Schrödinger Equation Revisited

Schrödinger equation revisited

Wolfgang P. Schleicha,b, Daniel M. Greenbergera,c, Donald H. Kobeb, and Marlan O. Scullyd,e,f,1

aInstitut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, D-89069 Ulm, Germany; bDepartment of Physics, University of North Texas, Denton, TX 76203-1427; cCity College of the City University of New York, New York, NY 10031; dTexas A&M University, College Station, TX 77843; ePrinceton University, Princeton, NJ 08544; and fBaylor University, Waco, TX 97096

Contributed by Marlan O. Scully, February 8, 2013 (sent for review November 8, 2012) The time-dependent Schrödinger equation is a cornerstone of quan- i) Starting from the nonlinear wave equation (Eq. 2), we as- tum physics and governs all phenomena of the microscopic world. sume that we search for a wave equation for a scalar wave However, despite its importance, its origin is still not widely ap- containing only a first-order derivative in time and a second- preciated and properly understood. We obtain the Schrödinger order derivative in space, and we establish a mathematical equation from a mathematical identity by a slight generalization identity involving derivatives of a complex-valued field. of the formulation of classical statistical mechanics based on the ii) A description (16–20) of classical statistical mechanics in Hamilton–Jacobi equation. This approach brings out most clearly terms of a classical matter wave whose phase is given by the fact that the linearity of quantum mechanics is intimately con- the classical action and is governed by the Hamilton–Jacobi nected to the strong coupling between the amplitude and phase equation (21), and whose amplitude is defined by the square of a quantum wave. root of the Van Vleck determinant (22) and satisfies a continuity equation (16–18, 22, 23), leads via our mathematical he birth of the time-dependent Schrödinger equation was identity to a nonlinear wave equation. iii perhaps not unlike the birth of a river. Often, it is difficult to ) However, a linear wave equation (i.e., the Schrödinger equa- T tion) emerges from our mathematical identity when we cou- locate uniquely its spring despite the fact that signs may officially ple amplitude and phase in a democratic way (i.e., the phase mark its beginning. Usually, many bubbling brooks and streams determines the dynamics of the amplitude, and vice versa). merge suddenly to form a mighty river. In the case of quantum mechanics, there are so many convincing experimental results that It is this mutual coupling between amplitude and phase that many of the major textbooks do not really motivate the subject. defines a quantum matter wave and ensures the linearity of the Instead, they often simply postulate the classical-to-quantum wave equation. Indeed, in the classical matter wave, this coupling rules as is broken; the phase still determines, via the continuity equation, the dynamics of the amplitude, but the equation of motion of the ∂ Z phase (i.e., the Hamilton–Jacobi equation) is independent of the E! Z p! V i ∂t and amplitude. It is for this reason that the wave equation is nonlinear. i Our article is organized as follows: we first establish the mathematical identity and then turn to the nonlinear wave equation E p Z ’ for the energy and momentum ,where is Planck scon- corresponding to a classical matter wave; the next section is π stant divided by 2 and operators are understood as acting dedicated to the task of arriving at the linear Schrödinger equa- ψ = ψ r t on the wave function ( , ). The reason given is that tion; and we conclude by summarizing our results and providing “ ” it works. an outlook. For example, the Schrödinger equation is then obtained (1, 2) To focus on the essential ideas, we have moved detailed cal- H ≡ p2 m + V from the classical Hamiltonian /(2 ) for a particle of culations or introductory material to an appendix. In the first mass m in a potential V = V(r, t)as section (Appendix), we use standard relations of vector calculus to obtain the building blocks of our mathematical identity. In the ∂ψ Z2 second section (Appendix), we provide more insight into the Van iZ = − V2ψ + Vψ: [1] ∂t 2m Vleck determinant by considering a 1D description. This approach is unfortunate. Many of us recall feeling dis- Mathematical Identity satisfied with this recipe. In this section, we formulate the problem and spell out our as- It was the left-hand side of Eq. 1 that was the sticking point for sumptions about the wave equation. We then obtain the mathe- Schrödinger (3–7). Wave equations usually involved second time matical identity, which is at the heart of our approach toward the derivatives. It is thus somewhat ironic that classical mechanics à Schrödinger equation. la the Hamilton–Jacobi equation yield a nonlinear wave equation Ourgoalistoobtainawaveequationformatter.Forthis that is similar to Eq. 1, namely, purpose, we consider the most elementary case of a scalar particle described by a single complex-valued function ð Þ 2 h i ∂ψ cl Z θðr;tÞ iZ = − V2ψ ðclÞ + Vψ ðclÞ + Q ψ ðclÞ ψ ðclÞ: [2] Z = Zðr; tÞ ≡ Aðr; tÞei ; [3] ∂t 2m

