Global Journal of Science Frontier Research: A Physics and Space Science Volume 20 Issue 2 Version 1.0 Year 2020 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Dirac Generalized Relativistic Quantum Wave Function Which Gives Right Electrons Number in Each Energy Level by Controlling Quantized Atomic Radius By Mubarak Dirar Abd-Alla Yagoub & Mohammed Idriss Abstract- Using generalized special relativistic energy-momentum relation the linear energy- momentum Dirac relation was obtained. This equation was used to find Dirac generalized relativistic quantum equation. This equation was used to find the probability and the number of particles for each atomic energy level. The wave function found using this equation can give an expression for the number of electrons in each energy levels if the atomic radius is quantized and satisfy a certain relation. Keywords: generalized special relativity, dirac equation, atomic energy levels, number of electrons in the energy level, atomic radius. GJSFR-A Classification: FOR Code: 020699 DiracGeneralizedRelativisticQuantumWaveFunctionWhichGivesRightElectronsNumberinEachEnergyLevelbyControllingQuantizedAtomicRadius Strictly as per the compliance and regulations of: © 2020. Mubarak Dirar Abd-Alla Yagoub & Mohammed Idriss. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Dirac Generalized Relativistic Quantum Wave Function Which Gives Right Electrons Number in Each Energy Level by Controlling Quantized Atomic Radius Mubarak Dirar Abd-Alla Yagoub α & Mohammed Idriss σ 2020 Abstract- Using generalized special relativistic energy- quantum laws, they suffer from some setbacks, for r ea momentum relation the linear energy-momentum Dirac relation example the wave function cannot give the correct Y was obtained. This equation was used to find Dirac number of electrons in each energy level [9,10]. It generalized relativistic quantum equation. This equation was cannot also explain quantum gravity [11, 12]. 311 used to find the probability and the number of particles for Fatma Osman, Mubarak Dirar and other studies each atomic energy level. The wave function found using this Quantum Equation for Generalized Special Relativistic equation can give an expression for the number of electrons in each energy levels if the atomic radius is quantized and satisfy Linear Hamiltonian, to solve some of these problems. a certain relation. They use generalized special relativistic energy – V Keywords: generalized special relativity, dirac equation, momentum relation a useful linear equation was II atomic energy levels, number of electrons in the energy obtained. They found that the coefficients and matrixes ue ersion I s level, atomic radius. resembles that of Dirac relativistic quantum equation s Anew quantum linear relativistic equation sensitive to the I I. introduction potential and the effects of fields was also obtained. XX uantum mechanics, as formulated by Bohr, This equation reduces to that of Dirac in the absence of Heisenberg, Schrödinger, Pauli, Dirac, and many fields [13]. others, is based on wave particle dual nature The solution of this equation predicts the Q ) of the atomic world. Schrödinger equation is based propagation of travelling wave inside fields without A in addition on the Newtonian energy-momentum relation attenuation. Thus it can describe the electromagnetic ( [1, 2]. wave propagation inside fields. It also predicts the According to quantum mechanics, particles do existence of biophotons as stationary waves that not have definite values of position and momentum at spreads themselves, instantaneously through the Research Volume the same moment. The square of the absolute value of surrounding media. It also shows that particles behave the wave function correspond to regions where the as harmonic oscillator inside atoms with rest mass particle is more likely to be found if a location energy equal to the zero point energy. These results measurement is done [3,4]. agree with observations [14]. Frontier The Schrödinger equation is the key equation of Ebtisam A. Mohamed and other studied Derivation of Statistical Physical Laws from Quantum Quantum mechanics. The first step in the development of a logically consistent theory of non relativistic Mechanics. Their work mainly aims to derive Maxwell – Science Boltzmann distribution, Fermi – Dirac and Bose – Quantum mechanics is to drive a wave equation which of can describe the particle, wave like behavior of a Einstein distribution laws by using the quantum wave quantum particle. This equation can describe function in the energy and momentum