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The Relativistic Wave Equation International Journal of Applied Physics and Mathematics, Vol. 3, No. 6, November 2013 The Relativistic Wave Equation Ilija Barukčić Meanwhile, there are several other wave equations (Weyl Abstract—In general, it is well known that the Schrödinger equation, Majorana equation, Breit equation, Proca equation, equation is not compatible with special relativity theory to a Rarita-Schwinger equation, Bargmann-Wigner equations et necessary extent. Thus far, there are already several trials to cetera), each of them with different strength and reliability. formulate versions of the Schrödinger equation to ensure But besides of all, a generally valid and invariant form of a compatibility with special relativity theory, the Klein-Gordon-equation or the Dirac equation are some of these relativity consistent wave equation or a fully self-consistent attempts. relativistic quantum theory is still not in sight. Material and Methods: In this paper, Einstein’s relativistic energy momentum relation is re-analyzed, a normalized relativistic energy momentum relation is derived. The derived II. MATERIAL AND METHODS normalized relativistic energy momentum relation together with the known Schrödinger equation is used as a starting point A. Einstein's Mass-Energy Equivalence Relation to establish a wave equation consistent with special relativity According to Albert Einstein [3], it is theory. Results: In this publication, based on Einstein’s relativistic energy momentum relation, the historical problem of the v² mm 2 1 (1) “particle-wave-duality” is solved. Furthermore, a special 0 r c² relativity theory consistent wave equation is derived. or equally Index Terms—Causality, quantum mechanics, special and general relativity, unified field theory. v² m c² m c ² 2 1 (2) 0 r c² I. INTRODUCTION where m0 denotes the “rest” mass, mr denotes the “relativistic” The famous, long lasting dispute between Leibniz mass, v denotes the relative velocity and c denotes the speed (conservation of vis viva or “kinetic” energy) and Newton of light in vacuum. (conservation of momentum) had its roots in a dispute B. The (Time-Dependent) Schrödinger Equation between Gottfried Wilhelm Leibniz [1] (1646-1716) and The Schrödinger equation [4] for any system, no matter followers of René Descartes (1596-1650) concerning whether relativistic or not, has the form Descartes' law of the quantity of motion, as discussed by Descartes in his Principia philosophiae of 1644. In fact, the (3) core of this controversy was the dispute about the H i t 0 t conservation of [mr × v × v] (Leibniz) versus the conservation of [mr × v] (Newton [2]) through changes. According to Roger Boscovich (1745) and Jean d’Alembert (1758) both where H is the Hamiltonian and (t) the wave function of the concepts are equally valid. Thus far, even today, it is a point quantum system. The form of the Schrödinger equation itself of dispute, what is preserved through changes. depends on the physical situation. Under certain circumstances, the momentum [mr × v] is C. The Energy Operator part of Schrödinger's equation too. The Schrödinger's In quantum mechanics, the Hamiltonian is a quantum equation is still of importance especially for quantum mechanical operator corresponding to the total energy of a mechanics. But it was Erwin Schrödinger himself who quantum mechanical system and usually denoted by H. recognized that his equation is not relativity consistent and By analogy with classical mechanics, the Hamiltonian is valid only under non-relativistic circumstances. In contrast to the sum of operators corresponding to the potential and this, the principle of relativity, which is the foundation of the kinetic energies (of all the particles associated with a system) theory of special relativity, should be valid in quantum of a quantum mechanical system and can take different forms mechanics too. In other words, can the principles of quantum depending on the situation. The total energy operator is mechanics be reconciled with the principles of special determined as relativity? The Klein-Gordon-equation established by Oskar Klein Hi (4) and shortly afterwards deduced by Walter Gordon is one of t these trials to reconcile relativity and quantum theory, the Dirac equation formulated by Dirac (1902-1984) another one. For our purposes, the (non-relativistic or relativistic) Hamiltonian is corresponding to the total energy of a Manuscript received June 2, 2013; revised December 23, 2013. Ilija Barukčić is with Horandstrasse, Jever, Germany (e-mail: quantum mechanical object. Thus far, using a suitable [email protected]). Hamiltonian operator (i.e. the problem of incorporating spin) DOI: 10.7763/IJAPM.2013.V3.242 387 International Journal of Applied Physics and Mathematics, Vol. 3, No. 