We derive a non-linear differential equation that must be satisfied by the quantum potential, in the context of the Madelung equations, in one dimension for a particular class of wave functions. In this case, we exhibit explicit conditions leading to the blow-up of the quantum potential of a free particle at the boundary of the compact support of the probability density

Devillanova, G.

Florio, G.

Maddalena, F.

English

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DOI 10.1393/ncc/i2013-11534-2Colloquia: MSQS 2012IL NUOVO CIMENTO Vol. 36 C, N. 3 Maggio-Giugno 2013Blow-up of the quantum potential for a free particlein one dimensionG. Devillanova(1), G. Florio(2)(3)(4)(∗) and F. Maddalena(1)(1) Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari70125 Bari, Italy(2) Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”Piazza del Viminale 1, I-00184 Roma, Italy(3) Dipartimento di Fisica and MECENAS, Universita` di Bari - I-70126 Bari, Italy(4) INFN, Sezione di Bari - I-70126 Bari, Italyricevuto il 18 Gennaio 2013Summary. — We derive a non-linear differential equation that must be satisfied bythe quantum potential, in the context of the Madelung equations, in one dimensionfor a particular class of wave functions. In this case, we exhibit explicit conditionsleading to the blow-up of the quantum potential of a free particle at the boundaryof the compact support of the probability density.PACS 03.65.-w – Quantum mechanics.PACS 03.65.Ca – Formalism.PACS 02.30.Hq – Ordinary differential equations.1. – IntroductionThe hydrodynamical formulation of quantum mechanics is based on the work of DavidBohm [1] and on the equations obtained by Madelung in his pioneering paper publishedin 1927 [2]. The lines of research related to this approach have grown during the last yearsand have found applications in several fields in the context of the Quantum TrajectoriesMethod [3] (for instance, for the analysis of semiconductor devices [4]). One of thecentral concepts in this case is the so-called Quantum Potential, the term appearing inthe quantum Hamilton-Jacobi equation (see sect. 2) which depends on the structure ofthe probability density function associated to the wave function. Of particular interest is(∗) E-mail: giuseppe.florio@ba.infn.itc© Societa` Italiana di Fisica 8384 G. DEVILLANOVA, G. FLORIO and F. MADDALENAthe problem related to the presence of singularities of the quantum potential. Typically,in the proximity of these points, due to the fact that the probability density goes tozero, the analysis of the problem is not easy and can require a particular care andproper strategies [5]. As a matter of fact, it would be interesting to find conditions interms of the probability density that guarantees the presence of blow-up of the quantumpotential.In this paper we will follow a particular approach to this problem. More precisely, wewill identify a class of wave functions whose squared modulus (in this case the probabil-ity density function of a single particle) can be directly expressed as a function of thequantum potential. We will confine our analysis to the case of one spatial dimension,then derive (at fixed time) a non-linear differential equation for the quantum potential.Finally, we will show that the solutions of this equation exhibit a blow-up behavior sothat the probability density must have a compact support. We will see that this behavioris strictly connected to the assumed relation between quantum potential and probabilitydensity.The paper is organized as follows. In sect. 2 we review the Madelung equationsobtained by the Schro¨dinger equation and define the Quantum Potential. In sect. 3 wedefine the