Journal of Modern Physics, 2012, 3, 1394-1397
http://dx.doi.org/10.4236/jmp.2012.310176 Published Online October 2012 (http://www.SciRP.org/journal/jmp)
Analysis of Characteristic of Free Particles:
Relativistic Concept
Emmanuel I. Ugwu1, Daniel Ugadu Onah1, D. Oboma1, V. O. C. Eke2
1Departments of Industrial Physics, Ebonyi State University, Abakaliki, Nigeria
2Departments of Computer Science, Ebonyi State University, Abakaliki, Nigeria
Received July 19, 2012; revised August 30, 2012; accepted September 10, 2012
ABSTRACT
A linear Hamiltonian in spatial derivative that satisfies Klein-Gordon equation was used starting from energy momen-
tum relation for free particle was solved in agreement with the matrices ˆ
ˆ

and

bearing in mind their suitability in
terms of anticommutation relations in parallel with the definition of algebraic matrices whose hermicity is fulfilled by
†
ˆˆˆ
i†
ˆˆ
i

  and

  and in turn linked up to explicit representation of the Dirac matrices. The wave packets of
plane Dirac wave obtained as a superposition of plane wave yielding a localized wave function was normalized consid-
ering only positive energy of plane wave in which the expectation value with respect to the wave packet resulted from

2
cp gr
v
E
was found to agree with the Ehrenfest theorem in relation to Schrodinger theorem as it relates to true
velocity of single particle. A comparison was made between the classical concept with Heisenberg representation from
where the combined effect of the positive and negative energy components was considered.
Keywords: Hamiltonian; Klein-Gordon Equation; Wave Function; Dirac Matrices Wave Packet; Heisenberg
Representation; Superposition; Energy; Group Velocity; Frequency
1. Introduction
The of the wave particle plays important role in under-
standing of the behavourial description of particle as re-
gards the electronic structure, energy spectrum and wave
characteristics [1]. With regard to this, many researchers
probe into this using classical approach which was of
course found to be insufficient in giving complete de-
scription particle. However this is because relativistic
effect which is of most important is in description of any
physical system is excluded. Indeed relativistic effect
plays a crucial role the description of electronic structure
of not only small particle but also heavy particles [1].
Sequel to the evolution of Dirac theory that led to for-
mulation Dirac equation, Dirac’s relativistic frame work
where one electron spinorial solution is a two vector
whose component are wave function has been used but
was found incomplete in giving description of particle
[2-4]. A survey of the current available analytical solu-
tion using relativistic one electron atoms has been carried
out.It has been made that reasonable description can only
be given by making use of quantum electrodynamics [5].
In this work we intend to analyze the wave character-
istics using relativistic approach by bringing together
momentum relation, vectorial relation and Dirac wave in
conjunction with Heisenberg representation observe in a
real sense the feasibility of existence of single particle.
2. Analytical Procedures
The analytical procedure is taken in three aspects via
momentum relation and matrices of free particle, Dirac
wave and Heisenberg representation.
2.1. Momentum Relation and Matrices of Free
Particle
We start with relativistic covariant wave equation of
Schrodinger form is written as


ˆ
H
t



 (1)
for free particle [5,6].
This is a linear time derivative. Now we construct a
Hamiltonian that is also linear in the spatial derivative
that is of the form
C
opyright © 2012 SciRes. JMP

E. I. UGWU ET AL. 1395
2
3
ˆo
mc
12 3
12
ˆˆˆˆ
f
c
Hi
ti xyz


 








 







(2)
Of which if every single component,


of the wave
function has to satisfy Klein-Gordon equation [4,5,7]
starting from energy momentum relation for free particle
22224
0
Epcmc (3)
In terms of Equation (1), we write
2
22
2
t

2224
o
cmc









(3)
This can be iterated to give

3
,1
ˆˆˆˆ
22 22
22
2
22
1,
ˆˆ
ˆˆ
ij ji
ij ij
oij
i
ij
c
tx
mc
ix
 

24
ˆo
x
mc














 








ˆ




This expression has to agree with the following re-
quirements for the matrices

and ˆ

to be suitable
for our application. These are
ˆˆ
ij ˆˆ2 ji ij

  
ˆˆˆ0
ii
  
22
ˆ
ˆi  
ˆ
(6a)
ˆ (6b)
(6c)
These anticommutation relations defines the algebraic
matrices whose hermicity is fulfilled if
†
ˆˆ
ii i

  and †
ˆˆ

 
1
with i and . This implies that the eigen-
values have the values ±1. One of the special features
exhibited by the eigenvalue is the fact that they are inde-
pendent of special representation and can be shown as
diagonal representation of the form with the eigenvalues
2
ˆ1  2
ˆ1
 
