7) is a partial differential equation of the first order in the three Cartesian coordinates x k with k = 1, 2, and 3 and time. Indeed, the Hamilton–Jacobi equation ( Eq. Here, S (cl) ≡ S ( cl)( r, α, t) denotes the classical action and the vector α combines three constants of the motion.

However, we now consider the more elementary case of a wave equation with only a first-order derivative with respect to time but second-order derivatives with respect to space.įor a nonrelativistic classical particle of mass m moving in a potential V = V( r, t), which may even be time-dependent. For example, the familiar wave equation for an electromagnetic wave involves second-order derivatives with respect to space and time. Wave equations contain derivatives with respect to the space and time coordinates. Indeed, our treatment will show that we use a complex function as a very efficient shorthand notation to combine two coupled real-valued equations into a single equation. However, this choice will become clear in the next section when we discuss classical matter waves. Here, r combines the Cartesian coordinates of the particle in three space dimensions and t is the coordinate time.Īt this point, the use of a complex, rather than a real, valued wave is by no means obvious. In the second section ( Appendix), we provide more insight into the Van Vleck determinant by considering a 1D description.Ĭonsisting of the real-valued and positive amplitude A = A( r, t) and the real-valued phase θ = θ( r, t). In the first section ( Appendix), we use standard relations of vector calculus to obtain the building blocks of our mathematical identity. To focus on the essential ideas, we have moved detailed calculations or introductory material to an appendix. Our article is organized as follows: we first establish the mathematical identity and then turn to the nonlinear wave equation corresponding to a classical matter wave the next section is dedicated to the task of arriving at the linear Schrödinger equation and we conclude by summarizing our results and providing an outlook. It is for this reason that the wave equation is nonlinear. Indeed, in the classical matter wave, this coupling is broken the phase still determines, via the continuity equation, the dynamics of the amplitude, but the equation of motion of the phase (i.e., the Hamilton–Jacobi equation) is independent of the amplitude. It is this mutual coupling between amplitude and phase that defines a quantum matter wave and ensures the linearity of the wave equation. Iii) However, a linear wave equation (i.e., the Schrödinger equation) emerges from our mathematical identity when we couple amplitude and phase in a democratic way (i.e., the phase determines the dynamics of the amplitude, and vice versa).

Schrodinger equation derivation pdf