class of wave functions that we want to analyze; as an example we will show thatthe Gaussian wave packet belongs to this class. Moreover, we will obtain a differentialequation to be satisfied by the quantum potential associated to the probability density.In sect. 4 we specialize this differential equation to the case of a simpler dependenceand in sect. 5 we will show that the solution of the simplified differential equation (thequantum potential) must exhibit a blow-up for a finite value of the position. In sect. 6we perform a qualitative analysis of the solutions in the proximity of the blow-up points.In sect. 7 we draw some conclusions.2. – Quantum hydrodynamics equations and the quantum potentialIn this section we briefly review the derivation of the Madelung equations [2] that arethe basis of the hydrodynamical interpretation of quantum mechanics and of the Bohmtheory [1].Let us consider a single particle with mass m > 0 moving under the influence ofa time-invariant potential described by the smooth function V : R → R (we restrictour analysis to the one-dimensional case). The associated time-dependent Schro¨dingerequation isi∂Ψ(q, t)∂t=(− 22m∂2∂q2+ V (q))Ψ(q, t),(1)where  is the reduced Planck constant and Ψ(q, t) is the normalized (square-integrable)complex-valued wave function that, in polar form, can be expressed byΨ(q, t) =√μ(q, t) eiS(q,t) ,(2)in terms of the probability density μ(q, t) and of the action function S(q, t) (which areboth real-valued).BLOW-UP OF THE QUANTUM POTENTIAL FOR A FREE PARTICLE IN ONE DIMENSION 85Inserting eq. (2) into eq. (1) and rearranging the terms, we obtain the coupled partialdifferential equations∂μ(q, t)∂t+∂J(q, t)∂q= 0,(3)∂S(q, t)∂t+12m(∂S(q, t)∂q)2+ V (q) + Φ(q, t) = 0,(4)where the terms J(q, t) and Φ(q, t) are respectively a probability currentJ(q, t) =1mμ(q, t)∂S(q, t)∂q,(5)which depends on both μ(q, t) and S(q, t) and the so-called quantum potentialΦ(q, t) = − 22m1√μ(q, t)∂2√μ(q, t)∂q2= − 24m[1μ(q, t)∂2μ(q, t)∂q2− 12μ2(q, t)(∂μ(q, t)∂q)2],(6)which only depends on μ(q, t). So, while eq. (3) is a continuity equation for the quantumprobability density μ(q, t) which, at time t flows at the point q with rate J(q, t), eq. (4)is the quantum Hamilton-Jacobi equation.In the following sections we analyze some particular cases of blow-up of the quantumpotential for a particular class of wave functions.3. – A non-linear equation for the quantum potential at fixed timeWe consider the case of a free particle, i.e. we set V (q) = 0, ∀q ∈ R. Moreover, wefix a time t = t0 and assume that the probability density exponentially depends on thequantum potential at time t0, i.e. has the formρ(q) := μ(q, t0) = Cαe−αf(φ(q)) with φ(q) = Φ(q, t0),(7)where α ∈ R, Cα is a normalization constant and f is a regular real valued function. Wenotice that eq. (7) is a particular case of the general dependenceρ(q) = g(φ(q)),(8)where g is a regular real valued function.3.1. Example: Gaussian wave packet . – We show that the probability density of aGaussian wave packet, for instance, exhibits this behavior for a suitable α < 0, f being,in this case, the identity map. To this aim we set, at time t0 = 0, the initial state of86 G. DEVILLANOVA, G. FLORIO and F. MADDALENAa Gaussian wave packetΨ(q, 0) =1(2πΔq2)1/4exp[− q24Δq2+ip0q],(9)μ(q, 0) =1(2πΔq2)1/2exp[− q22Δq2],where p0 is the average initial momentum and Δq2 is the variance (uncertainty) of theposition of the packet. Then, taking into account the analytic behavior obtained in [6]for the free evolution of a particle described by a Gaussian wave packet, we obtainμ(q, t) =1(2πΔtq2)1/2 exp[− (q − qt)22Δtq2],(10)whereΔtq2 = Δq2 +2t24m2Δq2and qt =p0mt.