysterious aspects of collapse will be expanded into new areas of investigation by considerations that simultaneously elucidate the nature of the quantum and of information. This work indeed is precisely concerned with the relationship of the classical, QM, and information, with the emergence of QM from the classical, and is set in phase–space. For an excellent and recent review of phase–space in physics is see Ref. [7]. Of course, an already classic exposition of phase–space is that of Arnold [8]. Other recent studies with related interests, but with very different methodologies, and results, often use the Wigner distribution function of phase–space. For instance, in Ref. [9], noting that within the Wigner operator–function correspondence there is no quantum state corresponding to a single phase–space point, the author nevertheless finds quantum states that can be considered the quantum counterpart of a point in classical phase–space. In the present communication we deal only with FIM’s classical phase–space version [10–12] and shed interesting new light on the well-established relation between FIM and Schroedinger’s equation [1,13–15] by recourse to phase–space considerations that have not been contemplated previously. The concomitant process we describe will take place entirely in phase–space and be carried out in strictly classical, non-quantal terms. Thus, we will be concerned here with applications of Fisher’s measure in classical phase–space, not in Hilbert’s space. For simplicity, our attention is focused on a two-dimensional phase–space instance, and we illustrate our approach with reference to the harmonic oscillator (HO). The general case is considered below. The HO is much more than a mere example, but rather an important system revealing insights that have a wide impact in many physical models. Nowadays it is of particular interest for the dynamics of bosonic or fermionic atoms contained in magnetic traps [16–18] as well as for any system that exhibits an equidistant level spacing in the vicinity of the ground state, like nuclei or Luttinger liquids. Let f (x, p) be a two-dimensional, classical probability distribution, with x, p independent variables. We have, for the associated classical Fisher measure I, the expression [10–12]

∫ I =

dxdp h

 f

cx2



∂ ln f ∂x

2 +

cp2



∂ ln f ∂p

2 

,

(5)

where h, (not necessarily Planck’s constant!), cx , cp are dimensionality constants and f obeys

∫

dxdp h

f (x, p) = 1.

(6)

We will show that if a particle’s motion is governed by Hamilton’s classical equations, then recourse to a minimization of I is all that is needed in order to get the usual stationary quantum trajectories, our only goal here. 2. The FIM variational problem in classical phase–space We now extremize the classical expression (5) subject to the normalization condition (6) and to some appropriate constraint that reflects our prior-information. It is of didactic value to choose, for the sake of specificity, to exemplify the procedure with reference to a mean energy-constraint, that of the harmonic oscillator (HO). The general instance will be tackled afterwards below. We are then led to,

∫

dxdp h

EHO (x, p)f (x, p) = ⟨EHO ⟩ ≡ E ,

(7)

where EHO (x, p) = p2 /2m + mω2 x2 /2. Thus, our variational problem becomes



δ I + λ0

∫

dxdp h

f (x, p) + λ

∫

dxdp h

EHO (x, p)f (x, p)



= 0,

(8)

where the λ’s are Lagrange multipliers. Expressing f now in terms of real amplitudes as f (x, p) = ψ 2 (x, p), and following the general tenets described in Refs. [13,19,20], we are immediately led to the Schroedinger-like equation cx2

2 ∂ 2ψ λ λ p2 λ0 2 2 2∂ ψ − m ω x ψ + c − ψ = ψ. p 2 2 ∂x 8 ∂p 8m 4

(9)

If we give ψ the form ψ(x, p) = ρ(x)η(p), Eq. (9) can be decoupled. Consequently, after setting

λ0 4 we obtain

= λx0 + λp0

(10)

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F. Olivares et al. / Physica A 390 (2011) 1926–1930

d2 ρ dx2 d2 η

− α 2 B2p p2 η =

dp2

λx0

− α 2 A2x x2 ρ =

cx2

λp0 cp2

ρ,

(11)

η,

(12)

with A2x = mω2 /8cx2 ;

B2p = 1/8mcp2 ;

and α 2 = λ.

(13)

Since we face here two HO-like Schroedinger equations, we know that, as shown for instance in (11), ρ is solved by (Hn are the Hermite polynomials)

ρn (x) = Cn e−αAx x

2 /2



Hn [ α Ax x],

(14)

with a similar form for η. In general, one would have for ψ 2 = fnl (x, p), after using (6) for normalization, solutions of the type

  (Ax Bp )1/2 α exp[−α(Ax x2 + Bp p2 )]Hn2 [ α Ax x]Hl2 [ α Bp p]. (15) n + l 2 n!l! π However, since the η-component describes just motion in the ‘‘impulse’’ subspace of classical phase–space, we will restrict ourselves, invoking Galilean invariance [21], to ‘‘static’’ solutions of the variational problem, choosing to fix η at its ‘‘lowest-lying’’ function (that with l = 0). Setting l = 0 we then obtain for ρ(x) the HO-quantum coordinate-probability distributions [22]. An equally productive alternative is, of course, to work with the coordinate-marginal of the distribution ψ 2 . fnl (x, p) = h

3. The general problem Confining ourselves to stationary solutions, as discussed above, we assume that our prior information is that of the mean value of a potential function V (x), so that the Fisher variational problem becomes



δ I + λ0

∫

dxdp h

f ( x, p ) + λ

∫

dxdp h

 V (x)f (x, p) = 0.