space beside the relation between the number of particles and chemical successfully the behavior of atoms including Hydrogen Journal atom [5, 6]. potential in addition to some thermodynamic relations Hydrogen atom consists of a positively charged concerning probability. The ordinary quantum proton and a negatively charged electron, moving in mechanical wave function for free particle was Global orbit under the action of centrifugal force and the differentiated with respect to energy and momentum. influence of their mutual attraction [6,7]. The fast The derivation of statistical laws from quantum wave electrons can be described by relativistic Klein- Gordon function shows the general nature of quantum laws [15]. or Dirac equation [8]. Despite the successes of These successes motivate to try to use GSR Dirac equation to solve the problem of the number of electrons in the energy states. This is done in section Author α σ: Department of physics, Faculty of Science, Sudan University of Science and Technology, Khartoum, Sudan. (2). Sections (3) and sections (4) are devoted for e-mail: [email protected] discussion and conclusion. ©2020 Global Journals Dirac Generalized Relativistic Quantum Wave Function Which Gives Right Electrons Number in Each Energy Level by Controlling Quantized Atomic Radius a) Time Independent Potential Dependent Special Relativistic Dirac Equation The full Dirac potential (GSR) equation takes the from 2 2 = 2 . V . + m 2 + m 2 V (1) t2 t o t o ∂ ψ ∂ψ ħ ∂ψ To find time independent−ħ Dirac ħequation,α ∇ � substitute� − α ∇��⃗ψ ħ β � � β ψ ∂ ∂ ∂ ( ) ( ) ( ) ( ) (2) , = = To get − 2 v 2 2 = + . + 2 + 2 v 2020 −ħ ħ From time dependent potentialħ the above equation can be rewrite asħ r ea 2 2 2 2 Y v (h ) = ( mo mo v)u 32 1 For simplify let ħ − ħ ħ − β ħ − β = (3) v E = ( 2 ħ2 2 v) = 2 v u (4) V 2 2 ħ − ħ −= − − (5) II For hydrogen atoms and hydrogen like atoms the potential is spherical up systemic, in the form ue ersion I − s s c0 (6) I V = XX Let − qq c = 0 (7) 0 4π Where q & charge of particles and r distance between the centers of particles. ) 0 A ε ( A direct substitution of equation (6) in (4) yields ≡ ≡ 2 c [v E ] = + 0 (8) Try now a solution Research Volume ħ − c2r c1e (9) − 2 c0 c0 c2 + c2E = + (10) Frontier Equating free terms and that having theħ power� ( 1), one� gets − c c = 2 c = 2 2 c Science 0 2 0 0 of ħ − (11) c2 = β 2 2 2 2 Journal − c2E = = ( α ħ ) = ( E)E ( 2) ħ − − (12) c2 = Global − Comparing (11) with (12) yields − ħ 2 = 2 2 (13) = 2 − Let 2 c3 = c0 , c4 = c0E , c5 = c0 (14) Sub in (8) to get ħ ħ ©2020 Global Journals Dirac Generalized Relativistic Quantum Wave Function Which Gives Right Electrons Number in Each Energy Level by Controlling Quantized Atomic Radius c c 3 c = + 5 u (15) r 4 r Multiply by (r) and use the fact that − − = (16) One gets c3 c4r = ( + c5) − − (c3 + c4r) =( + c5) (17) Let − 2020 r 1 y = c3 + c4r , dy = c r , = (18) ea 4 Y c 4 � � 331 Thus ( ) ( ) c4 = c3 + c5 c4 − � − � V c3 = + c5 II c4 c4 ue ersion I s s [ 1 � 2 + c−5] = [ 1 �+ 3] I c c 0 − 3 3 XX 1 = , 2 = , 3 = 5 2 = 5 (19) c4 c4 c4 3 1 −3 − 5 c4 = 1 + , = 1 + = 4 + (20) y 4 y y − ) − � � 1 �3 � � � A ( 4 = , 5 = (21) c4 c4 = 4 + 5 + 6 Research Volume ln �= +� ln + � (22) 4 5 6 5 ln ln = + Frontier 4 6 (23) − ln = + 5 4 6 Science � � 4 6 = (24) of 5 = 5 4 (25) 7 6 Journal 7 = (26) Thus where in view 7 of (18) yields ℎ 5 4( 3+ 4 ) Global = 7( 3 + 4 ) (27) In view of equations (21),(19) and (4) Let 5 = 0 Thus c3 4 5 2 3 = 0, 2 = 5 , 5 = , 0 = = c0 c4 3 c0 From (5) 2 2 = 2 − ©2020 Global Journals Dirac Generalized Relativistic Quantum Wave Function Which Gives Right Electrons Number in Each Energy Level by Controlling Quantized Atomic Radius 2 2 = 2 2 = 2 (28) Thus 4( 3+ 4 ) = 7( 3 + 4 ) (29) The probability of finding the particle at position 2 2 2 | | = 7 ( 3 + 4 ) (30) But experimentally it was found that the number of electrons in the energy level n is a nature number n0 Thus 2 2 2 2020 | | = 7 ( 3 + 4 ) = n0 (31) r ea n0 = 2,8,18,32,50,72,98 Y Thus 1 341 n 2 r = 0 3 (32 ) 7 4 4 − 2. David Tong (2007), Quantum Field Theory, II. Discussion V Department of Applied Mathematics and Theoretical II The GSR Dirac obtained by Fatma (Quantum Physics, Centre for Mathematical Sciences, Equation for Generalized Special Relativistic Linear ue ersion I Wilberforce Road, Cambridge, UK. s s Hamiltonian) has been exhibited in equation (1). The 3. David_J._Griffiths (2005), Introduction to Quantum I time independent part has been found in equation (4), Mechanics, Reed College. XX by assuming the time dependent part to be time 4.