(11)Rewriting eq. (6) with μ(q, t) given by eq. (10), we obtain the expression of the quantumpotential for a free particle described by a Gaussian wave packetΦ(q, t) =24mΔtq2(1− (q − qt)22Δtq2),(12)from which we deduce1− (q − qt)22Δtq2=4mΔtq22Φ(q, t).(13)So, setting at some fixed t > 0, α = −(4mΔtq2)/2 < 0, we get from eq. (10) that thebehavior assumed in eq. (7) is satisfied when f is the identity map andCα =1(2πe2Δtq2)1/2=√− 2mπe22α.(14)Finally, from eq. (10) we getμ(q, t) =√− 2mπe22αe−αΦ(q,t) = Cαe−αΦ(q,t) ,(15)as claimed.BLOW-UP OF THE QUANTUM POTENTIAL FOR A FREE PARTICLE IN ONE DIMENSION 873.2. Non-linear differential equation from eq. (7). – In what follows we obtain a dif-ferential equation describing the behavior of the quantum potential at a fixed time forwave functions satisfying eq. (7). At time t0 eq. (6) becomesφ(q) = − 24m[1ρ(q)d2ρ(q)dq2− 12ρ2(q)(dρ(q)dq)2].(16)We evaluate the first and second derivative of the function ρ(q) gettingdρ(q)dq=dg(φ)dφdφ(q)dq,d2ρ(q)dq2=dg(φ)dφd2φ(q)dq2+d2g(φ)dφ2(dφ(q)dq)2.(17)where we have abused the notation introduced in eq. (8) by writingg(φ) = g(φ(q)).(18)Moreover, from eq. (8) we getdg(φ)dφ= −αρ(q)df(φ)dφ,(19)d2g(φ)dφ2= −αρ(q)d2f(φ)dφ2+ α2ρ(q)(df(φ)dφ)2(20)= −αρ(q)[d2f(φ)dφ2− α(df(φ)dφ)2],where, as above, f(φ) means f(φ(q)).Inserting eqs. (19)-(20) into eq. (17) and then into eq. (16) we find, rearranging theterms, that the quantum potential φ associated to a probability density μ(q, t) given byeq. (7) at time t0 satisfies the following differential equation:df(φ)dφd2φ(q)dq2−[α2(df(φ)dφ)2− d2f(φ)dφ2](dφ(q)dq)2− 4mα2φ(q) = 0.(21)4. – A simplified form of the non-linear equationMotivated by the result obtained in the previous section for the Gaussian wave packet,we consider the case in which f is the identity map, so eq. (7) becomesρ(q) = Cαe−αφ(q).(22)We also assume that the probability density ρ(q) is symmetric around q = 0; this happenswhen φ is even (i.e. φ(−q) = φ(q)) as happens for the quantum potential relative to theGaussian packet at time t = 0 (at any time if p0 = 0), see eq. (12). The particulardependence of the probability density on the the quantum potential assumed in eq. (22)will be fundamental to obtain the main result of the paper. As a matter of fact, we will88 G. DEVILLANOVA, G. FLORIO and F. MADDALENA0.5 2410a1.0 0.5 0.0 0.5 1.0q246810φ qb1.0 0.5 0.0 0.5 1.0q0.20.40.60.81.0ρ qFig. 1. – (Color online) (a) Solution of eq. (23) for α = 0.5, α = 2, α = 4 and α = 10. (b) Plotof the normalized probability density ρ(q) corresponding to the solution of eq. (23) for α = 0.5(solid line), α = 2 (dashed line), and α = 10 (dotted line). We have set m = 1,  = 1, φ0 = 1and φ′(0) = 0.show that this dependence, under certain assumptions, implies a blow-up of the quantumpotential at finite position.Using eq. (22) we see that eq. (21) reduces toφ′′(q)− α2(φ′(q))2 − 4mα2φ(q) = 0,(23)where we have used the standard notationφ′(q) =dφ(q)dq, φ′′(q) =d2φ(q)dq2.(24)Then we assume φ(0) = φ0 > 0 and, being φ even, φ′ must be odd (i.e. φ′(−q) =−φ′(q)) and, as a consequence, φ′(0) = 0. By substituting these conditions into eq. (23)we immediately obtain that the solution hasφ′′(0) =4mα2φ0 > 0, with α > 0,(25)differently from the case of the Gaussian wave packet where α < 0. Therefore, forq  0, the quantum potential behaves as a confining potential. In fig. 1(a) we plot the(numerical) solution of eq. (23) for different values of the parameter α > 0; fig. 1(b)shows some examples of probability densities corresponding to the quantum potentialsobtained from eq. (23); in both cases we have set m = 1,  = 1 with the initial conditionsφ0 = 1, φ′(0)=0. The numerical integration indicates that the quantum potential exhibitsa blow-up for finite values of q and, therefore, the corresponding ρ(q)’s have a compactsupport and vanishes on the boundary.In order to analyze this property, it is useful to manipulate eq. (23). Let us definethe functionw(φ) = (φ′)2 + Bφ,(26)where A = α/2 and B = 4m/(α2). Thus, eq. (23) takes the formφ′′ = Aw(φ) + B(1−A)φ.