A new approach to solving the Schrödinger equation
aa r X i v : . [ qu a n t - ph ] J u l A new approach to solving the Schrödinger equation
Sergio A. Hojman
1, 2, 3, 4, ∗ and Felipe A. Asenjo † Departamento de Ciencias, Facultad de Artes Liberales,Universidad Adolfo Ibáñez, Santiago 7491169, Chile. Centro de Investigación en Matemáticas, A.C., Unidad Mérida, Yuc. 97302, México Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago 7800003, Chile. Centro de Recursos Educativos Avanzados, CREA, Santiago 7500018, Chile. Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago 7491169, Chile.
A new approach to find exact solutions to one–dimensional quantum mechanical systems is de-vised. The scheme is based on the introduction of a potential function for the wavefunction, and theequation it satisfies. The potential function defines the amplitude and the phase of any wavefunctionwhich solves the one–dimensional Schrödinger equation. This new approach allows us to recoverknown solutions as well as to get new ones for both free and interacting particles with wavefunctionsthat have vanishing and non–vanishing Bohm potentials. For most of the potentials, no solutionsto the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted familyof potentials allows the existence of particular solutions for which the Bohm potential vanishes.This family of potentials is determined, and several examples are presented. It is shown that someunexpected and surprising quantum results which seem to (but do not) violate the correspondenceprinciple such as accelerated Airy wavefunctions which solve the free Schrödinger equation, are dueto the presence of non–vanishing Bohm potentials. New examples of this kind are found and dis-cussed. The relation of these results to some of the unusual solutions to other wave equations isbriefly discussed.
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I. INTRODUCTION
The Madelung–Bohm approach to Schrödinger’s equa-tion, written in terms of the amplitude and the phase ofthe quantum wavefunction, gives rise to two real equa-tions. The first one is a Quantum Hamilton–Jacobi equa-tion (QHJE) that is similar to its classical counterpart[1, 2]. It differs from it by the addition of an extra term,the Bohm potential, which depends on the amplitude ofthe wavefunction only [1–3]. The second one is the con-tinuity (probability conservation) equation.The presence of the Bohm potential in the QHJEgives rise to unexpected solutions for the wavefunctionthat, in turn, behave in surprising ways that seem tobe at odds with their classical counterparts. One ofthe most striking results, accelerating “free” quantumparticles, was predicted by Berry and Balazs in 1979[4] and found experimentally, in an optical setting, bySiviloglou, Broky, Dogariu, and Christodoulides in 2007[5]. The best known free quantum particle wavefunc-tions have amplitudes whose Bohm potential vanishesand, therefore, their QHJEs are identical to the classi-cal Hamilton–Jacobi equations and they present no sur-prises. On the other hand, the Berry–Balazs solutionexhibits non–vanishing Bohm potential and a departurefrom its classical limit. This kind of unexpected behaviordue to the presence of the Bohm potential appears in any ∗ Electronic address: [email protected] † Electronic address: [email protected] wave equation irrespective of its classical or quantum ori-gin, as it does in the scalar wave equation, in Maxwell’sequations, in gravitational waves equations [3].The main aim of this work is to propose a mechanismto find exact solutions to the Schrödinger equation wherewe can explicitly identify the contribution of Bohm po-tential. In that way, we expect to obtain new solutions,with interesting features, for free particles as well as forsystems with non–trivial potentials. To contribute deep-ening the understanding of the role of Bohm potentialin Quantum Mechanics, we classify potentials accord-ing to whether or not they admit solutions with van-ishing Bohm potentials for