A solution of Dirac-like Equation for Electrons and the possible Dark objects & vice versa

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Volume 2012, Issue No.5, Dt.16 May 2012 Time: 8h 48 A.M.

A solution of Dirac-like Equation for Electrons and the possible Dark objects & vice versa

by

Professor Dr Kotcherlakota Lakshmi Narayana
{Retd. Prof. of Phys, SU, Kolhapur} 17-11-10, Narasimha Ashram, Official Colony,
Maharanipeta. P. O, Visakhapatnam - 530002. Mobile No: 9491902867

ABSTRACT

Dirac-like Equation solved under perturbation conditions to realize the possible solutions to describe the Physics of electrons and Dark positive electrons with the sea of negative electrons totally filled by the dark objects, of course, in the style of interpretation that is unique. The possible Dark background was realized by Late Prof K R Rao of Madras University, while carrying out his research for D.Sc.(Madras) during the years 1920 to 1924 at Andhra Pradesh in Viziyanagaram and Visakhapatnam.

INTRODUCTION:

We use in a standard formalism of P M A Dirac that
a⁰ = a₀;  a¹ =- a₁ ;  a²=-a₂ ; a³=-a₃      and
g⁰⁰ = 1;   g¹¹ = g²² = g³³ = -1; g ^μν  =  0          for μ ≠ ν and also
a^μ   = g ^μν a_ν   and

pr = -i*hc* ∂ /∂xr r=1,2,3 suffixes balanced and we have


pr = i*hc* ∂ /∂xr and extend to complete 4-vector as


pμ = i*hc* ∂ /∂xμ a new dynamical variable. Here we have,


p0 = i*hc* ∂ /∂x0 with p0 =E/c.

Here “hc” represents the “hcross” expression and not to be confused with the velocity of light symbol c.

Dirac states that “A perfect vacuum is a region where all the states of positive energy are unoccupied and all those of negative energy are occupied. We assume that these unoccupied negative-energy states are the positrons. Wave Equation of the form (11) with –e for e, and then supposed all the states of negative energy for the positrons are filled up, a hole in the distribution of negative energy positrons being then interpreted as an ordinary electron. The theory could be developed consistently with the hypothesis that all the laws of are symmetrical between positive and negative electric charge.”

MY MODEL

With momentum component p1=p2=zero we get the matrix for the electron and the positive dark matter. Here we have p0, p3, p4 and beta representing the momentum components and the last one as the term m*c where m is the mass of electron and as well of the Dark matter particle, of course, with the considerations of Dirac formalism. We expect all the theory consistently developed with the hypothesis that all laws are symmetrical between positive dark matter and negative electric charge.

I have introduced p4 to describe the perturbation


[ p0-beta, 0, -p3, -p4;


0, p0-beta, p4, p3;


-p3, -p4, p0+beta, 0;


p4, p3, 0, p0+beta]; ------------------------ (1)



Using p0-beta= p3*eta and p0+beta= p3/eta



we obtain the perturbed equations as


- p3*C3 + (p3* η)* C1 = p4*C4


p3*C4 + (p3*η)*C2= - p4*C3


-p3*C1 + ( p3/η)*C3 = p4*C2


p3*C2 + (p3/η)*C4 = – p4*C1----------------- (2)



To solve these equations of perturbation theory take p4 as a fraction f of p3 then


f= (-C3+ η*C1)/ C4 ---------------------(3)


f= (C4+ η*C2/)/ C3 ;---------------------(4)


compare and solve eq.3 and eq.4 equations to get


η = (C4^2+C3^2)/(C2-C3)*C1



f= (-C1+C3/eta)/C4; ------------------(5)


f= (C2+C4/eta)/C1-----------------------(6)


compare and solve to get from eq.5 and eq.6 as


η =( C4^2-C3*C1)/(C1^2+C4*C2)

An ambiguous case arises if one uses the set of eq.3 and eq.5 and again the set eq.4 and eq.6 to describe the electron and positive Dark Matter state respectively. Contradiction does not arise if one uses the perturbation theory and a judicious choice of equations to describe the particle sets.

