Dirac Generalized Relativistic Quantum Wave Function Which Gives Right Electrons Number in Each Energy Level by Controlling Quan

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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

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© 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

α σ Mubarak Dirar Abd-Alla Yagoub & Mohammed Idriss 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.

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 𝚌𝚌ħequation,α ∇ � substitute� − 𝚌𝚌 α ∇��⃗ψ 𝔦𝔦ħ β 𝚌𝚌 � � β 𝚌𝚌 ψ ∂ ∂ 𝔦𝔦 ∂ ( ) ( ) ( ) ( ) (2) , = = To get −𝑖𝑖𝑖𝑖𝑖𝑖 𝜓𝜓 𝑟𝑟 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑢𝑢 𝑟𝑟 𝑒𝑒 𝑢𝑢 𝑟𝑟 2 v 2 2 = + . + 2 + 2 v

2020 −𝑖𝑖𝑖𝑖ħ 𝚌𝚌𝚌𝚌 𝑖𝑖𝑖𝑖ħ 𝑜𝑜 𝑜𝑜 From time dependent potentialħ 𝜔𝜔the above equation𝛻𝛻𝛻𝛻 can be𝛼𝛼 rewrite𝛻𝛻𝛻𝛻 asħ𝜔𝜔 𝜔𝜔𝑚𝑚 𝑐𝑐 𝛽𝛽𝑚𝑚 𝑐𝑐 r 𝑢𝑢 𝑢𝑢 ea 2 2 2 2 Y v (h ) = ( mo mo v)u 32 1 For simplify let 𝑖𝑖𝑖𝑖ħ 𝛼𝛼𝛼𝛼𝛼𝛼 − 𝑖𝑖ħ𝜔𝜔 𝚌𝚌 𝛼𝛼𝛼𝛼𝛼𝛼 ħ 𝜔𝜔 − β ħ𝜔𝜔 𝚌𝚌 − β 𝚌𝚌 = (3)

2 2 2 2 v E = ( 𝐸𝐸 ħ𝜔𝜔 v) = v u (4)

V 2 2 𝑖𝑖𝑖𝑖ħ 𝛼𝛼𝛼𝛼𝛼𝛼 − 𝑖𝑖𝑖𝑖ħ 𝛼𝛼𝛼𝛼𝛼𝛼 𝐸𝐸 −= 𝛽𝛽𝛽𝛽𝑚𝑚𝑜𝑜 𝚌𝚌 − 𝛽𝛽𝑚𝑚𝑜𝑜 𝑐𝑐 𝑢𝑢 𝑏𝑏𝑜𝑜𝑢𝑢 − 𝛽𝛽𝑚𝑚𝑜𝑜 𝚌𝚌 (5) II For hydrogen atoms and hydrogen like atoms the𝑜𝑜 potential is𝑜𝑜 spherical up systemic, in the form ue ersion I 𝑏𝑏 𝐸𝐸 − 𝛽𝛽𝛽𝛽𝑚𝑚 𝑐𝑐

s s c0 (6) I V = XX Let − qq𝑟𝑟 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 𝑞𝑞 ≡ ≡ 2 c [v E ] = + 0 (8) 𝑜𝑜 Try now a solution 𝑜𝑜 𝛽𝛽𝑚𝑚 𝚌𝚌 Research Volume 𝑖𝑖𝑖𝑖ħ𝛼𝛼 𝛻𝛻𝛻𝛻 − 𝛻𝛻𝛻𝛻 𝑏𝑏c𝑢𝑢e c2r 𝑢𝑢 (9) 1 𝑟𝑟 − 2 c0 c0 c2 + c2E = + (10) Frontier 𝑜𝑜 𝛽𝛽𝑚𝑚 𝚌𝚌 Equating free terms and that having the𝑖𝑖𝑖𝑖ħ 𝛼𝛼power� ( 1), one� 𝑢𝑢 gets𝑏𝑏𝑜𝑜𝑢𝑢 𝑢𝑢 𝑟𝑟 𝑟𝑟 − c c = 2 c = 2 2 c Science 0 𝑟𝑟2 0 0