one-dimensional systems. Itis shown that the system can be completely solved byre–writing the wavefunction in terms of a potential field f . A non–linear integro–differential equation is derivedfor f . This equation, as far as we know, is new and itis equivalent to Schrödinger’s. It provides new insightsin the interrelation between the amplitude and the phaseof the wavefunction. In general, we solve exactly theSchrödinger equation for families of solutions with non–constant Bohm potentials for a wide variety of externalpotentials [6].Finally, we also discuss the pertinence of these resultsto the applicability of the Van Vleck–Morette determi-nant approximation [7, 8]. II. GENERAL APPROACH INONE–DIMENSIONAL QUANTUM MECHANICS
For a real potential V ( x, t ) , let us consider theSchrödinger equation in a one–dimensional space (cid:20) − ~ m ∂ ∂x + V ( x, t ) − i ~ ∂∂t (cid:21) ψ ( x, t ) = 0 , (1)for wavefunction ψ = ψ ( x, t ) , and its complex conjugatecounterpart. Therefore, a solution for the wavefunctionin terms of the polar form can be invoked [1, 2] ψ ( x, t ) = A ( x, t ) e iS ( x,t ) / ~ , (2)where now A ( x, t ) and S ( x, t ) are real functions of spaceand of time. This solution produces that the real andimaginary parts of Schrödinger equations become, re-spectively, m S ′ − ~ m A ′′ A + V + ˙ S = 0 , (3) m (cid:0) A S ′ (cid:1) ′ + ˙ A = 0 . (4)where and S ′ ≡ ∂ x S , and ˙ S ≡ ∂ t S (and similarly for A ).The first equation (3) is a modified version of the classi-cal Hamilton–Jacobi equation for the potential V [1, 2],while the second one (4) is simply the continuity (prob-ability conservation) equation. The classical Hamilton–Jacobi equation is modified by the addition of the Bohmpotential V B in one dimension V B ≡ − ~ m A ′′ A , (5)which can be interpreted as an internal energy [9].Rather than of solving the previous system, let us in-troduce the function f = f ( x, t ) , defined by the followingrelations A ( x, t ) = p f ′ ,S ( x, t ) = µ ( t ) − m Z ˙ ff ′ dx . (6)for an arbitrary time–dependent function µ ( t ) . The in-troduction of the potential field f transforms the continu-ity equation (4) in an identity. However, more important,it is the fact that knowing f , the fields A and S are found,and thus the wavefunction is completely determined.In order to find the evolution of field f , we need toevaluate a modified version of the classical Hamilton–Jacobi equation (3). This equation is now re-written as m ˙ f f ′ − ~ m (cid:18) f ′′′ f ′ − f ′′ f ′ (cid:19) + m Z ˜ x = x ˜ x =0 ˙ f ˙ f ′ f ′ − ¨ ff ′ ! d ˜ x + ˙ µ + V = 0 , (7) For a given potential V , this equation establishes the evo-lution of field f . However, this equation can also be readin the opposite way. Starting from any f , the equationdefines the potential that solves Schrödinger equation.Also, the gradient of Eq. (7) can be considered an equa-tion for the force F = − V ′ , by differentiating Eq. (7)with respect to x , m ˙ f f ′ − ~ m (cid:18) f ′′′ f ′ − f ′′ f ′ (cid:19)! ′ − F + m ˙ f ˙ f ′ f ′ − ¨ ff ′ ! = 0 . (8)It is important to remark that the above solutions for A and S are general for a one–dimensional system. Thisimplies that they provide a general solution for the ampli-tude which is, in general, different from the Van Vleck–Morette expression (VVM) [7, 8] for one–dimensional sys-tems A = s iπ ~ ∂ S∂x f ∂x i , (9)with initital x i and final x f positions. The VVM re-sult represents an approximation to the amplitude (inthe WKB approach sense) of the Schrödinger equationwavefunction. Below we show explicit examples for ex-act solutions (7) which coincide with the approximateVVM expression (9). On the other hand, we show otherexamples of exact solutions that do not coincide with theVVM expression for the amplitude of the wave function.It is clear that Bohm potential (5) plays a remark-able role in the solutions one can find for a given po-tential V . The Bohm potential is the origin to the dif-ference between classical and quantum dispersion rela-tions that emerge from the modified version of the clas-sical Hamilton–Jacobi equation [3]. For this reason, it isworth studying Eq. (7) for free and interacting particlesclassifying the solutions according to whether or not theyproduce a vanishing Bohm potential (5). We explore dif-ferent potentials V , for each case. We prove below thatthe condition of producing a vanishing Bohm potentialgives rise to a general family of time–dependent forces. III. VANISHING BOHM POTENTIAL