(cl) consisting of the real-valued and positive amplitude A = A(r, t) Here, Q is a nonlinear potential that depends on ψ (Eq. 14). and the real-valued phase θ = θ(r, t). Here, r combines the The basis for Eq. 2 is the fact that energy and momentum are – Cartesian coordinates of the particle in three space dimensions obtained from the Hamilton Jacobi theory by taking time and and t is the coordinate time. space derivatives of the action, as we discuss in the following. There are, of course, many ways (8–15) in which to obtain the

time-dependent Schrödinger equation, with the most prominent Author contributions: W.P.S., D.M.G., D.H.K., and M.O.S. designed research, performed being the one developed by Feynman (14) based on the path research, and wrote the paper. integral. In this article, we obtain the Schrödinger equation The authors declare no conﬂict of interest. (Eq. 1) and discuss the connection with Eq. 2 in three steps: 1To whom correspondence should be addressed. E-mail: [email protected].

5374–5379 | PNAS | April 2, 2013 | vol. 110 | no. 14 www.pnas.org/cgi/doi/10.1073/pnas.1302475110 Downloaded by guest on September 24, 2021 cl At this point, the use of a complex, rather than a real, valued S(cl) ≡ S( )(r, α, t) denotes the classical action and the vector α wave is by no means obvious. However, this choice will become combines three constants of the motion. clear in the next section when we discuss classical matter waves. Indeed, the Hamilton–Jacobi equation (Eq. 7) is a partial dif- Indeed, our treatment will show that we use a complex function ferential equation of the first order in the three Cartesian coor- as a very efficient shorthand notation to combine two coupled dinates xk with k = 1, 2, and 3 and time. Hence, we expect its real-valued equations into a single equation. solution to depend on 3 + 1 = 4 independent constants αl of in- (cl) Wave equations contain derivatives with respect to the space tegration. Because only derivatives of S enter into the differ- α and time coordinates. For example, the familiar wave equation ential equation, we can always add a constant 0 to any solution (cl) for an electromagnetic wave involves second-order derivatives of S and obtain another solution. When we disregard this trivial with respect to space and time. However, we now consider the constant of integration, we arrive at only three, that is, one for N fi more elementary case of a wave equation with only a first-order each coordinate. Obviously, in the case of coordinates, we nd N α derivative with respect to time but second-order derivatives with constants l of integration. respect to space. Most relevant for the present problem of obtaining a wave 7 – In Appendix, we establish the identities equation is the fact that Eq. implies (16 18, 22, 23) the Van Vleck continuity equation ∂Z ∂ ∂θ i 2 i = A − Z [4] ∂ VSðclÞ ∂t 2 A2 ∂t ∂t D + V · D = ; [8] ∂t m 0 and where V2A i 2 V2Z = V · A2Vθ − ðVθÞ + Z; [5] ∂2SðclÞ A2 A D ≡ [9] ∂xk∂αl which follow when we complete the differentiations on the right- hand sides of these equations, together with the representation denotes the Van Vleck determinant. (Eq. 3) of the complex-valued function Z in terms of the ampli- In Appendix, we motivate this conservation law by deriving it for tude A and the phase θ. a 1D motion. For the more complicated case of N coordinates and Because the first-order time derivative and the second-order N constants of integration, we refer to other sources (16–18, 22, 23). space derivative given by the Laplacian of Z have different − − dimensions, that is, [time] 1 vs. [space] 2, we need to introduce Nonlinear Wave Equation. Next, we note that the expression in the fi a constant β when we add the two derivatives. In this way, we rst square bracket on the right-hand side of the mathematical 6 arrive at the mathematical identity identity (Eq. ) is identical to the left-hand side of the Van Vleck continuity equation (Eq. 8), provided D ≡ A2 and 2βθ ≡ S(cl)/m. S(cl) ∂Z 1 ∂ This feature, together with the fact that the classical action i + βV2Z = i A2 + 2βV · A2Vθ defines wave fronts, suggests consideration (16–18) of a wave ∂t 2 A2 ∂t [6] 2 ∂θ V A ðclÞ i + − − βðVθÞ2 + β Z: ψ ðclÞ ≡ AðclÞeiθ ≡ D1=2exp SðclÞ [10] ∂t A Z