(27)BLOW-UP OF THE QUANTUM POTENTIAL FOR A FREE PARTICLE IN ONE DIMENSION 89By taking the derivative of eq. (26) with respect to q, we deduceBφ′ =dw(φ)dq− 2φ′φ′′ = dw(φ)dφφ′ − 2φ′φ′′,(28)and thereforeφ′(B − dwdφ+ 2φ′′)= 0.(29)Inserting eq. (27) into eq. (29) we obtainφ′(B − dwdφ+ 2Aw + 2B(1−A)φ)= 0.(30)Taking into account that φ ≡ 0 is the only constant solution of eq. (23) which does notsatisfy the initial condition φ(0) > 0, we will focus on the equationdw(φ)dφ= 2Aw(φ) + 2B(1−A)φ + B.(31)Equation (31) is a first order normal differential equation, so we get, by standard calcu-lations, that, for any C ∈ R,w(φ) = Ce2Aφ +BA(A− 1)φ− B2A2,(32)is a solution to eq. (31) (as it is possible to check by direct substitution).So, by eq. (26), we obtain a new first order equation for the quantum potential attime t0 relative to a probability density μ(q, t) which satisfies eq. (22) at fixed time t0|φ′(q)| =√Ce2Aφ(q) − BAφ(q)− B2A2=√Ceαφ(q) − 8mα22φ(q)− 8mα32.(33)In the following section we will analyze some properties of the solutions to eq. (33).5. – Blow-up at finite positionsWe will focus on the case α > 0 which is not covered by the quantum potential relativeto a Gaussian wave packet. In particular, we shall deal with1) φ(0) = φ0 > 0;2) φ even to have, by eq. (22), a probability density ρ(q) symmetric around q = 0; un-der such assumption φ′ must be odd (i.e. φ′(−q) = −φ′(q)) and, as a consequence,φ′(0) = 0;3) φ increasing on [0,+∞[ (i.e. φ′(q) = |φ′(q)| if q ≥ 0) and, as a consequence,decreasing on ]−∞, 0] (i.e. φ′(q) = −|φ′(q)| if q ≤ 0).90 G. DEVILLANOVA, G. FLORIO and F. MADDALENATherefore, we can find an even solution to eq. (23) with initial conditions φ(0) = φ0,φ′(0) = 0 by an even extension of the unique solution to the equationφ′(q) =√Ceαφ(q) − 8mα22φ(q)− 8mα32.(34)with initial condition φ(0) = φ0.We are now ready to show the existence of finite blow-up of the quantum potential,which is solution of eq. (34) and, after even extension, of eq. (23). This property ischaracterized by the following statement.Proposition 1. For every φ0 > 0 the solution to eq. (34) satisfying φ(0) = φ0 exhibitsa blow-up at the finite value q∗ < +∞ satisfying the following inequality:q∗ ≤√α√2mmax( √α3 + e−αφ0/2√Fα(φ0)− Fα(2φ0)e−αφ0/2,√α +e−αφ0√Fα(φ0)− Fα(2φ0)),(35)where, for all x ∈ R, Fα(x) = (αx + 1)e−αx.Proof. By imposing the condition φ′(0) = 0, or equivalently the “pairing” conditionw(φ0) = −Bφ0, we get from eq. (34)C =8mα22(φ0 +1α)e−αφ0(36)and thereforeφ′(q) =√8mα32eαφ/2√(αφ0 + 1)e−αφ0 − (αφ + 1)e−αφ.(37)Then, we define the functionFα(x) = (αx + 1)e−αx,(38)which results to be strictly decreasing ∀x ≥ 0 so that eq. (37) takes the formφ′(q) =√8mα32eαφ/2√Fα(φ0)− Fα(φ) =: θ(φ),(39)where θ(φ) is well defined for φ ≥ φ0. Notice that φ → θ(φ) is a positive and continuousfunction. To prove the thesis we are going to show that the Osgood condition [7] issatisfied, namely thatq∗ =∫ +∞φ01θ(φ)dφ < +∞.(40)To this aim, since the funcion 1/θ(φ) is unbounded near φ0, we additively split theintegral in eq. (40) into two terms and proceed to estimate them. We setq∗1 =∫ 2φ0φ01θ(φ)dφ and q∗2 =∫ +∞2φ01θ(φ)dφ.(41)BLOW-UP OF THE QUANTUM POTENTIAL FOR A FREE PARTICLE IN ONE DIMENSION 91In order to evaluate q∗1 we have to estimatelimω→φ+0∫ 2φ0ω1θ(φ)dφ.