of 𝑜𝑜 𝑜𝑜 𝑖𝑖𝚌𝚌ħ𝛼𝛼 𝛽𝛽𝑚𝑚 𝑐𝑐 −𝑖𝑖 𝛽𝛽𝑚𝑚 𝚌𝚌 (11) c2 = 𝑖𝑖β 𝑚𝑚𝑜𝑜 𝚌𝚌 2 2 2 2

Journal − c2E = = ( α ħ ) = ( E)E 𝑜𝑜 ( 𝑜𝑜 2) 𝑜𝑜 𝑖𝑖𝚌𝚌𝛼𝛼 ħ 𝑏𝑏 𝐸𝐸 − 𝛽𝛽𝛽𝛽𝑚𝑚 𝚌𝚌 𝑖𝑖 𝛽𝛽𝑚𝑚 𝚌𝚌 − (12) c2 = Global 𝐸𝐸 − 𝛽𝛽𝑚𝑚𝑜𝑜 𝚌𝚌 Comparing (11) with (12) yields −𝑖𝑖 𝚌𝚌ħ𝛼𝛼 2 = 2 2 (13) 𝑜𝑜 = 2 𝑜𝑜 𝐸𝐸 − 𝛽𝛽𝛽𝛽𝑚𝑚 𝚌𝚌 𝛽𝛽𝑚𝑚 𝚌𝚌

Let 𝐸𝐸 𝛽𝛽𝑚𝑚𝑜𝑜 𝚌𝚌 2 c3 = c0 , c4 = c0E , c5 = c0 (14)

Sub in (8) to get 𝚌𝚌ħ𝛼𝛼 𝚌𝚌ħ𝛼𝛼 𝛽𝛽𝑚𝑚𝑜𝑜 𝑐𝑐

c c 3 c = + 5 u (15) r 4 r Multiply by (r) and use the fact that −𝑖𝑖 𝛻𝛻𝛻𝛻 − 𝑖𝑖 𝛻𝛻𝛻𝛻 𝑏𝑏𝑜𝑜𝑢𝑢

= (16) 𝑑𝑑𝑑𝑑 One gets 𝛻𝛻𝛻𝛻 𝑑𝑑𝑑𝑑 c3 c4r = ( + c5) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 −𝑖𝑖 − 𝑖𝑖 𝑏𝑏𝑜𝑜𝑟𝑟 𝑢𝑢 (c3 + c𝑑𝑑4𝑑𝑑r) =𝑑𝑑(𝑑𝑑 + c5) (17)

Let 𝑑𝑑𝑑𝑑 𝑜𝑜 −𝑖𝑖 𝑏𝑏 𝑟𝑟 𝑢𝑢 2020

𝑑𝑑𝑑𝑑 r 1 y = c3 + c4r , dy = c r , = (18) ea

4 Y c 4 𝑑𝑑𝑑𝑑 � � 𝑑𝑑𝑑𝑑 331 Thus

(c4) = ( c3) + c5 c4 𝑑𝑑𝑑𝑑 𝑏𝑏𝑜𝑜

−𝑖𝑖𝑖𝑖 � 𝑦𝑦 − � 𝑢𝑢 V 𝑑𝑑𝑑𝑑 c3 = + c5 II c4𝑜𝑜 c𝑜𝑜4 𝑏𝑏 𝑏𝑏 ue ersion I s � 𝑦𝑦 − � 𝑢𝑢 s [ 1 2 + c5] = [ 1 + 3] I c c 0 𝑏𝑏 𝑦𝑦 − 𝑏𝑏 3 𝑢𝑢 𝑏𝑏 𝑦𝑦 𝑏𝑏 𝑢𝑢 3 XX 1 = , 2 = , 3 = 5 2 = 5 (19) c4 c4 c4 𝑏𝑏 𝑏𝑏𝑜𝑜 𝑏𝑏𝑜𝑜 𝑏𝑏 𝑏𝑏3 𝑏𝑏1 𝑐𝑐 −3 𝑏𝑏 𝑐𝑐 − 5 c4 = 1 + , = 1 + = 4 + (20) y 4 y y