Because A and θ are real, the expressions in the two square whose amplitude A(cl) ≡ D1/2 is the square root of the Van Vleck brackets on the right-hand side of Eq. 6 are also real. Moreover, determinant and whose phase θ(cl) ≡ S(cl)/Z is the classical action they bear a great similarity to a continuity equation and the S(cl) divided by a constant Z with the dimension of an action. This Hamilton–Jacobi equation of classical mechanics, respectively. In constant ensures that the phase of the wave is dimensionless. Need- the next two sections, we shall use physical arguments to identify less to say, there is no justification to identify this constant with the the constant β, simplify the right-hand side of Eq. 6, and obtain in reduced Planck’s constant. Indeed, we could have chosen any quan- this way the two wave equations corresponding to classical and tity with the dimension of an action to make the phase dimension- quantum mechanics. less. Nevertheless, our analysis brings out the critical role of this constant in the transition from classical to quantum mechanics. Classical Matter Waves When we substitute the wave ansatz in Eq. 10 into the math- So far, we have only used mathematics. To make contact with ematical identity (Eq. 6), we arrive at physics, we recall in the present section the Hamilton–Jacobi fi equation (21). We then de ne a classical matter wave whose phase Z ∂D β is given by the classical action and whose amplitude follows from + 2 V · DVSðclÞ i D ∂t Z an appropriate combination of second derivatives of the action, 2 that is, from the Van Vleck determinant. Finally, the mathematical " # identity will show that a so-defined classical matter wave satisfies 2 ðclÞ ∂SðclÞ β 2 V ψ [11] ðclÞ ðclÞ a nonlinear wave equation. + − − VS + Zβ ψ ∂t Z ψ ðclÞ Classical Mechanics as Field Theory. Central to the present section is the Hamilton–Jacobi equation (21) ∂ψ ðclÞ = iZ + ZβV2ψ ðclÞ: ∂t ∂SðclÞ VSðclÞ 2 − = + V [7] The choice ∂t 2m Z PHYSICS for a nonrelativistic classical particle of mass m moving in a po- β ≡ [12] tential V = V(r, t), which may even be time-dependent. Here, 2m