N
A
A
100
010
001










ˆ
were
ˆˆ
i
   (5)
From the anticommutation relation of Equation (6) it is
generalize that trace, i.e. the sum of the diagonal ele-
ments of the matrices of each

and ˆ

has to be zero
representation.
i.e.
ˆ
ˆˆ
i

 
ˆˆ ˆˆ
tr tr
(7)
As
A
A

 
ˆˆ
ˆˆˆ tr trtr
ii i
(8)
which invariably indicates that

  
ˆ tr 0
i
(9)
Indicating that
(10)


This agrees with the practice that the trace of a matrix
[8,9] is always equal to the sum of its eigenvalues. i.e.
1
2
00
tr 00
N
A
A
A







 
1
1
ˆ ˆˆ
tr
N
K
K
A
uu 




1
ˆˆ tr tr
i
uu
(11)

i



211
KK
KA A
(12)
where


ˆ 0
ˆˆ0
i
i
(13)
Since the explicit representation of the Dirac matrices
are





11 0
ˆ
011





ˆi
(14a)
(14b)

being Pauli’s matrices and 11 unit matrix With
ˆ
ˆi

and

can be written especially as shown below.
1
0001
0010
ˆ0100
1000







2
000
00 0
ˆ000
000
i
i
i
i









3:
00 1 0
000 1
ˆ100 0
0100









10 00
00 01
ˆ
001 0
0001










(15)
(16)
(17)
(18)

E. I. UGWU ET AL.
1396
2.2. Dirac Wave
By considering only a free particle we set
ˆ0 p (19)


This makes Equation (2) to become
2
0
ˆMC
t






 (20)
of which we can write four solutions of wave function
thus


2
0
1
iMC t




10exp
0
0







(21)


2
0
iMC t



21exp
0
0







0

(22)


2
0
iMC t



30exp
1
0







0

(23)


2
0
iMC t





40exp
0
1






0


(24)
of which the first two correspond to positive energy val-
ue while the last two are negative energy values. In addi-
tion to this unique properties exhibited by Dirac equation,
it also depicts covariance properties. This on the other
hand explains that the solution appears to possess the
correct behaviour in the non relativistic limit in agree-
ment with the considered the wave packet. The wave
packets of plane Dirac waves are superpositions of plane
wave yield localized wave functions in space and time
since Dirac equation is a linear wave equation that
marches a wave packet of a plane wave with positive
energy [3,7]. They are of the form.


1
000
2 3
50
d,,
ipp x
MC
p
xtbpsup s e





,bps

13
2πs
E

(25)
Here the amplitude determines the admixture
of the plane wave ,i upsexh 





3
,,d1xtt x 
 

to the wave packet
and the plus indicates that a superposition of only posi-
tive energy plane wave is considered.
This is normalized as
(26)
with this normalization [7], the expectation value with
respect to the wave packet of positive energy is written is
written as
 
22
3
d,
ii
igr
i
S
pcc p
Jp bpsV
EE





 (27)
Similarly the mean current of an arbitrary wave packet
of plane wave of positive energy from non relativistic
concept is equal to the expectation value the classical
group velocity,
2
gr cp
VE  (28)
This corresponds to the Ehrenfest theorem on relation
to Schrödinger theory and invariably agrees with the true
velocity of single particle that constitutes. 2ˆ
p
cpE
and
2ˆ
p
cp
E
 for free solutions of the Dirac equation. This
assertion agrees with the classical picture.
2.3. Heisenberg Representation
On the other hand, for waves with the same characteris-
tics [7,10,11], the result exhibits paradoxical nature as
their velocity is directed against their momentum which
explains the fact that those particles with negative energy
apparently appears to behave as if they have a negative
mass. Further clarification can be made using the equa-
tion of motion in the Heisenberg representation.
ˆ
d1 ˆ
ˆ,
df
x
x
Hc
tih






(29)
This can be solved by comparing it with the classical
equation of motion of a rapidly oscillating motion

ˆ
0exp2
f
f
cp
iiHt
H

 (30)
Whose amplitude and frequency of these additional
oscillations are of the other 0 2mc and 2
0
2mc
 re-
spectively. This oscillating motion as mentioned already
is known as Zitterbewegung which vanishes if wave
packets with exclusively positive or negative energy are
considered. This implies that interference effect resulting
from the combined effect of the positive and negative
energy components of a wave packet and invariably de-
monstrates that in real sense a single particle theory is
not absolutely possible in practice but can only be ap-
proximately obtained if the associated wave packet is
restricted to one energy range. The most interesting as-