The WKB approximation in Quantum Mechanics andthe eikonal approach which leads to Geometrical Opticsdeal with solutions which yield a negligible Bohm poten-tial [10], i.e., a Bohm potential which is approximatelyequal to zero (either because ~ or the wavelength of lightare considered small, in the quantum mechanical or op-tical cases, or the amplitude is assumed to vary slowlyin either case). In this section we will find the most gen-eral one–dimensional potential V which admits solutionswhich give rise to a Bohm potential (5) which is exactlyequal to zero.For a vanishing Bohm potential (5), the function f must satisfy A ′′ A ≡ (cid:18) f ′′′ f ′ − f ′′ f ′ (cid:19) = 0 . (10)The general solution for function f ( x, t ) reads f ( x, t ) = a ( t ) x + a ( t ) b ( t ) x + b ( t ) x + c ( t ) , (11)for arbitrary functions a ( t ) , b ( t ) and c ( t ) . In this case,Eq. (7) reduces to m ˙ f f ′ + m Z ˜ x = x ˜ x =0 ˙ f ˙ f ′ f ′ − ¨ ff ′ ! d ˜ x + ˙ µ + V = 0 , (12)The force F = − V ′ , derived from the above potential,for vanishing Bohm poential, is readily obtained as F ( x, t ) = m ˙ f f ′ ! ′ + m ˙ f ˙ f ′ f ′ − ¨ ff ′ ! = − xa ( t ) + b ( t )) · [ mx (cid:0) − a ( t ) ˙ a ( t ) + 6 a ( t ) ¨ a ( t ) (cid:1) + mx ( − a ( t ) b ( t ) ˙ a ( t ) − a ( t ) ˙ a ( t )˙ b ( t ) +27 a ( t ) b ( t )¨ a ( t ) + 9 a ( t ) ¨ b ( t )) + mx ( − a ( t ) b ( t ) ˙ a ( t ) − a ( t ) b ( t ) ˙ a ( t )˙ b ( t ) +45 a ( t ) b ( t ) ¨ a ( t ) + 45 a ( t ) b ( t )¨ b ( t )) + mx ( − b ( t ) ˙ a ( t ) − a ( t ) b ( t ) ˙ a ( t )˙ b ( t ) − a ( t ) b ( t )˙ b ( t ) − a ( t ) ˙ a ( t ) ˙ c ( t ) +33 a ( t ) b ( t ) ¨ a ( t ) + 81 a ( t ) b ( t ) ¨ b ( t ) +9 a ( t ) ¨ c ( t )) + mx ( − b ( t ) ˙ a ( t )˙ b ( t ) − a ( t ) b ( t ) ˙ b ( t ) − a ( t ) b ( t ) ˙ a ( t ) ˙ c ( t ) + 9 b ( t ) ¨ a ( t ) +63 a ( t ) b ( t ) ¨ b ( t ) + 27 a ( t )2 b ( t )¨ c ( t )) + mx ( − b ( t ) ˙ b ( t ) − b ( t ) ˙ a ( t ) ˙ c ( t ) +18 b ( t ) ¨ b ( t ) + 27 a ( t ) b ( t ) ¨ c ( t )) + m ( − b ( t ) ˙ b ( t ) ˙ c ( t ) + 18 a ( t ) ˙ c ( t ) +9 b ( t ) ¨ c ( t ))] . (13)This force (or its associated one–dimensional potential V ) is the most general one whose Schrödinger equationsadmit solutions with vanishing Bohm potential, i.e., suchthat V B ( x, t ) = 0 . This depends on the different andarbitrary choices of the time–dependent functions a ( t ) , b ( t ) and c ( t ) . On the other hand, every time that f doesnot have the form (11), Bohm potential does not vanish.Below we study different exact solution that exemplifythose cases. IV. FREE PARTICLES WITH VANISHINGBOHM POTENTIAL
Several solutions for the quantum free particle, with V = 0 , produce a vanishing Bohm potential. A. Free particle as a plane wave
Using (11), let us choose a ( t ) = 0 , b ( t ) = 1 , c ( t ) = − km t , (14)where k is a constant with units of inverse length. Thisfunctions allow us to study a free particle with vanishingBohm potential. This produces the amplitude and phase A ( x, t ) = 1 ,S ( x, t ) = kx − k m t , (15)where µ ( t ) = k t/ m . This quantum solution representsa free particle as a plane wave, with a wave phase velocityequal to k/ m . B. Non–separable solution for the quantum freeparticle
The above solution can be easily found by traditionalapproaches by separating the functionality of space andtime in the wavefunction. However, there are other solu-tions for quantum free particles, where this is not possi-ble.Let us choose f from (11), such that a ( t ) = r α ( t − t i ) , b ( t ) = r βt − t i , c ( t ) = γ , (16)with constant α , β , γ , and initial time t i . Then, theamplitude and phase (6) becomes A ( x, t ) = √ α x ( t − t i ) / + r βt − t i ,S ( x, t ) = m x t − t i ) , (17)with µ = 0 . This is a solution for the free particle, with V = 0 . V. FREE PARTICLES WITH NON–VANISHINGBOHM POTENTIAL
