Simple Models for Classical Electron Radius and Spin

Transcription

1 Simple Models for Classical Electron Radius and Spin 1 Abstract Vedat Tanrıverdi Physics Department, METU tvedat@metu.edu.tr Some simple models about classical electron radius and spin are considered. These simple models are considered for a better understanding spin and its relation with other electron properties, i.e. charge, inertia, energy. These models have different inconsistencies with current theories, however they are still helpful for understanding. 2 Introduction In a previous work, history of electron spin is reviewed [1] and some problems are underlined related with it. In physics, electron is considered as a point particle and it has spin and magnetic moment. However, there is not any explanation why a point particle have magnetic moment and angular momentum. It is said that spin is an intrinsic property. It is just escaping from the question instead facing it. More importantly, to define a point particle one could have 3 independent variable, however electron is defined with 4 quantum numbers. These are most basic conceptual problems related with electron spin. Yet, there is not any consistent and explanatory theory on it. These basic problems are related with how we treat electron spin. Some of basic problems are conceptual and some of them are qualitative. In the conceptual part, most of the answers addressed to one word intrinsic, there is 1

2 no underlying answers or relations. Other than intrinsic, we have some comments on experimental results and relations. These relations are basically to treat experimental results, not to explain spin in a causal way. Qualitative parts are related with basically experimental observations. Experimental extractions from observations related with electron can be considered having mass, charge, 4 quantum number, spin angular momentum, magnetic moment; and any qualitative model should be consistent with the observations in the Stern-Gerlach experiment, having two distinct trace; in the sequential Stern-Gerlach experiment, not depending on the previous observation if spin is measured in a different perpendicular direction; and also should give correct experimental measurements. A successful model should explain these experimental results in a qualitative way. In this work, some simple models related with electron are considered and aim of this paper is not to explain all things in a qualitative way. The ones related with classical electron radius calculations are reconsidered and two other simple model is also considered. These are only toy models and they are studied to understand the problem in a better way. These models are considered in the concepts of angular momentum, energy and inertia and analyzed by using some simple relations. 3 Models Here, 4 different models are considered; spherical shell model, spherical model, two point particle model and three point particle model. In the first one, the electron is considered as a spherical shell, this is a model that is considered in some quantum mechanics books[2, 3]. This model is also considered in the calculations of classical electron radius[4]. In this model, electron is considered as a spherical shell with radius a carrying charge e and having mass m. Total charge will be always equal to the electron charge e = C, and mass m will be equal to electron mass m = kg except rotational energy consideration. For the spherical model, electron is considered as a sphere instead of shell in a similar way. For the two and three particle model, electron is considered as constituent either from two or three particle, having charge e/2 or e/3 and mass m/2 or m/3, Figure 1. Total of these constituent particles will be equal to mass of the electron, m, except rotational energy consideration. The radius a will be calculated by using angular momentum and energy. 2

3 We will mainly consider that angular momentum of the shell is the source of spin angular momentum of the electron, and the energy of the shell in different ways corresponds to electron mass. In a case, v r /c speed at rim to the speed of the light ratio will also be calculated. For these four model we will apply three calculation scenario; the first one is the view of angular momentum as a source of spin, the second one is similar to classical electron radius calculations and the final one is a consideration of mass as a result of total of rest energy of the system and its rotational kinetic energy. In the last case, differently from classical electron radius calculations, source of the mass or the inertia is considered as rotational kinetic energy plus rest energy. While doing this, we defined m as the mass of shell, sphere or total rest masses of sub-particles. Since we have an extra parameter, instead of calculating v r /c ratio we take it as 1, as a limit value, and calculated electron radius due to these considerations. Due to changes in moment of inertia or electrostatic energy, there will be some changes among the model results. Angular momentum In this consideration, spin angular momentum of the electron is taken simply equal to the angular momentum of rotating shell. Rotating shell has moment of inertia I = 2 3 ma2 and angular momentum L = Iw and spin angular momentum has a magnitude S = S = 3 h. In the v 2 r/c = 1 case, here v r = wa speed at the rim, if L = S one can find a = m. For spherical model only change comes from the difference in moment of inertia I = 2 5 ma2, then in the v r = c case a = m. For the two or three particle model, we consider 2 or 3 particle with masses m/2 or m/3 rotating in a plane around the center of mass, also in a circular path with constant radius a. Then moment of inertia of the systems will be same, I = ma 2. Then in the v r /c = 1 case, a = m for both case. Electrostatic energy In the classical electron radius calculations, electron is considered as a spherical shell or sphere, electron s charge and mass distributed on or within either of these uniformly. For such cases, electrostatic potential energy is calculated for a radius a, and this energy is considered as the source of inertia. After equating the electrostatic energy to the rest energy of the electron, electron s radius are calculated and this radius is named as classical electron radius. For uniformly distributed charge, the electrostatic energy for the spherical shell is U ES = 1 1 e 2 2 4πɛ 0. If one considers this energy as the source of inertia, a 3

