Journal of Modern Physics, 2013, 4, 559-560
http://dx.doi.org/10.4236/jmp.2013.44078 Published Online April 2013 (http://www.scirp.org/journal/jmp)
The Dirac Equation with the Scattered Electron
Including Extra Potential Energy Comes
fr om the Virial Theorem
Hasan Arslan
Physics Department, Bingöl University, Bingöl, Turkey
Received January 29, 2013; revised February 28, 2013; accepted March 9, 2013
Copyright © 2013 Hasan Arslan. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The scattering of electron by a photon is a well-known reaction in physics. In this study, the change in the electron’s
energy after the scattering is taken into account. The previous works are searched. In order to take into account this
change in the electron’s energy in the equation of motion of the electron, the Dirac equation is used with the virial
theorem. The scattered electron kinetic energy which is given to the electron by the loss in photon’s energy is related to
the potential energy of the electron by the virial theorem which states that the potential energy is two times of the ki-
netic energy in minus sign. A first time application of the virial theorem on a scattered electron by a photon is included
to the Dirac equation.
Keywords: Dirac Equation; Virial Theorem; Scattered Electron
1. Introduction
The Dirac equation, virial theorem and photoelectric ef-
fect are searched from the previous works. The calcula-
tions needed for our purpose are done. The paper is de-
signed as follows. In Section 2, the calculations are given.
In Section 3, the results of these calculations are conclud-
ed.
2. Calculations
The scattered electron by a photon is simply given by the
reaction
ee


h
  (1)
The photon initially has an energy of

and after
the reaction it has an energy of h



Kh
. The loss in photon’s
energy is the gain in the electron’s kinetic energy. There-
fore, the gained kinetic energy of the electron can be
written as


 
(2)
This process is widely defined in [1]. Here, we con-
sider an electron in the electromagnetic interaction before
scattering.
The Dirac equation for a particle interacting with an
elecromagnetic field is derived in [2,3] as
2
ie cmce
tc






 
 


pA 
10 0
,
01 0






(3)
where,


(4)
and

are the 2 × 2 Pauli matrices. By using the virial
theorem, we can add to the Equation (3) an extra poten-
tial energy term given to the electron by the photon.
Since the Equation (3) describes the electron before scat-
tering we must add the potential energy term after scat-
tering. The kinetic energy given in the Equation (2) is
half of the potential energy of the scattered electron due
to the virial theorem [4-6]. So, we denote the potential
energy lost by the electron after scattering as

22VKh


   (5)
Then, the Dirac equation given by the Equation (3) can
be written as

2
i4π
e
cmce
tc







 
 


p 
(6)

can be written in two-compo- The wave function
C
opyright © 2013 SciRes. JMP

H. ARSLAN 560
nent [2] as
2
i
emc t






 (7)
Putting Equation (7) in Equation (6) we get the equa-
tion

i
4π
e
c
tc
e


20
2mc












  

  

 
  
pA


(8)
By writing the small component

of the wave func-
tion in terms of the large component of the wave function

by the equation [2];
2
e
c
mc






pA
iab ab


(9)
and using the Pauli spin matrices identity [2]
 ab

 (10)
also defining the terms
1
, , , a
2
 BAABrLr1 nd 2 pS

(11)
we can write the Equation (8) into two equations after
some calculations. The equation for an electron in the
electromagnetic interaction is

2
i2
22
ee
tmmc


4π

 



 


 

pLSB
0ˆ
VeA


(12)
where B is the weak uniform magnetic field, L is the
orbital angular momentum, and S is the electron spin.
In [7], the interaction of a field with an electromag-
netic potential for an electron is given by

 
0
AIe A 
(13)
where
ˆ
Ve (14)
here I is a 4 × 4 unit matrix and

represents the Dirac
matrices. The second term on the right of Equation (14)
is written in the Equation (12). The first term on the right
of this equation can be replaced instead of the term e

.
Doing so we obtain
 
0
i24π
22 eA I
tmmc
2e







 





LSB
(15)
In Equation (15), p is the momentum operator, m
m
p
is the
ass of the electron, e is the electron charge, c is the
speed of light, L is the angular momentum of the electron,
S is the spin operator of the electron, B is the weak uni-
form magnetic field, eA0 is the potential energy of the
electron before scattering,  is the Planck’s constant,
and

and


are the initial and final frequencies of
the poton reectively. By comparing this equation to
the previous works, it can be seen that this equation is the
Schrödinger-Pauli wave equation when the potential en-
ergy absence during scattering is added.
3. Conclusion
h sp
in this study to combine the Dirac
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