Schleich et al. PNAS | April 2, 2013 | vol. 110 | no. 14 | 5375 Downloaded by guest on September 24, 2021 for the free parameter β allows us now to take advantage of the not a characteristic feature of quantum mechanics but, rather, Van Vleck continuity equation (Eq. 8), and the first square bracket reflects the fact that the underlying dynamics rest on two equa- in Eq. 11 vanishes. Moreover, due to the value of β given by Eq. 12, tions rather than one: the continuity equation and the Hamilton– the Hamilton–Jacobi equation (Eq. 7)replacesthefirst two terms Jacobi equation. At this point, it is of no importance that the latter in the second square bracket in Eq. 11 by the potential V,andwe implies the former. Therefore, complex numbers are just a useful find (16–20) for ψ(cl) the wave equation tool to combine two real equations into a single complex equation. We conclude our discussion of classical matter waves by noting ðclÞ 2 h i ∂ψ Z ð Þ ð Þ ð Þ ð Þ that the appearance of i in quantum mechanics as a mathemati- iZ = − V2ψ cl + Vψ cl + Q ψ cl ψ cl [13] ∂t 2m cal convenience rather than a necessity is also confirmed by the formulation in terms of the Wigner phase space distribution with function (29). Indeed, this quantity is always real. We will return to this point in Conclusions and Outlook. 2 ð Þ h i Z2 V ψ cl Q ψ ðclÞ ≡ : [14] Quantum Matter Waves m 2 ψ ðclÞ So far, we have achieved a wave description (16–20) of classical statistical mechanics, and the corresponding wave equation is As a result, the wave equation (Eq. 13) for ψ(cl) (i.e., for a nonlinear. In this section, we obtain by a special choice of the mu- classical matter wave) is very close to the Schrödinger equation tual coupling between the amplitude and phase of the wave a linear fi of quantum mechanics but is nonlinear. The nonlinearity is due wave equation. Because the wave equation of the so-de ned wave to the quantity Q, which involves the ratio of the Laplacian of the that follows from our mathematical identity is the Schrödinger amplitude jψ(cl)j of the classical wave and jψ(cl)j. Moreover, it is equation, we refer to this type of wave as a quantum matter wave. proportional to the square of Z. Despite the appearance of Z in Linear Wave Equation. To make the transition from the nonlinear this nonlinear wave equation, we are still in classical mechanics. classical wave equation to the linear Schrödinger equation, that Indeed, it is exactly the potential Q that enforces the classicality 13 is, from classical to quantum physics, we first note that due to the of Eq. ; that is, it ensures that the two basic equations of this 13 wave description of classical mechanics, the Hamilton–Jacobi nonlinearity, Eq. does not allow standing waves, that is, a su- equation for S(cl) and the Van Vleck continuity equation for the perposition of a right-going wave and a left-going wave. For this density jψ(cl)j2 = D, are free of Z. For this reason, we shall refer purpose, waves of the type to this potential as the classicality-enforcing potential. i We also note that Q will play an important role in the quantum ψ ðqÞ ≡ AðqÞ SðqÞ [15] exp Z description, where it appears in the dynamical equation for the action and carries the name Madelung–Bohm quantum potential linear (24–26). However, in this equation, it enters with the opposite must satisfy a wave equation that follows from the mathemat- 6 sign, as we shall show in the next section. ical identity (Eq. ) when we absorb the classicality-enforcing poten- Q fi 14 In this approach, we have used the classical Hamilton–Jacobi tial de ned by Eq. , which is the last term of the second square 6 S(q) equation (Eq. 7), together with the Van Vleck continuity equa- bracket in the mathematical identity (Eq. )intheaction . θ = S(q) Z β = Z m tion (Eq. 8), to obtain from the mathematical identity (Eq. 6) the Indeed, with / and the choice /(2 ) given by Eq. 12 6 nonlinear wave equation (Eq. 13). Needless to say, it is also , the mathematical identity (Eq. ) takes the form possible to move in the opposite direction. Indeed, the nonlinear ∂ψ ðqÞ Z2 wave equation implies the two equations we have started from. iZ + V2ψ ðqÞ ∂t 2m ( Probability Interpretation and the Imaginary Unit. The analysis lead- 13 i i ∂ 2 2VSðqÞ ing to Eq. teaches us three important lessons: ( )Theclassical = AðqÞ + V · AðqÞ [16] S(cl) fi – 7 2 action as de ned by the Hamilton Jacobi equation (Eq. ) 2ðAðqÞÞ ∂t m represents a major portion (1, 2) of the phase of a quantum wave; (cl) 1/2 (cl) ii A ≡ D ψ 9 ðqÞ h i ( ) the amplitude of the wave given by Eq. ∂S 1 ð Þ 2 ð Þ ð Þ depends on the phase S(cl)/Z, but the Hamilton–Jacobi equation + − − VS q + Q A q ψ q ; ∂t 2m (Eq. 7) determining S(cl) is independent of D;and(iii) the Van Vleck continuity equation (Eq. 8), together with the wave repre- where we have recalled the definition (Eq. 14) of the potential Q. sentation (Eq. 10), provides us with the identification of the (cl) (cl) 2 (cl) Motivated by the fact that classical matter waves satisfy the classical density ρ ≡ D ≡ jψ j and the classical current j ≡ 8 ρv ≡ D VS(cl) m Van Vleck continuity equation (Eq. ), we now postulate that ( / ). a similar equation should also hold true for quantum matter The last feature yields the interpretation (1, 2) of jψ(cl)j2 as a waves; that is, we assume the relation density. Indeed, we have already reached a statistical theory, but of classical mechanics only. In this framework, we consider an ∂ 2 2VSðqÞ ensemble of particles moving along classical trajectories in phase AðqÞ + V · AðqÞ = : [17] ∂t m 0 space given by Newton’s equation. The nonlinear wave equation 13 ψ(cl) (Eq. ) for is completely equivalent to classical statistical 16 mechanics. Because Eq. 13 is complex, it contains two real- As a consequence, Eq. reduces to – 7 valued equations, namely, the Hamilton Jacobi equation (Eq. ) ðqÞ 2 ðqÞ h i S(cl) 8 D ∂ψ Z ðqÞ ∂S 1 ðqÞ 2 ðqÞ ðqÞ for and the Van Vleck continuity equation (Eq. ) for . ih + V2ψ = − − VS + Q A ψ : In the past, Stückelberg (27), Wheeler (28), and many others ∂t 2m ∂t 2m have addressed the question of why the imaginary unit appears so [18] prominently in quantum mechanics but not in standard formula- tions of classical mechanics. However, the complex-valued func- So far, we have not specified the equation of motion of the tion ψ(cl), which obeys the nonlinear Schrödinger equation (Eq. quantum action S(q). In the case of the classical action S(cl), the 13), demonstrates that the appearance of the imaginary unit i is Hamilton–Jacobi equation (Eq. 7) plays this role and brings