E. I. UGWU ET AL.
1397
pect observed in this analysis is that the true velocity of
single particle appears to constitute both the negative and
positive terms of energy which brings about the idea that
a particles with negative energy apparently tends to have
negative mass as their velocity is directed against their
momentum.
General observation shows that the mean current of an
arbitrary wave packet plane energy from non relativist
concept agrees with the expectation value of the classical
group velocity
2
gr cp
VE




which corresponds to the
Ehrenfest theorem in relation to Schrodinger theory [3]
further clarification during the analysis using Heisenberg
representation brings about frequency of additional os-
cillations of the order
0
2mc and
2
2
0
2mc
 (31)
3. Discussion
In this paper we have analytically studied the fundamen-
tal wave characteristics of free particle based on equation
constructed from relativistically covariant wave equation
of Schrodinger equation that satisfies Klein-Gordon equ-
ation. The expression for anticommutation relations that
defined the algebraic matrices with special representation
showing the diagonal matrices A, to AN in conjunction
with the explicit representation of the Dirac matrices are
seen in Equations (14a) and (14b) with their explicit val-
ue presented in Equations (15)-(18). The wave func- tion
relating to wave packet of the plane wave associated with
any particle being described using Dirac equation is pre-
sented in Equations (21)-(24) as wave packets. The
packet explains the fact that Dirac waves are superposi-
tion plane waves and yields localized wave functions in
space and time in which when considered with only posi-
tive energy of a particle is compactly written as in Equa-
tion (35) and normalized as in Equation (26). The nor-
malization enabled us to write the expectation value with
respect to the wave packet as seen in Equation (27) is
known as Zitterbewegung that vanishes if the wave
packet associated exclusively to both positive and nega-
tive energy. This implies that Zitterbewegung is as a re-
sult of the combined effect of negative and positive en-
ergy components a situation that confirmed the fact that
in a real sense of it, the idea of a single particle is feasi-
ble only when it is considered with restriction to one en-
ergy range.
REFERENCES
[1] A. L. A. Fonseca, D. L. Nascimento, F. F. Monteiro and
M. A. Amato, “A Variational Approach for Numerically
Solving the Two-Component Radial Dirac Equation for
One-Particle Systems,” Journal of Modern Physics, Vol.
No. 4, 2012, pp. 350-354.
[2] R. Franke, “Numerical Study of the iterated solution of
one electron Dirac Equation based on ‘Dirac Perturbation
theory’,” Chemical Physics Letters, Vol. 264, No. 5, 1997
pp. 495-501. doi:10.1016/S0009-2614(96)01361-9
[3] S. McConnel, S. Fritzsch and A. Surzhykoy, “Dirac: A
New Version of Computer Algebra Tools for Studying
the Properties and Behaviour of Hydrogen-Like Ions,”
Computer Physics Communication, Vol. 181, No. 3, 2010,
pp. 711-713.
[4] A. Surzhykoy, P. Koval and S. Fritzsch, “Algebraic Tools
for Dealing with Atomc Shell Model. 1. Wavefunctions
and Itegrals for Hydrogen-Like Ions,” Computer Physics
Communication, Vol. 165, No. 2, 2005, pp. 139-156.
doi:10.1016/j.cpc.2004.09.004
[5] A. Zee, “Quantum Field Theory in Nutshell,” Princeton
University Press, Princeton, 2010.
[6] P. W. Atkins, “Molecular Quantum Mechanics,” Oxford
University Press, Oxford, 1983.
[7] R. A. C. Dirac, “The Lagrangian in Quantum Mechan-
ics,” Physikalisch Zeitchrift der Sowjetunion, Vol. 3,
1933, pp. 62-72.
[8] R. Ballan and J. Zinn-Justin, “Methods in Field Theory,”
North Holland Publishing, Amsterdam and World Scien-
tific, Singapore City, 1981.
[9] P. A. M Dirac, “Principle of Quantum Mechanics,” Ox-
ford University Press, Oxford, 1935.
[10] E. G. Milewski, “Vector Analysis Problem Solver,” Re-
search and Education Association, New York, 1987.
[11] L. H. Ryder, “Quantum Field Theory,” Cambridge Uni-
versity Press Cambridge, 1996.