6, November 2013 is of fundamental importance. m ² v² 0 1 (11) mcr ² ² III. RESULTS A fundamental insight of special relativity is the Rearranging the equation above yields relativistic energy-momentum relation. Even particles which propagate at velocities comparable to the speed of light c will mc² 4 v² m m c² 0 rr 1 (12) obey this natural law. Thus far, extracting some mathematical m² c4 c² m m c² and physical consequences from the relativistic r r r energy-momentum relation is of use while establishing a In general, the “rest energy” is E ² = m ²× c²× c² while the generally valid relativistic wave equation. But a logically 0 0 consistent relativistic description of quantum mechanical “relativistic energy” is Er² = mr²× c²× c² and the “relativistic systems will not be achieved only by inserting the energy momentum” is pr² = v² × mr². Thus far, the equation before operator and momentum operator into the relativistic can be simplified as energy–momentum relation. E ² pc² ² A. The Normalized Relativistic Energy Momentum 0 r 1 (13) EE² ² Relation rr Thus far, let m denote the “rest mass”, let E = m × c² 0 0 0 The energy of the electromagnetic wave E is known to denote “the rest energy”, let m denote the “relativistic mass”. Wave r be determined as E = p × c. Based on this insight, the In this context, let E = m × c² denote “the relativistic energy”, Wave r r r general form of the normalized relativistic energy momentum let v denote the relative velocity, let c denote the speed of relation follows as light in vacuum. Further, let pr = mr × v denote the “relativistic momentum”. In the following, let EWave = pr × c EE0² Wave ² (14) denote the energy of the electromagnetic wave. 1 EErr² ² Claim. Q. e. d. In general, it is A fundamental property of objective reality and thus far EE² ² one of the most important features of quantum theory is that 0 Wave 1 (5) quantum mechanical objects or matter as such can exhibit EErr² ² both, wave and particle properties, simultaneously. Under Proof. different circumstances, either the particle aspects or the Starting with wave aspects of a quantum mechanical object may manifest themselves more strongly. 11 (6) The simultaneous observation of wave and particle properties of quantum mechanical objects may be difficult as it is equally such. The question whether is it possible at all to measure simultaneously (collapse of the wave function) the extent, to what a quantum mechanical object has to be regarded as a 11mm (7) 00 particle or as a wave, is not the topic of this publication. These questions have long been debated and continue to be Based on Einstein's mass-energy equivalence (1) it follows debated. Yet and besides of all, any single quantum that mechanical object is determined by (14) too. Thus far, in order to consider the contradictory nature of v² the particle-wave duality of quantum mechanical objects in mm 2 1 (8) 0 r c² greater detail, equation (14) show us to what extent something must be regarded as a wave and to what extent the Rearranging this equation, we obtain same something must be regarded as being a particle. According to (14), the more a quantum mechanical object is a m v² “particle” the less it is a “wave” and vice versa. It is 0 2 (9) 1 worthwhile to state here that the one (i.e. the extent to which a mcr ² quantum mechanical object is a particle) can be calculated, as Let us perform a square operation of this equation. The soon as its own other (i.e. the extent to which a quantum result is mechanical object is a wave) is measured. The historic problem of the particle-wave duality, deeply embedded into the foundations of quantum mechanics, is m0 ² v² (10) 1 solved by (14). mcr ² ² B. The Relativistic Wave Equation or the normalized relativistic energy momentum relation, a probability theory consistent formulation of the particle-wave In general, however, just as proofed before, a “classical” or duality, as relativity based concept of “particle” and “waves” fully 388 International Journal of Applied Physics and Mathematics, Vol. 3, No. 6, November 2013 describes the behavior of quantum-scale objects too. determined as To put it slightly differently, nearly all physicists and ^ Let E denote the quantum mechanical operator of the philosophers will agree that the total energy of a physical p system S cannot be different in quantum theory and in special relativistic potential energy. For our purposes it is important relativity theory. to note that the relativistic potential energy is the energy of a (high speed) particle (or of a system) due to the position of 1) Definitions this (high speed) particle (or of this system). Thus far, in what follows, let us consider the following. Let The definition of the relativistic potential energy Ep is m0 denote the “rest mass”,
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