E. I. UGWU ET AL.

Copyright © 2012 SciRes. JMP

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.

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