Scale invariant quantum potential leading to globally self-trapped wave function in Madelung fluid
aa r X i v : . [ phy s i c s . g e n - ph ] A ug Scale invariant quantum potential leading toglobally self-trapped wave function in Madelungfluid
Agung Budiyono
Institute for the Physical and Chemical Research, RIKEN, 2-1 Hirosawa, Wako-shi,Saitama 351-0198, JapanE-mail: [email protected]
Abstract.
We show in spatially one dimensional Madelung fluid that a simplerequirement on local stability of the maximum of quantum probability density will,if combined with the global scale invariance of quantum potential, lead to a class ofquantum probability densities globally being self-trapped by their own self-generatedquantum potentials, possessing only a finite-size spatial support. It turns out to belongto a class of the most probable wave function given its energy through the maximumentropy principle. We proceed to show that there is a limiting case in which thequantum probability density becomes the stationary-moving soliton-like solution ofthe Schr¨odinger equation.PACS numbers: 03.65.Ge,03.65.Ca,03.65.Vf
1. Madelung fluid: global scale invariant quantum potential
Let us consider a spatially one dimensional Madelung fluid [1] for a single free particlewith mass m . The state of the system is then determined by a pair of fields inposition space q as: { ρ ( q ) , v ( q ) } , where ρ ( q ) is a normalized quantity called as quantumprobability density and v ( q ) is velocity field. The temporal evolution of both at time t is then assumed to satisfy the following coupled dynamical equation: m dvdt = − ∂ q U, ∂ t ρ + ∂ q ( ρv ) = 0 . (1)Here, U ( q ) is the so-called quantum potential generated by the quantum amplitude R = ρ / as U ( q ) = − ¯ h m ∂ q RR , (2)For the case of a spatially one dimensional fluid, the velocity field can always bewritten as the spatial gradient of a scalar function S ( q ) as v ( q ) = ∂ q S/m. (3)One can then use this new quantity to define a complex-valued wave function ψ = R exp( iS/ ¯ h ) to show that the Madelung fluid dynamics given in Eq. (1) is equivalentto the Schr¨odinger equation for a single free particle as follows: i ¯ h∂ t ψ ( q ; t ) = − ¯ h m ∂ q ψ ( q ; t ) . (4)Notice that while the equation on the right of Eq. (1) is but the conventionalcontinuity equation which guarantees the conservation of probability flow, the leftequation takes the form of Newtonian dynamical equation with a classically absencenew term appears on the right hand side. In this regards, the term F = − ∂ q U iscalled as quantum force. This fact suggests that the quantum force is the new quantitywhich is responsible for the nonclassical behaviors of Schr¨odinger equation. Thus, it isreasonable to pay serious attention to the the property of the quantum potential.Let us mention an important properties of quantum potential that will playimportant roles in our discussion later. First is multiplying the quantum probabilitydensity with a constant will not change the profile of the quantum potential. Namely,the quantum potential is invariant under global rescaling of the quantum probabilitydensity, namely its own source, as: U ( cρ ( q )) = U ( ρ ( q )) , (5)where c is constant. This shows that the quantum potential only cares about the formof the quantum probability density and is independent from the strength of the latter[2]. It is as if the quantum potential considers the quantum probability density as acode in telecommunication system in case of which only the profile of the sequence ofthe binary wave is important, the strength of the received wave itself is of no use. Inthis sense, the quantum potential is of informational nature. For an interesting andstimulating discussion concerning this matter see [2].Among the consequences of the above invariant property is that, first rescaling thequantum probability density by the mass of the particle, ˜ ρ = mρ will not change thedynamical equation on the left of Eq. (1). On the other hand, the continuity equationon the right of Eq. (1) becomes ∂ t ˜ ρ + ∂ q ( v ˜ ρ ) = 0 , (6)which can now be read as the equation for the conservation of mass density, rather thanthe conservation of quantum probability density. The other consequence of the scaleinvariance property of quantum potential is that even at points where the strength ofthe quantum probability density is very low, the quantum potential that it generatesat that point might be very high. In this paper, we shall be interested in a class ofquantum probability densities with this specific property.Next, let us mention another property of quantum potential U ( q ) that the averageof the quantum force, F = − ∂ q U , over the quantum probability is vanishing [3] Z dq ∂ q U ( q ) ρ ( q ) = 0 . (7)This can be proven easily by assuming ρ ( ±∞ ) = 0. Imposing this into the dynamicalequation of Eq. (1), one reproduces the Ehrenfest theorem [3] m d ¯ vdt = 0 , (8)where ¯ v is the average value of the velocity field defined as ¯ v ≡ R dq v ( q ) ρ ( q ).