The previous cases are known free particle solutions.What is probably not as well–known is that they give riseto a vanishing Bohm potential. This opens the possibilityto look for other free particle solutions with V = 0 anda non–vanishing Bohm potential.In this section we show some of them for time–dependent amplitudes. In these cases, the function f does not satisfy (11). This is an important fact since thenon–vanishing Bohm potential introduces unexpected ef-fects such as self–acceleration in exact solutions, as weshow below. A. Free particle as a non–plane wave witharbitrary velocity
We show in Sec. IV A that a free particle as a planewave must have a vanishing Bohm potential. A non–plane wave has a variable amplitude, and then, producesBohm potential different from zero.For the current case, let us consider the function f ( x, t ) = 1 λ exp (cid:18) λ x − ~ λ km t (cid:19) , (18)where λ , and k are arbitrary constants with units of in-verse length. This f produce a wavefunction that solvesthe free particle problem with V = 0 .In this case, we can generate the amplitude and phaseof the wavefunction A ( x, t ) = s exp (cid:18) λ x − ~ λ km t (cid:19) ,S ( x, t ) = ~ kx + µ ( t ) ,µ ( t ) = − ~ k t m (cid:18) − λ k (cid:19) . (19)This wave is different to the plane wave (15), as it has aconstant Bohm potential (5), given by V B = − ~ λ m . (20)A vanishing Bohm potential is obtained only in the case λ = 0 , thus recovering a plane wave solution. On theother hand, the phase velocity of this wave is ~ k m (cid:18) − λ k (cid:19) (21)which can be as small as it is required, when λ → k .However, this solution has a constant velocity, as no force(and thus no acceleration) is applied on the evolution ofthe particle. This can also be understood as, for thiscase, V ′ B = 0 . B. Accelerating Airy wave packets
It is interesting to show how the very well–known Airywave packets [4, 5, 11–15] is obtained in our formal-ism. We prove below that the acceleration experiencedby these wave packets is due to the Bohm potential. Let us consider f ( x, t ) = Z (cid:20) A i (cid:18) β ~ / (cid:18) x − β m t (cid:19)(cid:19)(cid:21) dx , (22)with a non–zero constant β . For this case A ( x, t ) = A i (cid:18) β ~ / (cid:18) x − β m t (cid:19)(cid:19) ,S ( x, t ) = β t m (cid:18) x − β m t (cid:19) , (23)with µ ( t ) = − β t / (12 m ) . This solution for free parti-cles, with V = 0 , has a non–vanishing time– and space–dependent Bohm potential given by V B ( x, t ) = − β m (cid:18) x − β m t (cid:19) . (24)The Bohm potential is responsible for the constant ac-celeration a Airy experienced by the Airy wave packet a Airy = − V ′ B m = β m . (25)Note that this result is consistent with the velocity ofthe Airy package v Airy = p Airy /m where the momentum p Airy is given by p Airy = ∂S ( x, t ) /∂x . Therefore, thereis a solution to the free Schrödinger equation which hasa constant acceleration given by (25) in spite of being inthe presence of a vanishing (external) force.This solution represents a wavepacket constructed bya non–normalizable superposition of constant velocityplane wave solutions for free particles, but a normaliz-able solution can be constructed from it [16], which canbe shown to travel without acceleration. C. A different solution for a free particle