(16)

According to Ref. [13], we are then immediately led to the Schroedinger equation for a particle moving in the potential well V (x), whose solution is then the exact quantum-one for such a well. This enables us to state our main conclusion. Trajectories in the coordinate-subspace of classical phase–space points that minimize Fisher’s measure are identical to their quantum mechanical counterparts. In other words, out of the infinitude of HO-phase–space trajectories, minimizing FIM selects out just the quantum trajectory subset. This is discussed further in the conclusions. In order to illustrate further interesting details we return again to the HO-problem. 4. The Fisher-HO instance Let us elucidate further details concerning the stationary l = 0 solutions of the classical phase–space FIM-variational problem. We have

−λx0 = α Ax (2n + 1),

(17)

p 0

−λ = α Bp .

(18)

Adding up (17)–(18) and considering (10) we find

−

λ0 4

= α[Ax (2n + 1) + Bp ].

(19)

We are then in a position to deal with (7) E =

1

∫

2m

dxdp h

fnl (x, p) p + 2

mω 2 2

∫

dxdp h

fnl (x, p)x2 ;

to be reconciled with E (input-datum) = h¯ ω(n + 1/2),

(20)

the well-known value for the HO-energy [22]. After integration one gets E=

1

1

4m α Bp

+

mω2 1 2

α Ax

(n + 1/2).

(21)

F. Olivares et al. / Physica A 390 (2011) 1926–1930

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√ √ 2(n+1/2) Since (cf. Eq. (13)) A2x = mω2 /8cx2 ; B2p = 1/8mcp2 ; α = λ = h¯ ω and specifying, à la [23], cx2 = 1/mω2 and n +1 cp2 = m we are led to  1/2  1/2 λ λ . [(l + 1/2) + (n + 1/2)] = (n + 1), (22) h¯ ω/2 = E =

2

2

because we take l = 0. There appears then a classical ‘‘zero-point’’ energy, for n = 0 Eo =



λ/2 ≡ h¯ ω/2! (quantum-‘‘vacuum’’).

(23)

We then reach an important conclusion. The HO-energy of x-trajectories minimizing FIM are quantized exactly as its quantum mechanics’ counterpart, with the correct zero-point energy. By inspection of (17)–(18) we realize now that

− λ0 = 2λE ,

(24)

expressing proportionality between the normalization-multiplier λ0 and the system’s energy. Now turning our attention to Eq. (20), we see that the two integrals yield, respectively, the kinetic and potential energies’ mean values.

−λx0 = −λp0 =

λ 2

λ 2

⟨U ⟩f ,

(25)

⟨K ⟩f .

(26)