2. Local stability and globally self-trapping quantum potential
Let us now show that local geometrical restriction on the maximum point of the quantumprobability density, if combined with the scale invariance of the quantum potential willdetermine the global geometrical property of the quantum potential, thus the quantumprobability density as well. First, since the quantum probability density is vanishingat infinity, ρ ( ±∞ ) = 0, non-negative and normalized, it must at least have one localmaximum point. Let us denote this maximum point by q = Q . It thus satisfies ∂ q ρ | Q = 0 , ∂ q ρ | Q < . (9)One first observes that at this point the quantum potential is positive definite U ( Q ) = − ¯ h m (cid:16) − (cid:16) ∂ q ρρ (cid:17) + ∂ q ρρ (cid:17)(cid:12)(cid:12)(cid:12) Q > . (10)Before proceeding, let us write a useful formula for later discussion ∂ nq ρ s ρ s (cid:12)(cid:12)(cid:12) Q = s ∂ nq ρρ (cid:12)(cid:12)(cid:12) Q , (11)which can be shown easily by utilizing the left equation in (9) to be valid for any positiveinteger n .Now, let us put a local restriction on a class of quantum probability densities ρ ( q )so that its maximum point stays at the minimum point of the quantum potential U ( q )which it generates through Eq. (2). One therefore imposes ∂ q U | Q = 0 , ∂ q U | Q ≥ . (12)Dynamically we are thus looking for a class of quantum probability densities in whichat least its maximum is temporally stable. Next, let us show that the restrictions givenby Eqs. (12) will uniquely determine the form of U ( q ) as a function of ρ ( q ). First, fromthe global scale invariant property of the quantum potential, then the quantum force isalso global scaling invariance; so that one has ∂ q U ( cρ ) = ∂ q U ( ρ ) for any real constant c . It is therefore reasonable to write the quantum force to take the following non-trivialform: ∂ q U ( ρ ) = 1 ρ s ( a + a ∂ q + a ∂ q + a ∂ q + . . . ) ρ s , (13)where s and a i , i = 0 , , , . . . , are arbitrary real number. Evaluating at q = Q andusing the fact of Eq. (11) one has ∂ q U ( ρ ( Q )) = a + sρ ( a ∂ q + a ∂ q + a ∂ q + . . . ) ρ (cid:12)(cid:12)(cid:12) Q . (14)The left equation in (12) imposes the right hand side of Eq. (14) to be vanishing.Keeping in mind Eqs. (9) and the fact that ∂ nq ρ | Q , for n ≥
3, are fluctuating betweenpositive and negative value, ∂ q U | Q = 0 can then be accomplished by imposing a = 0, a j = 0 for j ≥
2, and a is arbitrary, yet non-vanishing. One therefore has ∂ q U ( ρ ) = a ∂ q ρ s ρ s . (15)Next, let us rewrite Eq. (15) as follows ∂ q U ( ρ ) = a ∂ q ln ρ s = a s∂ q ln ρ = a s ∂ q ρρ = a s ∂ q ρρ , (16)where we have denoted a s = a s . Now, taking spatial derivation on both sides of theabove equation and using the left equation in Eq. (9), one gets ∂ q U (cid:12)(cid:12)(cid:12) Q = a s ∂ q ρρ (cid:12)(cid:12)(cid:12) Q − a s (cid:16) ∂ q ρρ (cid:17) (cid:12)(cid:12)(cid:12) Q = a s ∂ q ρρ (cid:12)(cid:12)(cid:12) Q . (17)Comparing this fact to the right inequality in (12) and keeping in mind the fact that ∂ q ρ | Q <
0, one concludes that a s must be non-positive. One can then verify thatany quantum probability density that satisfies Eq. (16) satisfies all the requirementsthat we set at the beginning. Moreover, assuming ρ ( ±∞ ) = 0, Ehrenfest theorem isautomatically satisfied Z dq ρ∂ q U = a s Z dq ∂ q ρ = 0 . (18)To proceed, for simplicity of notation, let us rewrite Eq. (16) as follows ∂ q U = − a ∂ q ρρ , a ≥ . (19)It can be readily integrated to obtain ρ ( q ; a ) = 1 Z ( a ) exp (cid:16) − a U ( q ; a ) (cid:17) , (20)where Z ( a ) = R dq exp( − U/a ) is a normalization constant independent of q . Weshall show later that U ( q ) can be interpreted as internal energy density. Bearingthis in mind, then the quantum probability density given in Eq. (20) resembles inform with the Maxwell-Boltzmann-Gibbs (MBG) canonical distribution in equilibriumthermodynamics. It is thus suggestive to apply thermodynamics formalism to furtherstudy the property of quantum probability density given in Eq. (20) [5].Next, let us recall that in quantum mechanics ρ ( q ) gives the essential informationon the position of the particle [2, 6, 7]. In the so-called pragmatical approach of quantummechanics, ρ ( q ) is given meaning as the probability density that the particle will be foundat q if a measurement is performed. On the other hand, in the ontological approach, ρ ( q ) is argued as the probability density that the particle is at q regardless of anymeasurement. It is thus reasonable to quantify the randomness encoded in ρ ( q ). Oneobvious way is then to use the