We can find a different set of solutions for V = 0 inthe following way. Consider f ( x, t ) = Z " p σ ( t ) Z (cid:18) xσ ( t ) (cid:19) dx , (26)where σ ( t ) is an arbitrary time–dependent function, and Z is also an arbitrary function of argument x/σ ( t ) . Thisfunction generates the following one–dimensional ampli-tude and phase A ( x, t ) = 1 p σ ( t ) Z (cid:18) xσ ( t ) (cid:19) ,S ( x, t ) = m ˙ σ ( t )2 σ ( t ) x + µ ( t ) . (27)For this case, Eq. (11) becomes m ¨ σ σ x + ˙ µ = ~ mσ Z d Zdy = − V B , (28)with derivatives respect to the argument y ≡ x/σ . Theonly form to solve Eq. (28) in a general fashion is when Z d Zdy = ζ x σ + ζ . (29)for arbitrary constants ζ and ζ . In such case, the func-tions σ and µ are completely determined by the equations σ = r αt + βt + β α + ~ ζ αm ,µ = ~ ζ √ ζ arctan (cid:18) m ~ √ ζ (2 αt + β ) (cid:19) , (30)for constants α and β . Thereby, the problem of a free par-ticle is solved. Notice that Eq. (29) implies that Bohmpotential (28) is non–zero in general. With these re-sults, phase (27) results to be a generalization of phaseof Sec. IV B.Any function Z satisfying (29) produces solutions. Oneexample is a Gaussian wavepacket Z = exp (cid:18) − q x σ (cid:19) , (31)for some constant q . that gives ζ = 4 q , and ζ = − q .These solutions are for free particles, and they do nothold, for example, if the quantum system has diffusion[17].Other example is the families of functions that pro-duces ζ = 0 and ζ = constant. Some of those funcionsare sin , cos , sinh − , cosh − , among others.The most general functions satisfying (29) are Weberfunctions, also called Parabolic cylinder functions [18].Once, the complete solution for the Parabolic cylinderfunctions is proposed for arbitrary ζ and ζ , the wholeproblem is solved. VI. PARTICLES UNDER NON–ZEROEXTERNAL POTENTIAL WITH VANISHINGBOHM POTENTIAL
In the presence of an external potential, when theBohm potential vanishes, we can use Eq. (11) to con-struct a solution to the quantum problem. We show howto proceed for standard potentials.
A. Simple solution for the quantum harmonicoscillator
The best known solutions for the wavefunctions of aquantum harmonic oscillator (written in terms of Her-mite polynomials) have non–vanishing Bohm potentials.One usual solution to the Schrödinger equation for theharmonic oscillator potential V = mω x / (with con-stant frequency ω ) can be recovered from the solution (11) for vanishing Bohm potential, when a ( t ) = 0 , and b ( t ) = r α sin ( ω ( t − t i )) , c ( t ) = − αx i tan ( ω ( t − t i )) , (32)for constant α and t i . This solution allows us to find theamplitude and phase of the wavefunction A ( x, t ) = r α sin ( ω ( t − t i )) ,S ( x, t ) = mω (cid:0) x + x i (cid:1) ω ( t − t i )) − mωxx i sin ( ω ( t − t i )) . (33)These are amplitude and the phase for the one standardwavefunction for a particle subject to a harmonic oscil-lator potential [19]. In this case, the amplitude coincideswith the one prescribed by the VVM expression (9), whentaking α = imω/ (2 π ~ ) . B. A different solution for the harmonic oscillator
It is not trivial to show that a particle subject to theharmonic oscillator potential V = mω x / can have an-other solution completely different to the previous one,with vanishing Bohm potential. In Eq. (11), let us take a ( t ) = 0 = c ( t ) , and b ( t ) = cos − / ( ω ( t − t i )) . (34)Thereby, we can calculate A ( x, t ) = cos − / ( ω ( t − t i )) ,S ( x, t ) = − mωx ω ( t − t i )) . (35)It can be straightforwardly proved that this is an exactsolution for the harmonic oscillator, with vanishing Bohmpotential. Solution (35) is different from (33), and theexact amplitude is not described by the VVM expression(9). C. Another different solution for the harmonicoscillator
Another solution can be found for the harmonic oscil-lator by taking, in solution (11), the conditions b ( t ) =0 = c ( t ) , and a ( t ) = cos − / ( ω ( t − t i )) . (36)For this case, we get A ( x, t ) = x cos − / ( ω ( t − t i )) ,S ( x, t ) = − mωx ω ( t − t i )) . (37)This is also an exact solution for the harmonic oscillatorwith vanishing Bohm potential. The exact amplitude is not given by the VVM expression (9).Notice that solutions VI B and VI C have the samephase, but different amplitudes, and therefore those twopackets can be differentiated from each other. D. Time–independent forces
