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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 con- tinuity 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 PHYSICS 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 math- ematical identity and then turn to the nonlinear wave equation E p Z ’ for the energy and momentum ,where is Planck s con- 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 Our goal is to obtain a wave equation for matter. For this 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]. www.pnas.org/cgi/doi/10.1073/pnas.1302475110 PNAS Early Edition | 1of6 Downloaded by guest on September 28, 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 À Á 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 ∂ À Á 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.
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