differential entropy or the so-called Shannon informationentropy [8] over the quantum probability density given by H [ ρ ] = − Z dq ρ ( q ) ln ρ ( q ) . (21)It gives the degree of localization of the wave function in position space.One can then show that the canonical quantum probability density of the formgiven in Eq. (20) maximizes the Shannon entropy provided that the average quantumpotential is given by [9]¯ U = Z dq U ( q ) ρ ( q ) . (22)Hence, the quantum probability density developed in the previous section satisfies theso-called maximum entropy principle [10]. It has been argued that the maximum entropyprinciple is the only method to infer from an incomplete information, which does notlead to logical inconsistency [11]. The self-trapped quantum probability density can thenbe seen as the most probable quantum probability density given its average quantumand kinetic energy [12].Combined with the definition of quantum potential given in Eq. (2), Eq. (20)comprises a differential equation for ρ ( q ) or U ( q ) subjected to the condition that ρ ( q )must be normalized, R dqρ ( q ) = 1. In term of quantum potential, one has the followingnonlinear differential equation ∂ q U − a ( ∂ q U ) − ma ¯ h U = 0 . (23)Figure 1 shows the solution of Eq. (23) with the boundary conditions: ∂ q U (0) = 0and U (0) = 1 for a = 1. All numerical solutions in this paper are obtained by putting m = ¯ h = 1. The quantum potential is shifted down so that its global minimum isvanishing. One can first see that the maximum point of the quantum probability densityand the minimum point of the corresponding quantum potential coincide, thus satisfiesour requirement. Yet, what makes even interesting is that, though we only requiresthe quantum potential to trap the area of the quantum probability density around itsmaximum, it turns out that the resulting quantum potential is convex everywhere andbecomes the global trapping potential for its own source: quantum probability density. Figure 1.
The profile of quantum probability density and its corresponding quantumpotential which satisfies Eq. (23).
Next, one can also see that the solution plotted in Fig. 1 possesses blowing-uppoints at q = ± q m , namely U ( ± q m ) = ∞ [4]. Let us first prove that the blowing-up willcertainly occur at finite point from the origin as along as U (0) ≡ X is not vanishing.To do this, Let us define a new variable u = ∂ q U . The nonlinear differential equationof Eq. (23) then transforms into ∂ q u = 12 a u + 4 ma ¯ h U, (24) Figure 2. ˜ u ( q ) and u ( q ). See text for detail. Moreover, the boundary condition translates into u (0) = ∂ q U (0) = 0. Let us nowconsider the following nonlinear differential equation ∂ q ˜ u = 12 a ˜ u + 4 ma ¯ h X, (25)where X ≡ U (0) with ˜ u (0) = 0. Since U ( q ) ≥ U (0) = X , then it is obvious that | u ( q ) | ≥ | ˜ u ( q ) | .On the other hand, one can solve the latter differential equation of Eq. (25)analytically to have˜ u ( q ) = d tan( gq ) , d = 2 a ¯ h √ m, g = 1¯ h √ mX. (26)It is then clear that at q = ± ˜ q m = ± π/ (2 g ), ˜ u is blowing-up, namely ˜ u ( ± ˜ q m ) = ±∞ .Recalling the fact that | u ( q ) | ≥ | ˜ u ( q ) | , then u ( q ) is also blowing-up at point q = ± q m , u ( ± q m ) = ±∞ , where q m ≤ ˜ q m . See Fig. 2. Putting this into the original nonlineardifferential equation of Eq. (23), one concludes that U ( q ) is also blowing up at q = ± q m , U ( ± q m ) = ∞ . Notice that even though u ( − q m ) = −∞ is blowing-up to minus infinity, U ( − q m ) = ∞ is obviously blowing up into positive infinity. Next, it is clear that theblowing-up is due to the existence of the nonlinear term on the right hand side ofEq. (23). Hence, finally one can safely say that for any non-vanishing U (0) = X ,the corresponding quantum probability density possesses only a finite range of support, q ∈ [ − q m , q m ]. The case when U (0) = 0 will give the trivial solution U ( q ) = 0 for thewhole space q so that ρ ( q ) is unnormalizable. The above fact also confirms our assertionin Section I that the quantum potential might take large value even at points where thecorresponding quantum probability density is very small.In Fig. 3 we plot the blowing-up point q = q m , namely half length of the spatialsupport of the quantum probability density against the value of the quantum potentialat the global minimum: U (0) = X . One observes that q m is decreasing as we increase X for fixed a = 1. This can be understood directly from Eq. (26). One can also confirmthat the occurrence of blowing-up is the case only when X = 0, namely lim X → q m = ∞ . Figure 3.
