IJSR - INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCH 77 Volume : 5 | Issue : 5 | May 2016 • ISSN No 2277 - 8179 | IF : 3.508 | IC Value : 69.48 Research Paper Physics Jose Garrigues Baixauli Derivation of the Schrodinger Equation from Classical Physics KEYWORDS : 4-d space-time discrete, Schrödinger equation, Planck length, Quantifcation of space–time, minimal length, imaginary time Introduction In 1923, De Broglie suggested that all matter should behave as waves with a wavelength given by , where λ is the wavelength of the particle, h is Planck’s constant, m is the mass of the particle and v its velocity. Motivated by the hypothesis of De Broglie, in 1926 Erwin Schrödinger conceived an equation as a way to describe the wave behaviour of particles of matter. Te equation was later called the Schrödinger equation. On the one hand, despite much debate, it is accepted that the square of the wave function at a point represents the probabil- ity density at that point. Max Born gave the wave function a dif- ferent probabilistic interpretation than that given by De Broglie and Schrödinger, an interpretation that Einstein never shared. On the other hand, Schrödinger published two attempts to de- rive the equation that takes his name [1, 2]. Tere have also been attempts by other authors to obtain Schrödinger’s equation from diferent principles [3–9]. Discrete space–time General relativity implies that space–time is continuous. How- ever, there is no experimental evidence for it. Are space and time continuous? Or are we only convinced of that continuity as a result of education? In recent years, both physicists and math- ematicians have asked if it is possible that space and time are discrete. If we could probe to size scales that were small enough, would we see “atoms” of space, irreducible pieces of volume that cannot be broken into anything smaller? [10]. Minimum volume, length or area are measured in units of Planck [10]. Planck’s constant, h, which represents the elemen- tary quantum of action, has an important role in quantum me- chanics. Tere are several theories that predict the existence of a minimum length [11–12]. Besides black hole physics, these the- ories are related to quantum gravity, such as string theory and double special relativity [13–15]. Quantifcation of space–time maintains the relativistic invari- ance [16] and causation and allows us to distinguish elementary particles from themselves in a simple and natural way [17]. Discrete space–time is used as a model by other authors to pre- sent the solution of the Schrödinger equation for a free particle [18] or for electromagnetic waves and the Helmholtz equation [19]. Heisenberg said that physics must have a fundamental length scale, and with Planck’s constant h and the speed of light, allow the derivation of the masses of the particles [20–21]. Planck’s length has been considered as the shortest distance having any physical meaning. It is shown that Planck’s length is a lower bound to a proper physi- cal length in any space–time. It is impossible to construct an ap- paratus which will measure length scales smaller than the Planck length [22]. Electron wave particle duality Suppose the universe is made of atoms of space–time and dis- crete particles have four spatial dimensions, with radius equal to the Planck diameter r p . To simplify the drawing, only three di- mensions are considered r(x,y) y u. Figure 1. Rotations of the particle Te particle may rotate both in three-dimensional space and the fourth dimension (u, Fig. 1), which gives rise to the following combinations: • 0 rotations. • 1 spatial rotation: w e . • 1 rotation in the fourth dimension: w u . • 2 rotations, one space: w e , and the other in the fourth dimen- sion: w u . If we suppose that we have a particle of mass m, which rotates at velocity ω e , the potential of the gravitational feld at the dis- tance r, will be : where G is the gravitational constant, and v the velocity. Let us assume that this is the linear speed of rotation of the particle. Te atoms of space and time are united by Planck’s force, so that turning one of them, will drag it to ad- jacent atoms, so that the linear velocity of rotation (Fig. 2) will increase as we move away from the rotat- ing atom, to the speed of light c, in the distance r, then : y (2) ABSTRACT In this work, the Schrödinger equation is deduced in a very simple manner. Te starting point is the assumption that the Universe and particles are formed by four-dimensional Planck atoms. Te wave function is the ratio between the kinetic energy that the electron has when it is unobserved and the energy that it acquires due the observation.