5376 | www.pnas.org/cgi/doi/10.1073/pnas.1302475110 Schleich et al. Downloaded by guest on September 24, 2021 the potential V into the wave equation for ψ(cl). Moreover, it Conclusions and Outlook leads to the nonlinear wave equation (Eq. 13) because the clas- In summary, we have obtained the Schrödinger equation start- (cl) sicality-enforcing potential Q[A ] that is already presented in ing from a mathematical identity (Eq. 6). In our argument, the the mathematical identity (Eq. 6) is not eliminated. Because our Hamilton–Jacobi equation of classical mechanics has played a (q) goal is to achieve a linear wave equation, we can attach Q[A ]to key role. Indeed, it not only suggests a specific choice of the (q) the dynamical equation of the quantum action S and postulate parameter β in Eq. 6 connecting space and time derivatives, but the equation of motion: it shows us that we can linearize the nonlinear wave equation Q h i when we include the potential in the equation of motion for ∂SðqÞ 2 − = 1 VSðqÞ + V − Q AðqÞ : [19] the action rather than in the equation of motion for the wave. At ∂t 2m the same time, we require a continuity equation similar to the one of Van Vleck describing classical statistical mechanics. In (q) With the help of this definition of the dynamics of S ,we this way, the amplitude of a quantum wave depends on its phase, arrive at the familiar Schrödinger equation: and vice versa. This rather symmetrical dependence is broken in classical mechanics, giving rise to the nonlinear wave equation. ðqÞ 2 ∂ψ Z ð Þ ð Þ Our analysis also shines some light on the old question: Why iZ = − V2ψ q + Vψ q : [20] ∂t 2m the imaginary unit? The complex wave function, which depends on the position variable and time as a parameter, is just an effi- We emphasize that we have achieved this linear wave equation cient way of combining two coupled real equations. The square Q by removing the classicality-enforcing potential from the wave of the amplitude of the wave determines the probability density, and associating it with the action. Indeed, in comparison to the and the derivative of the phase yields the momentum. In contrast, (cl) classical Hamilton–Jacobi equation (Eq. 7) governing S , the the Wigner function (29) is always real and lives in phase space. (q) dynamics of the quantum action S given by Eq. 19 depend on Therefore, it depends on the position as well as the momentum the classicality-enforcing potential Q. In contrast to the non- variable. In this sense the two degrees of freedom of the complex- linear wave equation (Eq. 13), where Q enters with a positive valued wave function, amplitude and phase, are connected to the sign, it appears in Eq. 19 with a minus sign. More importantly, real-valued Wigner function of the two variables position and the system of Eqs. 17 and 19, consisting of the quantum conti- momentum. nuity and the quantum Hamilton–Jacobi equations, contains It is also remarkable that the potential V enters the scene only (q) Planck’s constant explicitly. Hence, the potential −Q[A ] in the through the Hamilton–Jacobi equation. Needless to say, we could (q) equation of motion for the quantum action S plays a similar have also added on both sides of the mathematical identity the (cl) (cl) role as +Q[A ] does in the nonlinear wave equation for ψ .In term γVZ with another constant γ to match the different dimen- (q) Eq. 19,itdefines the quantum nature of the equation for S in sions. In this case, we would have obtained an expression on the (cl) the same way as Q[A ] determines the classical nature of the non- left-hand side of Eq. 6, which is even closer to the Schrödinger linear equation. It is interesting that Z appears in the nonlinear equation. wave equation despite the fact that it is of classical nature. This However, one might then wonder why not add on both sides a property stands out most clearly in the classical Hamilton–Jacobi nonlinear function