The half length of support q m plotted against the variation of the globalminimum U (0) = X .
3. Solitonic wave function
Now let us proceed to study the behavior of the quantum potential as one varies thenon-negative parameter a . Figure 4 gives the variation of the blowing-up point, q m ,thus the size of the range of the support against the variation of the parameter a .This is obtained by solving the differential equation of Eq. (23) with fixed boundaryconditions: ∂ q U (0) = 0 and U (0) = 1. One first observes that as a is increased, q m decreases and eventually vanishing for infinite value of a . This shows that thequantum probability density is becoming narrower for larger a while kept normalized;and eventually collapsing onto Dirac delta function for infinite value of a . Figure 4.
The half length of the support against the variation of a . A very interesting phenomena is observed as one decreases the parameter a towardzero. One finds that the blowing up point q m is increasing and eventually convergingtoward a finite value q for a = 0,lim a → q m ( a ) = q . (27)This suggests to us that the quantum potential and the corresponding quantumprobability density are also converging toward certain functions for vanishing valueof a : lim a → U ( q ; a ) = U ( q ) , lim a → ρ ( q ; a ) = ρ ( q ) . (28)Let us discuss this situation in more detail. In Figure 5 we plot the profile of thequantum probability density and the corresponding quantum potential for several smallvalues of parameter a with fixed boundary conditions: ∂ q U (0) = 0 and U (0) = 1. Onecan then see that as a is decreased, the quantum potential is becoming flatterer insidethe support before blowing-up at q = ± q m ( a ). One might then guess that at the limit a = 0, the quantum potential is perfectly flat inside the support and is infinite at theblowing-up points, q = ± q . Let us show that this guess is correct. To do this, let usdenote the assumed constant value of the quantum potential inside the support as U c .Recalling the definition of quantum potential given in Eq. (2), one has − ¯ h m R ( q ) = U c R ( q ) , (29)where R ≡ ρ / . This has to be subjected to the condition that R ( ± q ) = 0. Figure 5.
The profile of self-trapped quantum probability density and itscorresponding quantum potential for several small values of parameter a . We alsoplot the case when a = 0 obtained analytically in Eq. (30). Next, solving equation (29) one obtains R ( q ) = A cos( k q ) , (30)where A is a normalization constant and k is related to the quantum potential as k = q mU c / ¯ h . (31)The boundary condition imposes k q = π/
2. In Fig. 5, we plot the above obtainedquantum probability density, ρ ( q ). One can see that as a is decreasing toward zero, ρ ( q ; a ) obtained by solving the differential equation of (23) is indeed converging toward ρ ( q ) given in equation (30). This confirms our guess that at a = 0 the quantumpotential takes a form of flat box with infinite wall at q = ± q .Let us now take { ρ ( q ) , v ( q ) } as the initial state of the dynamics. Here v ( q ; 0) = v c is a uniform velocity field with non-vanishing constant value only inside the support.Since at a = 0 the quantum potential is flat, then inside the support the quantum forceis vanishing: F = − ∂ q U = 0. Inserting this into the dynamical equation of Eqs. (1), onehas dv/dt = 0. Hence the velocity field at infinitesimal lapse of time, t = ∆ t , remainsconstant and uniform. This in turn will not change the initial probability density, butshift it in space by ∆ q = v c ∆ t : ρ ( q ; ∆ t ) = ρ ( q − v c ∆ t ; 0). Accordingly, the support isalso shifted by the same amount. This will repeat in the next infinitesimal time lapseand so on and so forth so that at finite lapse of time t , both the initial velocity field andquantum probability density remains unchanged but are shifted by an amount ∆ q = v c t .One thus has ρ ( q ; t ) = ρ ( q − v c t ; 0) = A cos ( k q − ω t ) , (32)where we have put ω = k v c .Before proceeding, let us give physical meaning to the average quantum potential.To do this, let us calculate the ordinary quantum mechanical energy given by h E i ≡ R q − q dq ψ ∗ ( q )( − ¯ h / m ) ∂ q ψ ( q ) . Writing the wave function in polar form one has h E i = Z q − q dq (cid:16) − ¯ h m R∂ q R + 12 m R ( ∂ q S ) − i ¯ hm R∂ q R∂ q S − i ¯ h m R ∂ q S (cid:17) . (33)The first term on the right hand side is equal to the