An exciting possibility then opens up. Nothing prevents us from assuming that these expectation values depend upon temperature, which paves the way for an immediate extension to statistical mechanics. 5. Conclusions and discussion We have proven that those classical phase space point trajectories, or flows, that minimize Fisher’s measure exactly coincide with their quantum counterparts. Thus, our main conclusion simply states that trajectories in the classical coordinatesubspace of phase–space that minimize Fisher’s measure are identical to their quantum mechanical counterparts. FIM is here an integral over phase–space x, p, coordinates. Notice that there is a physically measurable denumerable rational number infinite subset of the mathematically ideal non-denumerable real number continuum of the entire classical phase–space. This is the case as the classical trajectories can only be measured to a finite precision, therefore automatically excluding all except rational numbers. Fisher’s minimization with energy constraint (MFI) is able to extract just the infinite denumerable set of quantum trajectories, denumerable as they reflect the set of the integers. The emergence of quantum mechanical ‘‘spacing’’ between trajectories is proven here in general, and illustrated via the harmonic oscillator, that important, ubiquitous building block of many physical theories. The passage from continuum trajectories to discrete ones was historically called ‘‘quantization’’, an effect that stems from the very nature of Fisher’s information (FI). Quantum probabilities emerge from FI’s rather non-classical way of defining the probability p(x) of any PDF as the square of an amplitude, say ψ (cf. Eq. (2)). As Fisher’s information measure (FIM) is given in terms of amplitudes (cf. Eq. (2)), so is any differential equation coming from MFI, including the Schroedinger-like equation we derived here. Those solutions are functions of amplitudes ψ , not expressed directly in terms of ψ 2 probabilities [2]. When solutions for ψ are in the form of superposition integrals, and they are then squared to transform the results into probabilities, non-classical cross-terms arise, the identifying hallmarks of Quantum Mechanics. Previous FI derivations of Schroedinger’s equation using MFI [13] and the extreme physical information (EPI) [2], as well as previous EPI derivations of the more general quantum equations of Dirac, and Klein Gordon, also were set in entirely classical scenarios, which supports our previous assertion that FIM is inherently ‘‘quantal’’ in nature, or at least potentially so. In this way one can explain how our initial scenario with seemingly classical a priories in FIM and phase space could result in quantization effects. At first sight we seem to give an a priori quantum input for getting a zero-point HO energy, (although not for obtaining the quantum orbitals in general, for any potential V (x)). Notice however that for the HO the zero-point energy would appear automatically merely from the fact that in order to have proper dimensions in the Fisher measure one needs to introduce some constant ‘‘h’’ (in action units) within the pertinent integral (cf. Eq. (5)). This paper in its FIM-related quantum derivations, takes for them the novel phase–space point of view. Also new is the way of eliminating momentum from consideration in order to only deal with coordinates alone when deriving the nonrelativistic Schroedinger equation from MFI. Many previous derivations of quantum mechanics using FI performed a unitary transform, i.e. a Fourier transform to handle the coordinates. (See for examples Ref. [2, 2004, Ch 4, Ch 10, and Appendix D].) Appendix. Time dependence Our treatment prima facie does not consider the question of time dependence. There are two important points to note about the extension of our results to include time. The reader should keep in mind the general result of Liouville’s theorem [7,8].

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F. Olivares et al. / Physica A 390 (2011) 1926–1930

In the variational equation (16) in Section 3, including time in our treatment would entail simply changing f (x, p) to f (x, p, t ). However, since V (x) is time-independent, absolutely none of our results in Section 4 would change in the slightest for all times t. On the other hand for a time-independent Hamiltonian H, it is well-known, that if {ψn } are the eigen-functions of H at any time t (0), then the time-dependent Schroedinger equation’s most general solution Ψ (t ) at any other t is, calling ∆(t ) = t − t (o, )

Ψ (t ) =

−

Cn exp[−iH ∆(t /h¯ )]ψn ,

(27)

n

and can be tackled without undue effort. However to work out the details would exceed the scope of the present effort, and remains to be investigated further. References [1] B.R. Frieden, B.H. Soffer, Phys. Rev. E 52 (1995) 2274. [2] B.R. Frieden, Physics from Fisher Information, Cambridge University Press, England, Cambridge, 1998, Science from Fisher information (England: Cambridge University Press, Cambridge, 2004). [3] A.R. Plastino, A. Plastino, Phys. Rev. E 54 (1996) 4423. [4] A. Plastino, A.R. Plastino, H.G. Miller, Phys. Lett. A 235 (1997) 129. [5] M.J.W. Hall, Phys. Rev. A 62 (2000) 012107. [6] W.H. Zurek, Rev. Modern Phys. 75 (2003) 715. [7] D.D. Nolte, Phys. Today 38 (2010) 33. [8] V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, NY, 1989. [9] A. Luis, Phys. Rev. A 69 (2004) 052112. [10] F. Pennini, A. Plastino, G.L. Ferri, F. Olivares, Phys. Lett. A 372 (2008) 4870. [11] F. Pennini, A. Plastino, G.L. Ferri, Felipe Olivares, M. Casas, Entropy 11 (2009) 32. [12] F. Olivares, F. Pennini, A. Plastino, Phase–space distribution by variation of information measures, Physica A 389 (2010) 2218. [13] R. Frieden, A. Plastino, A.R. Plastino, B.H. Soffer, Phys. Rev. E 60 (1999) 48. [14] P.J. Hueber, Robust Statistics, John Wiley & Sons, New York, NY, 1981. [15] M. Reginatto, Phys. Rev. A 58 (1998) 1775. [16] M.H. Anderson, et al., Science 269 (1995) 198. [17] K.B. Davis, et al., Phys. Rev. Lett. 75 (1995) 3969. [18] C.C. Bradley, C.A. Sackett, R.G. Hulet, Phys. Rev. Lett. 78 (1997) 985. [19] A. Hernando, C. Vesperinas, A. Plastino, Phys. Lett. A 374 (2009) 18. [20] A. Hernando, C. Vesperinas, A. Plastino, Physica A 389 (2010) 490. [21] I.M. Yaglom, An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity, Springer-Verlag, NY, 1979. [22] A. Messiah, Quantum Mechanics, Dover Publications, NY, 1999. [23] F. Pennini, A. Plastino, Phys. Lett. A 349 (2006) 15.