The problem of the non-relativistic state in the limit of η-->zero remains as an illusion. This assumption surmounts the contradictions.

CONCLUSIONS

A plane new wave runs in the z direction. It shall be obvious that avoidance of suppressed equations, to illustrate the unique existence of electron and a dark matter particle would give a new insight of the physics of ordinary and the dark matter considerations in the realm of Cosmology and Particle Physics.

ACKNOWLEDGEMENT

I am indebted to Late Professor K. R Rao D.Sc. (Madras) D.Sc. (London) for his inspiring guidance and helping me to formulate new thoughts in the subjects of human endeavor. Please see the reference to Dark background by Late Prof K. R. Rao D.Sc.(Madras) during the years 1920-1924.

References

1. Quantum Mechanics several books available in the Andhra University Library, Visakhapatnam since 1930 when Late Professor K. R Rao D.Sc. (Madras) D.Sc. (London) joined the University.

2. Prof KRRao D.Sc.(Madras) D.Sc. (London)


        D.Sc (Madras) thesis extract on Dark Background:

 Experiments were therefore made to study the fluorescence of

Thallium vapour when excited by different radiations.

The vessel used for these experiments was in the

form of a cross made of seamless steel tubing 1" in

diameter having welded joints at the cross. A sketch of

the apparatus is given in Fig. 6. The end D is permanently

closed with a welded joint so as to serve as a perfectly

black background. The primary or exciting beam

passes along and is brought to a focus at the center of

the cross and fluorescence was observed in the direction

DC against the dark background. With a view to

photograph even the faint fluorescent light, it became

necessary to avoid all scattered light and so between

the slit of the observing spectrograph and the end of the

cross there were placed a number of coaxial diaphragms.

The dotted portion was subject to preliminary heating

and was exhausted to a pressure of 1 m.m at about 200 C

and then the temperature was raised to about 500 C ,when

observations were made in the lateral direction by illuminating

the vapours with different radiations. The table gives a

summary of the observations. These radiations are obtained

by using different colour filters.

 Colour filter                    Region of Result of Interposing            Transparency Colour filter.


Ni SO4 (N/10)                     whole of ultra-
 
Solution                               violet but opaque                                     No fluorescence.


                                            to l 3700 - l 4200.


CoCl 2 (N/10)


Solution                                 l 3000- l 4000                                             Green fluorescence.


Colour filter                              Region of Result of interfering            Transparency Colour filter


Dark Green                                 Yellow-                                                    No fluorescence.


Glass Green


NiCl 2 (N/5)                                l5000 - 4400 "


Solution



The observations indicate that the vapour is excited
 to fluorescence only when illuminated by radiations of
l 3775 and shorter than this. l 3775 therefore represents
the minimum excitation energy required to make the

 vapour fluoresce. Further when illuminated by l3775

 the valence electron is thrown out from 1p₂   to 1s and

 from 1s  it jumps either to 1p₂  or to 1p₁. Corresponding to

 these  two transitions the vapour emits l 3775 and l5350.
The experiments not only give the minimum excitation


energy but also indicate in a striking manner that
1p₂  is the ground orbit of the valence electron in the neutral

atom of Thallium.

 It may be interesting to mention here that similar

 experiments conducted on Bismuth27 vapour in this

laboratory  have led to the clear conclusion that l 4722 of the

Arc Spectrum of the element, although it is a 'raie ultime'
according to de Gramont and a resonance line, seemed

to result as a transition not to the normal state but to

one of the excited states. The further recent experiments

confirmed this view for the underwater spectrum of Bismuth

showed this line l 4722 distinctly in emission while

Several other lines ll3067, 3025, 2993, 2989, 2938,

2898, 2810, etc. are absorbed and therefore considered to

be due to transitions to the lowest energy level.