average quantum potential,¯ U = R dqU ρ . Next, defining kinetic energy density as K ( q ) = ( m/ v ( q ), the secondterm is equal to the kinetic energy ¯ K = R dqKρ of the Madelung fluid, which for ourstationary state is given by ¯ K = ( m/ v c . Further, for a uniform velocity field, the lastterm is vanishing, ∂ q S = m∂ q v c = 0. Again for a uniform velocity field, since R ( q ) isan even function and ∂ q R ( q ) is an odd function then the third term is also vanishing.Hence, in total, the quantum mechanical energy of a self-trapped wave function for a = 0 moving with a uniform velocity field can be decomposed as h E i = ¯ U + ¯ K. (34)One can then conclude that ¯ U must essentially be interpreted as the rest energy of thesingle particle. Namely it is the energy of the particle when it is not moving so that¯ K = 0. Moreover, since inside the support the quantum potential is flat given by U c ,one has ¯ U = R q − q U ( q ) ρ ( q ) = U c . Recalling Eq. (31), one finally obtains h E i = ¯ h k m + 12 mv c . (35)Let us now give the corresponding complex-valued stationary wave function ψ ( q ).One thus needs to calculate the quantum phase S which can be obtained by integrating ∂ q S = mv c to give: S ( q ; t ) = mv c q + σ ( t ), where σ ( t ) depends only on time. Onetherefore has ψ st ( q ; t ) = A cos ( k ( q − v c t )) exp (cid:16) ( i/ ¯ h )( mv c q + σ ( t )) (cid:17) , where q ∈ M t ≡ [ v c t − q , v c t + q ]. Inserting this into the Schr¨odinger equation of Eq. (4) and using Eq.(35) one has σ ( t ) = −h E i t modulo to some constant. Putting all these back, one finallyobtains the following solution: ψ st ( q ; t ) = A cos ( k ( q − v c t )) exp (cid:16) i ¯ h ( mv c q − h E i t ) (cid:17) , (36)where q ∈ M t .Equation (36) is of soliton type. This suggests a direct association of the wavefunction to a particle by considering the wave function as a physical field . In this regard,the continuity equation of mass density of Eq. (6) is becoming relevant. On the otherhand, we have also shown in the previous section that the localized-stationary-travelingsolution belongs to a class of wave function which maximizes Shannon entropy, whichsuggests that it is a probabilistic wave field . Hence, one arrives at one of the old problemof quantum mechanics concerning the physical status of the wave function.
4. Conclusion and Interpretation
By exploiting the scaling invariant property of the quantum potential, we show thatthe requirement of local stability on the maximum of quantum probability density leads1us to a class of quantum probability densities which is globally trapped by its ownquantum potential with finite-size spatial support. It turns out that they belong to aclass of wave function which maximizes Shannon entropy given the average quantumpotential. Further, we show that for a single free particle quantum system, there isan asymptotic limit in which the self-trapped wave function is traveling while keepingits form unchanged. This fact thus suggests to us to associate the localized-travelingquantum probability density as a real particle. In contrast to this, in conventionalformalism of quantum mechanics, one usually choose a plane wave to represent a freesingle particle.In our formalism of a single particle as a localized and self-trapped wave function,we showed that a particle possesses an internal energy which is absence if one use a planewave instead. It is equal to the quantum potential. On the other hand, we showed inthe beginning of the paper that the quantum potential is invariant under the globalrescaling of its own source, namely the quantum probability density. It depends onlyon the form of the latter. It is a surprising fact, since, usually energy is an extensivequantity with respect to its source. Hence, the internal energy is of different naturefrom the ordinary one. Since it only records the profile of its own source, one mightconclude that its nature is informational, rather than material.An interesting point is left unexplored. On one hand, we showed that the canonicalform of quantum probability density given in Eq. (20) is the consequence of the scaleinvariant property of the quantum potential. On the other hand, we also showed thatit maximizes Shannon information entropy given its average quantum potential. Onemay then expect that these two facts are related in a nontrivial way.
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