Analysis of Characteristic of Free Particles: Relativistic Concept

doi:10.4236/jmp.2012.310176

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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

Email: [email protected]

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



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





 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



cp gr

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











 (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

E. I. UGWU ET AL. 1395

ˆo

12 3

ˆˆˆˆ

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

Epcmc (3)

In terms of Equation (1), we write



2224

cmc

















(3)

This can be iterated to give



ˆˆˆˆ

ˆˆ

ij ji

ij ij

oij

 



ˆo





























 

























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

 



ˆi





(6a)

ˆ (6b)

(6c)

These anticommutation relations defines the algebraic

matrices whose hermicity is fulfilled if

†

ˆˆ

ii i





 and †

ˆˆ





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

ˆ1



2

ˆ1





A

100

010

001











were

ˆˆ



  (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.

ˆˆ







ˆˆ ˆˆ

tr tr

(7)





ˆˆ

ˆˆˆ

tr trtr

ii i

(8)

which invariably indicates that



 



tr 0

(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.

tr 00









 

ˆˆ











ˆˆ

tr tr

(11)







211

KA A

(12)

where





ˆˆ0

(13)

Since the explicit representation of the Dirac matrices

are











11 0

011











ˆi

(14a)

(14b)



being Pauli’s matrices and 11 unit matrix With

ˆi



and



can be written especially as shown below.

0001

0010

ˆ0100

1000













000

00 0

ˆ000

000

















00 1 0

000 1

ˆ100 0

0100



















10 00

00 01

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

ˆMC













 (20)

of which we can write four solutions of wave function

thus



iMC t









10exp











(21)



iMC t







21exp













(22)



iMC t







30exp













(23)



iMC t









40exp















(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.





000

d,,

ipp x

xtbpsup s e











,bps



2πs

E



(25)

Here the amplitude determines the admixture

of the plane wave ,i

upsexh







,,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



igr

pcc p

Jp bpsV









 (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,

gr cp

 (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

cpE

and

2ˆ

 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.

d1 ˆ

ˆ,

tih













(29)

This can be solved by comparing it with the classical

equation of motion of a rapidly oscillating motion



0exp2

iiHt



 (30)

Whose amplitude and frequency of these additional

oscillations are of the other 0

2mc and 2

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

gr cp









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

2mc and

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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