A whole family of time–independent forces can beproved to solve Schrödinger equation with vanishingBohm potential. In Eq. (11), let us consider the case a ( t ) = a e − µt , b ( t ) = b e − µt , c ( t ) = c e − µt , (38)where a , b , c , and µ are constants. These choices producetime–independent forces (13). Let us point out that thezero force case as well as (a repulsive) Hooke’s force areincluded among them.We can analyze few special cases. Take a = c = 0 toget F = 4 mµ x , (39)while for b = c = 0 , the force is F = 49 mµ x . (40)Both cases are (repulsive) Hooke forces. If we consideronly a = 0 , we obtain F = 4 m ( cb + x ) µ , (41)a (repulsive) Hooke force plus a constant force. On theother hand, for b = 0 , we get F = 4 mµ ( − ac − a cx + a x )9 a x , (42)a (repulsive) Hooke force plus (or minus) a centrifugalbarrier force plus a force proportional to x − .Finally, when µ = 0 , one gets a free particle with F =0 . VII. PARTICLES UNDER NON–ZEROEXTERNAL POTENTIAL WITH NON–ZEROBOHM POTENTIAL
Several different potentials can be found in a straight-forward form as exact solution for systems with non–vanishing Bohm potential. Below we show families ofsuch potentials that allows to solve Schrödinger equationin an exact manner
A. Harmonic oscillator and /x potential Consider f ( x, t ) = x n cos − n ( ω ( t − t i )) , (43)for a constant n , and frequency ω . For this case A ( x, t ) = p n x n − cos − n ( ω ( t − t i )) ,S ( x, t ) = − mωx ω ( t − t i )) . (44) These amplitude and phase correspond to wavefunctionwith a Bohm potential V B ( x, t ) = − ~ ( n − n − mx . (45)which vanishes only in the cases n = 1 and n = 3 . Noticethat those two cases correspond to the ones studied inSecs. VI B and VI C. Also notice that this solution havethe same phase than those in Secs. VI B and VI C, butdifferent amplitude.In this way, solution (44) solves equation (7) for thetotal potential V ( x, t ) + V B ( x, t ) = mω x / of the har-monic oscillator, and thus, for the external potential V ( x, t ) = 12 mω x − ~ ( n − n − mx . (46)Therefore, any harmonic oscillator minus a quantum po-tential x − [20] can be solved by the presented solution. B. Position–independent forces
In this case, we are looking for a solution for a potential V ( x, t ) = − F ( t ) x , with a time–dependent force F ( t ) = − V ′ . Let us consider f ( x, t ) = Z (cid:20) G (cid:18) β ~ / x + ζ ( t ) (cid:19)(cid:21) dx , (47)for arbitrary functions G and ζ ( t ) , and constant β .Thereby, the amplitude is in terms of an arbitrary func-tion, while the phase is A ( x, t ) = G (cid:18) β ~ / x + ζ ( t ) (cid:19) ,S ( x, t ) = − ~ / mβ x ˙ ζ ( t ) + µ ( t ) . (48)In order to solve Eq. (7) for the potential V ( x, t ) = − F ( t ) x , we need to choose that d G ( y ) dy = ± y G ( y ) , (49)for the argument y ≡ βx/ ~ / + ζ ( t ) of the function G.This implies that this function is an Airy function. In thisway, this solution has clearly a non–zero Bohm potential V B ( x, t ) = ∓ ~ / β m (cid:18) β ~ / x + ζ ( t ) (cid:19) , (50)whose time dependence is through ζ . With all these con-ditions, and by chosing µ ( t ) = Z (cid:18) − ~ / m β ˙ ζ ± ~ / β m ζ (cid:19) dt , (51)Eq. (7) is solved for the force F ( t ) = ~ / mβ ¨ ζ ( t ) ± β m , (52)which is determined by ζ .Notice that this wave solution is always acceleratingor decelerating, independent of ζ , with constant accel-eration or deceleration (depending the chosen solution)given by the spatial derivative of Bohm potential − V ′ B m = ± β m . (53)In this way, any solution (48) with property (49) produceaccelerating or decelerating wave packets under position–independent forces.In vacuum, F ( t ) = 0 , ζ ( t ) = ∓ β t / m ~ / , and µ = − β t / m , thus recovering the accelerating Airy wavepacket of Sec. V B, appropriately choosing the upper signsolution [4]. C. Attractive or repulsive harmonic oscillatorswith non–vanishing Bohm potential