𝑑𝑑𝑑𝑑 𝑏𝑏 𝑑𝑑𝑑𝑑 − 𝑏𝑏 𝑏𝑏 )

−𝑖𝑖 �𝑏𝑏 �𝑢𝑢 1 �𝑏𝑏3 � 𝑑𝑑𝑑𝑑 𝑖𝑖 �𝑏𝑏 � 𝑑𝑑𝑑𝑑 A 𝑑𝑑𝑑𝑑 𝑢𝑢 𝑖𝑖𝑖𝑖 ( 4 = , 5 = (21) c4 c4 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏

= 4 + 5 + 6 Research Volume 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ln �= 𝑖𝑖 +� 𝑏𝑏 𝑑𝑑𝑑𝑑ln +𝑖𝑖𝑏𝑏 � 𝑏𝑏 (22) 𝑢𝑢 4 5 6 𝑦𝑦 5 ln ln = + Frontier 𝑢𝑢 𝑖𝑖𝑏𝑏 𝑦𝑦 𝑖𝑖𝑏𝑏4 𝑦𝑦6 𝑏𝑏 (23) 𝑖𝑖𝑏𝑏 𝑢𝑢 − 𝑦𝑦 ln 𝑖𝑖𝑏𝑏 𝑦𝑦= 𝑏𝑏 4 + 6 5

𝑢𝑢 Science � 𝑖𝑖𝑏𝑏 � 𝑖𝑖𝑏𝑏 𝑦𝑦 𝑏𝑏 4 6

= 𝑦𝑦 (24) of 5 𝑢𝑢 𝑖𝑖𝑏𝑏 𝑦𝑦 𝑏𝑏 𝑖𝑖𝑏𝑏 5 4 𝑒𝑒 = 𝑒𝑒7 (25) 𝑖𝑖𝑏𝑏 𝑦𝑦 𝑦𝑦 𝑖𝑖𝑏𝑏 𝑦𝑦

6 Journal 𝑢𝑢 7 𝑏𝑏=𝑒𝑒 𝑒𝑒 (26) 𝑏𝑏 Thus where in view 7 of (18) yields 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑏𝑏 𝑒𝑒

5 4( 3+ 4 ) Global 𝑏𝑏 = 7( 3 + 4 ) (27) 𝑖𝑖𝑏𝑏 𝑖𝑖𝑏𝑏 𝑐𝑐 𝑐𝑐 𝑟𝑟 In view of equations (21),(19) and (4) 𝑢𝑢 𝑏𝑏 𝑐𝑐 𝑐𝑐 𝑟𝑟 𝑒𝑒 Let 5 = 0 Thus 𝑏𝑏 c3 4 5 2 3 = 0, 2 = 5 , 5 = , 0 = = c0 c4 3 c0 𝑏𝑏𝑜𝑜 𝑐𝑐 𝑐𝑐 𝐸𝐸 From (5) 𝑏𝑏 𝑏𝑏 𝑐𝑐 𝑐𝑐 𝑏𝑏 𝛽𝛽𝑚𝑚𝑜𝑜 𝑐𝑐 2 2 = 2 𝑐𝑐

2 = 2 2

2 𝐸𝐸 = 2𝛽𝛽 𝑚𝑚 𝑜𝑜 𝑐𝑐 𝐸𝐸 (28)

Thus 𝑜𝑜 𝐸𝐸 𝛽𝛽𝑚𝑚 𝑐𝑐 4( 3+ 4 ) = 7( 3 + 4 ) (29) 𝑖𝑖𝑏𝑏 𝑐𝑐 𝑐𝑐 𝑟𝑟 The probability of finding the particle at position𝑢𝑢 𝑏𝑏 𝑐𝑐 𝑐𝑐 𝑟𝑟 𝑒𝑒 2 2 2 | | = 7 ( 3 + 4 ) (30)