of the amplitude of the wave, such as κA2Z, equation (Eq. 7) and in the continuity equation (Eq. 8), which are and arrive at a nonlinear Schrödinger equation of the Gross– independent of Z. In contrast, in the corresponding quantum Pitaevskii type. The reason is because it is the classical Hamilton– equations, it does not drop out. Nevertheless, for Z → 0, the Jacobi equation that brings in V, which does not contain such quantum potential −Q[A(q)] vanishes and the quantum Hamil- terms to begin with. They do not emerge in quantum theory ei- ton–Jacobi equation reduces to the classical equation, in com- ther as confirmed by landmark experiments (30–32) verifying that plete agreement with the correspondence principle. the quantum mechanics of noninteracting particles are linear. In the same vein, we could add to the first-order time de- Difference Between Classical and Quantum. Eqs. 17 and 19 are mo- rivative given in Eq. 4 a first-order space derivative, such as VZ, tivated by three principles: (i) conservation of matter as suggested rather than second-order space derivatives. However, in this by the Van Vleck continuity equation (Eq. 8) of classical me- case, we not only have the problem of the different dimensions of chanics, (ii) an appropriate generalization of the classical Ham- the two derivatives but the fact that they are of different vectorial ilton–Jacobi equation (Eq. 7) dictated by the goal to achieve a natures. Indeed, the time derivative is a scalar, whereas the space linear wave equation, and (iii) a smooth classical-quantum tran- derivative involves a gradient and is thus a vector. Therefore, this sition in accordance with the correspondence principle. Needless addition of the two derivatives will require another vector. More- to say, Eqs. 17 and 19 also follow from the Schrödinger equation over, we might want to move from scalar waves to vector waves. (Eq. 20) and the decomposition equation (Eq. 15) into amplitude We suspect that one might also get some deeper insight into the and phase, which is the path taken by Madelung (24) in 1926 and Weyl equation or the Dirac equation in this way. Bohm (25, 26) in 1952. In this article, we have only addressed the Schrödinger equa- Although Eqs. 17 and 19 seem to differ only slightly from their tion corresponding to a single nonrelativistic particle without classical counterparts, Eqs. 8 and 7, there is a dramatic con- internal degrees of freedom. Thus, we have not touched the ceptual difference. Indeed, the phase S(q) influences through its phenomenon of entanglement, which Schrödinger called the trait gradient in the continuity equation (Eq. 17) the amplitude A(q) of of quantum mechanics. Entanglement arises from the interaction the quantum wave ψ(q).Inturn,A(q) enters through the poten- of several particles and manifests itself in a single wave function tial −Q[A(q)] appearing in the quantum Hamilton–Jacobi that depends on the coordinates of all particles but is not sepa- equation (Eq. 19) into the phase S(q). This coupling scheme rable. This feature suggests that entanglement is best described between the amplitude and phase of a quantum wave is in sharp in configuration rather than in phase space. The connection be- contrast to the corresponding coupling scheme of a classical tween entanglement and Born’s rule has been illuminated by cl wave. Here, the classical action S( ) determines the amplitude Zurek (33), and it would be fascinating to build a bridge between A(cl) ≡ D1/2 through the definition (Eq. 9) of the Van Vleck our approach and that of Zurek (33). (cl) 2 determinant. However, because D ≡ (A ) does not appear in Unfortunately, this question, as well as the derivation of the PHYSICS the classical Hamilton–Jacobi equation (Eq. 7), the classical Weyl or Dirac equation, is beyond the scope of the present ar- action S(cl) is independent of A(cl). ticle and will be the topic of a future publication.