In Sec. VII A we obtain a solution for harmonic os-cillator potential that require the appearance of a /x potential. In this section we show a wave solution fora pure attractive or repulsive harmonic oscillator, thatpresents non–constant acceleration.Let us again start from the solutions (47) and (48).However, now let us choose the arbitrary function G sat-isfying d G ( y ) dy = ± y G ( y ) . (54)Thus, the function G are Weber functions. The Bohmpotential is now given by V B ( x, t ) = ∓ ~ / β m (cid:18) β ~ / x + ζ ( t ) (cid:19) , (55)Eq. (7) is solved for the attractive (+) or repulsive ( − ) harmonic oscillator potentials V ( x ) = ± mω x , (56)for a frequency ω = β /mh / , with the function ζ ( t ) fulfilling ¨ ζ ± ω ζ = 0 , (57)and µ ( t ) = Z (cid:18) − ~ ω ˙ ζ ± ~ ω ζ (cid:19) dt . (58) In general, the above solution has the time–dependentacceleration (or deceleration) − m ( V ′ + V ′ B ) = ± β m ζ ( t ) . (59)The harmonic oscillator is obtained by choosing theupper sign solution with ζ ( t ) = ζ cos( ωt ) . In this case,we are describing a wave packet that has a non–constantacceleration produced by force (59), explicitly given as ζ β cos( ωt ) /m . This wave packet oscillates.On the other hand, the repulsive harmonic oscillatoris obtained with the lower sign solution, when ζ ( t ) = ζ exp( − ωt ) . The wave packet experiences a decelera-tion given by − ζ β exp( − ωt ) /m . With this, we havegeneralized the results presented in [21]. D. General solution with non–vanishing Bohmpotential
We can explore a general solution using the solutions(47) and (48), and an arbitrary function satisfying d G ( y ) dy = ± y n G ( y ) . (60)for integer n > . The Bohm potential in this case, thatagain produces non–constant acceleration, is V B ( x, t ) = ∓ ~ / β m (cid:18) β ~ / x + ζ ( t ) (cid:19) n . (61)By choosing µ ( t ) = Z (cid:18) − ~ / m β ˙ ζ ± ~ / β m ζ n (cid:19) dt , (62)Eq. (7) is solved for any potential with the form V ( x, t ) = ± ~ / β m (cid:20)(cid:18) β ~ / x + ζ ( t ) (cid:19) n − ζ ( t ) n (cid:21) + ~ / mβ x ¨ ζ ( t ) . (63)General potentials with the polynomial form (63) canbe solved through accelerating wave packets. The spe-cific form of the potential is given by ζ ( t ) which can bechoosen freely. For this solution, it is not possible toconstruct a time–independent potential for n > .Finally, all these wave packets experience a generaltime–dependent acceleration given by − m ( V ′ + V ′ B ) = − ~ / β ¨ ζ ( t ) , (64)and therefore, the functionality of ζ determines the evo-lution of the wave packet. VIII. DISCUSSION AND OUTLOOK
We have shown that our approach allows us to find ex-act solutions to the Schrödinger equations for several dif-ferent external potentials, with either vanishing or non–vanishing Bohm potentials for both free and interactingparticles. The main procedure described by Eq. (7) for anone–dimensional configuration can be extended to two–and three–dimensional systems [6, 22].Several of our solutions are, to the best of our knowl-edge, new or generalizations of previous known ones.However, one important result of this work is the re-alization on how a non–vanishing Bohm potential has anon–trivial impact on the evolution of the wavefunctions.This can be seen in the case of free particle solutions, forinstance, or in accelerating wavepackets solutions, where the Bohm potential is, at least, partially responsible forproducing the acceleration.It should be also emphasized that the Bohm poten-tial plays a remarkable role in any wave equation irre-spective of its classical or quantum character, produc-ing non–geodesic wave propagation in vacuum as well ason the presence of gravitational fields, birrefringence inanisotropic spacetimes and coupling of polarization withrotation of the gravitational backgrounds, among otherunexpected effects (see, for instance [22] and referencestherein).We are currently carrying out research to extend theresults presented here to multiple dimensions, to createnew applications of wavefunctions with non–vanishingBohm potentials and to construct quantum propagatorsusing the techniques developed here. [1] P. R. Holland,
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