But experimentally it was found that the number𝑢𝑢 𝑏𝑏 of electrons𝑐𝑐 𝑐𝑐 𝑟𝑟 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 𝑢𝑢 𝑏𝑏 𝑐𝑐 𝑐𝑐 𝑟𝑟 Thus 1 341 n 2 r = 0 3 (32 ) 7 4 4 𝑐𝑐 − II. Discussion 𝑏𝑏 𝑐𝑐 𝑐𝑐2. David Tong (2007), Quantum Field Theory, 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. B.H. Bransden and C. J. Joachain (2000), Quantum oscillating. mechanics. Second edition, Essec CM 202 For hydrogen like atoms the potential is given England. ISBN 0582-35691. by equation (6). The GSR Dirac equation for hydrogen 5. Phillips A. C (2003) Introduction to Quantum

) atom is given by (8). Rearranging for simplification is

A Mechanics, New York, ISBN 0-470-85323-9

( found by defining new variably in equation (19). The (Hardback), 0-470-85324-7 (Paperback). solution of this equation is a complex wave function given by equation (25) and (27). In its general form this 6. Andrew lgla (2017), Quantum Mechanics solution is purely complex. To make the wave function Schrodinger Schrodinger’s Equation Transmission

Research Volume (r) dependent the energy should be proportional to the and Reflection at barrier the tunnel effect Compton rest mass energy as shown by equation (28). This is Effect and university physics, ISBN: soft cover 798- quite natural as well as the stable atom corresponds to 1-5245-9260-8 e Book 978-1-5245-9259-2.

Frontier minimum non existed atomic state [see equation (29)]. 7. Raymond A. Serway and John W. Jewett (2014), The probability or the square of the wave function can Physics for Scientists and Engineers with Modern be made to be equal to the number of atoms in each Physics, Compositor: Lachina Publishing Services 2014, 2010, 2008 by Raymond A. Serway, Library of

Science level [see equation (31)] by adjusting the atomic radius(r) to be dependent on the number of electrons in Congress Control Number: 2012947242, ISBN-13: of a certain energy level. This expression [see equation 978-1-133-95405-7, ISBN-10: 1-133-95405-7. (32)] shows that (r) increases and quantized. This 8. Greiner, W. (2000) Relativistic Quantum Mechanics, conforms to observation. Springer, Berlin, 3rd edition. Journal 9. Hilo, M.H.M., et al. (2011), Using of the generalized III. onclusion C special relativity (GSR) in estimating the neutrino

Global The GSR Dirac equation can successfully masses to explain the conversion of electron explain the mystery of the electrons quantum number if neutrinos. NS, 3, 334-338. doi: 10.4236/ns.2011. the atomic radius is quantized and satisfies a certain 32020. relation. It shows also that the stable atom corresponds 10. P. Dirac (1928), The quantum theory of the electron. to the minimum relativistic Einstein energy. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, References Références Referencias The Royal Society. 1. Michele Maggiore (2005), A Modern Introduction to 11. David McMahon (2008), Quantum Field Theory Quantum Field Theory, Oxford University Press. Demystified, McGraw-Hill Companies.

12. Hobson, E fstathion and Lasen by (2006), General relativity An Introduction for physicists, ISBN -13 978-0521-82951-9 published in New York. 13. Fatma Osman, Mubarak Dirar, Ahmed Alhassen, Musa Ibrahim and Sawsan Ahmed, (2019), Quantum Equation for Generalized Special Relativistic Linear Hamiltonian, IJRERD, ISSN: 2455-8761. 14. Fatma Osman, Mubarak Dirar, Ahmed Alhassen, Musa Ibrahim and Sawsan Ahmed, (2019), Time Independent Generalized Special Relativity Quantum.

15. Mubarak Dirar Abd Allah, Ebtisam Abd-Alla 2020

Mohamed, Ibrahim Adam Ibrahim Hammad, Ibrahim r ea

Alfaki (2017), Derivation of Statistical Physical Laws Y From Quantum Mechanics, American Journal of Innovative Research and Applied Sciences. ISSN 351 2429-5396, www.american-jiras.com

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