4 (a) 2 point particle model (b) 3 point particle model Figure 1: 2 and 3 point particle models then it should be equal to E = mc 2. Then one can obtain a = 1 1 and 2 mc 2 its numerical value is a = m. In this case, by using L = S speed at the rim to the speed of the light ratio can be calculated as v r /c = 3 3 h 4 mca and its numerical value is v r /c = If similar assumptions are considered, one can obtain a = m and v r /c = for spherical model; a = m and v r /c = for two particle model; and a = m and v r /c = for three particle model. For spherical model charge is considered as uniformly distributed within the sphere, for two and three particle model charge is equally split amongst the particles. Rotational motion and energy 4πɛ 0 e 2 If one considers total of rest energy of the shell and its rotational kinetic energy as the source of inertia, then E = mc I L2 should be equal to E = mc 2, here m is the mass of the shell and m is the mass of the electron. Since total of rest energy of the shell and its rotational kinetic energy is considered as the source of the inertia, m is different than mass of electron m. Then one can obtain the relation mc 2 = mc 2 (1 + 1 vr 2 ). If v 3 c 2 r /c is taken as 1, m = 3m; so m > 3m. If v 4 4 r/c is taken as 0 (very close to zero), m = m; so m < m. Hence m > m > 3 m. If one considers L = S, then 4 m = 3 3 h and if one considers v 4 v ra r/c as 1, then from previous inequality one can obtain 3 3 h < a < 3 h. From this, one can reach the numerical 4 mc mc values m < a < m. If one follow similar steps, then one can obtain m < a < m for spherical model, m < a < m for two and three particle models. 4

5 4 Comparison Four different model is resulted in different numerical values due to their considerations. Smallest radius is obtained in two point particle model from the electrostatic energy consideration as a = m. Biggest radius is obtained spherical model rotational energy consideration as a = m as an upper limit. The other results are between these values. However all these are much bigger than previous limiting studies[5], even most results are bigger than charge radius of the proton, m [6]. Smallest values for radius are obtained from the electrostatic energy considerations, however these considerations are resulted in very high v r /c ratio. For the smallest radius value a = m, the biggest v r /c ratio obtained as 947. This is really huge number for v r /c ratio, and none of our current physical theories support such a thing. If one eliminates results with v r /c > 1, we left with results at the order m. These are really big radius values for electron. From these results, one can say that in this format these models are away from the experimental measurements. 5 Conclusion and discussion The above models are simple ones that are considered to understand spin problem in a better way. These classical models are only dealt with radius and rotation velocity in terms of angular momentum and energy. Magnetic moment of the electron and its relation with the magnetic field should also be considered for such cases for better understanding. Now let us discuss the assumptions that have been done in this work. One of the assumption is that electrostatic energy is considered as source of inertia as saying mc 2 = c 1, where c is a constant depending on the model, r is the radius of the electron in the model. Is it a valid assumption? We have theories on some bound systems like nuclei and mesons. In the nuclei, mass comes from masses of nucleons and mass of formed nuclei is always smaller than total masses of protons and neutrons in the nuclei. The energy difference is considered as binding energy and this is one of the cases that a bound system has less mass than the constituted particles total mass. In the nuclear theory we do not have any contribution of the charge to the mass. So if we want to use knowledge of nuclear theory we can write a theory e 2 4πɛ 0 r 5

6 whose constituent particles total mass bigger than the constituent particle. Also we should not include charge as the contributor to the mass. Also in meson theory, total mass of the constituent particles can be bigger than the meson[7]. In that theory charge does not contribute to the mass. So, the relation mc 2 = c 1 is not supported by nuclear theory and meson theory. Hence, we can say that classical electron radius calculations are not true if we consider the successes of the meson and nuclear theory. In meson theory, rest energies of constituent quarks and kinetic energy of them contribute to mass of the meson. The interaction energy, i.e., color, spin orbit and spin-spin, in general reduces the mass of the meson. Similar reduction is also valid for the nuclear theory. So mc 2 = mc 2 + L 2 /2I is not valid always. In the mc 2 = mc 2 + L 2 /2I assumption, the biggest value that m can take is m. So we can modify this equation by adding some negative constants. This negative constant can not be found in a deterministic way, since we do not have a theory that describes such interactions. One of the possibility is preon models to explain such interactions, however they are not advanced enough to determine this negative constant. These toy models can be improved in different ways and these improvements can be helpful to understand spin in a better way. References e 2 4πɛ 0 r [1] V. Tanrıverdi, vixra: [2] J. L. Powell and B. Crasemann, Quantum Mechanics. Addison-Wesley, (1961). [3] W. M. Wilcox, Qunatum Principles and Particles. CRC Press, (2012). [4] Wikipedia, Classical electron radius. Web. 30 Sep [5] H. Dehmelt, Physica Scripta. T22, 102 (1988). [6] CODATA Internationally recommended 2014 values of the Fundamental Physical Constants, NIST, Web. 30 Sep [7] S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985). 6