Schleich et al. PNAS | April 2, 2013 | vol. 110 | no. 14 | 5377 Downloaded by guest on September 24, 2021 Appendix Van Vleck Continuity Equation. In this section, we motivate the Van 8 Operator Identities. In this section, we provide more details about Vleck continuity equation Eq. by considering the case of a m x the derivation and building blocks of the mathematical identity single particle of mass moving along the axis. Here, the Van (Eq. 6). Here, we consider differentiations of the complex- Vleck determinant reduces to valued field ∂2SðclÞ ∂2SðclÞ D ≡ ≡ ; [30] θðr;tÞ Zðr; tÞ ≡ Aðr; tÞei ; [21] ∂x∂α ∂α∂x S(cl) = S(cl) x α t – defined by its real-valued and positive amplitude A = A(r, t) and where ( , , ) follows from the Hamilton Jacobi real-valued phase θ = θ(r, t). In particular, we consider the first- equation order derivative with respect to time and the second-order de- ∂SðclÞ ∂SðclÞ 2 rivative with respect to space. For this purpose, it is convenient − = 1 + V: [31] to use Eq. 21 to express A in terms of Z, that is, ∂t 2m ∂x

A = Ze−iθ: [22] When we differentiate D as deﬁned by Eq. 30, interchange the order of differentiations, and take advantage of the Hamilton– We start our discussion by establishing the relation Jacobi equation (Eq. 31), we ﬁnd ∂Z i ∂ ∂θ ∂D ∂2 ∂SðclÞ ∂ ∂ ∂SðclÞ 2 i = A2 − Z [23] 1 ∂t A2 ∂t ∂t = = − + V : [32] 2 ∂t ∂x∂α ∂t ∂x ∂α 2m ∂x Z for the time derivative of by noting the identity Because the potential V does not depend on the constant α of integration, we obtain ∂ ∂Z − θ − θ∂θ ∂Z 1 ∂θ A2 = 2 A e i − iZe i = 2 A2 − i ; [24] ∂t ∂t ∂t ∂t Z ∂t ∂D ∂ ∂SðclÞ ∂2SðclÞ = − 1 ; [33] ∂t ∂x m ∂x ∂α∂x which follows directly from Eq. 22. Next, we verify the formula that is, the Van Vleck continuity equation V2A 2 i 2 2 ∂D ∂ ∂SðclÞ V Z = V · A Vθ − ðVθÞ + Z; [25] + D 1 = [34] A2 A ∂t ∂x m ∂x 0

which involves second derivatives with respect to position. fi fi for one dimension. Here, we have recalled the de nition in Eq. For this purpose, we rst note that 30 of D. We emphasize that the corresponding derivation for N space V · A2Vθ = 2AVA · Vθ + A2V2θ; [26] coordinates xk and N constants αl of integration is slightly more complicated and needs as an additional ingredient the Jacobi which leads with the identity equation (34) VA = VZ −iθ − AVθ [27] − e i dM = MTr M 1dM [35] 22 following from Eq. to for the differential of a determinant M of a matrix M in terms of − − the trace Tr of the product M 1dM consisting of the inverse M 1 2 −iθ 2 2 2 2 V · A Vθ = 2AVZ · Vθe − 2iA ðVθÞ + A V θ: [28] and the differential dM of M. For the details of this derivation, we refer the reader to other sources (16–18, 23). Moreover, we find from Eq. 27 the relation ACKNOWLEDGMENTS. We thank L. Cohen, J. P. Dahl, J. Dalibard, W. D. V2A = V2Ze−iθ − 2iVZ · Vθe−iθ − AðVθÞ2 − iAV2θ: [29] Deering, M. Fleischhauer, R. F. O’Connell, H. Paul, E. Sadurni, A. A. Svidzinsky, and W. H. Zurek for many fruitful discussions. D.M.G. is grateful to the 28 29 Alexander von Humboldt Stiftung which made this project possible. M.O.S. When we substitute Eqs. and into the right-hand side of acknowledges the support of the National Science Foundation (Grant PHY- Eq. 25, we indeed arrive on the left-hand side. 1241032) and the Robert A Welch Foundation (Award A-1261).

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