(42)We firstly observe that,∃ limφ→φ0√Fα(φ0)− Fα(φ)√φ− φ0= α√φ0 e−αφ0/2 =: l0,(43)so that, not only√Fα(φ0)− Fα(φ)  l0√φ− φ0 = α√φ0 e−αφ0/2√φ− φ0 as φ → φ0,(44)but, a posteriori, alsoddφ√Fα(φ0)− Fα(φ)  l0 ddφ√φ− φ0 as φ → φ0.(45)Therefore, settingη0 = max(l0√2φ0 − φ0√Fα(φ0)− Fα(2φ0), 1)= max(αφ0e−αφ0/2√Fα(φ0)− Fα(2φ0), 1)(46)we easily obtain the following boundq∗1 =√α328m∫ 2φ0φ0e−αφ/2√Fα(φ0)− Fα(φ)dφ≤√α328m∫ 2φ0φ0e−αφ0/2η0α√φ0e−αφ0/2√φ− φ0dφ(47)≤√α228mη0√φ0∫ 2φ0φ01√φ− φ0dφ =√α222mη0 < +∞.In order to estimate q∗2 we argue as follows. Since x → Fα(x) is strictly decreasing onR+, for every φ > 2φ0 we have1√Fα(φ0)− Fα(φ)<1√Fα(φ0)− Fα(2φ0).(48)Therefore we findq∗2 =√α328m∫ +∞2φ0e−αφ/2√Fα(φ0)− Fα(φ)dφ≤√α328m1√Fα(φ0)− Fα(2φ0)∫ +∞2φ0e−αφ/2dφ(49)=√α22m1√Fα(φ0)− Fα(2φ0)e−αφ0 < +∞.Then, by setting q∗ = q∗1 + q∗2 and employing eq. (47) and eq. (49) we get the thesis.92 G. DEVILLANOVA, G. FLORIO and F. MADDALENA6. – Qualitative analysis of the quantum potential in the proximity of theblow-upWe can now perform a qualitative analysis of the features of the solutions of eq. (34)in proximity of the blow-up points.Using the function Fα(x) introduced in eq. (38) by which C = 8mFα(φ0)/(α32),eq. (34) becomesφ′(q) =√Ceαφ(q)/2√1− Fα(φ)Fα(φ0).(50)Taking into account that Fα(φ)/Fα(φ0) tends to zero as φ → +∞, eq. (50) takes theformφ′(q) √Ceαφ(q)/2 =√8m(αφ0 + 1)α32exp[−α(φ− φ0)2].(51)for φ large enough. Equation (51) admits a solution of the formφ(q)  − 2αln(−α2√Cq + D),(52)where D is an integration constant. In this approximation the quantum potential exhibitsa blow-up when q+ = 2D/(α√C), with D,C < +∞. The points q+ and q− = −q+represent, in the above approximation, the boundary of the support of the even solutionto eq. (51).7. – ConclusionsIn this paper we have defined a class of wave functions whose probability density(assumed symmetric with respect to q = 0) can be directly expressed in terms of theassociated quantum potential. Moreover, we have found that it is possible to derive adifferential equation for the quantum potential whose solution exhibits a blow-up behav-ior forcing the probability density to have a compact support. This kind of analysis is ofinterest to predict the existence of singularities of the quantum potential and, therefore,the identification of probability densities with compact support. It would be interestingto extend these results to a more general functional dependence of the probability densityon the quantum potential going beyond the identity map. Moreover, this analysis havebeen performed fixing a certain value of time t0. One could also extend this work in orderto study the evolution of the quantum potential during the propagation of the particle.∗ ∗ ∗The authors thank S. Solimini for useful discussions. GF acknowledges supportfrom the University of Bari through the Project IDEA and from Istituto Nazionale diAlta Matematica and Gruppo Nazionale per la Fisica Matematica through the ProjectGiovani GNFM.BLOW-UP OF THE QUANTUM POTENTIAL FOR A FREE PARTICLE IN ONE DIMENSION 93REFERENCES[1] Bohm D., Phys. Rev., 85 (1952) 166; 180.[2] Madelung E., Z. Phys., 40 (1927) 322.[3] Wyatt R. E., Quantum Dynamics with Trajectories: Introduction to Quantum Hydrody-namics (Springer, New York) 2005.[4] Gardner C. L., SIAM J. Appl. Math., 54 (1994) 409.[5] Wyatt R. E., J. Chem. Phys., 121 (2004) 9230.[6] Shankar R., Principles of Quantum Mechanics, 2nd edition (Plenum) 1994.[7] Osgood W. F., Monatsh. Math. Phys., 9 (1898) 331.

Blow-up of the quantum potential for a free particle in one dimension

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Blow-up of the quantum potential for a free particle in one dimension

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