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

iab ab

(9)

and using the Pauli spin matrices identity [2]

ab

(10)

also defining the terms

1

, , , a

2

BAABrLr1

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

REFERENCES

[1] A. Beiser, “Csics,” 4th Edition,

e-

ics,” Addison-

nton, “Classical Mechanics of

echanics,” Addison-Wesley

tic Spin-½ and

One approach is done

equation and the virial theorem. The electron scattered by

a photon gains some kinetic energy and this kinetic en-

ergy is related to the loss in potential energy according to

the virial theorem. This extra potential energy is added to

the Dirac Equation to describe the motion of the particle.

oncepts of Modern Phy

McGraw-Hill International Editions, New York, 1987.

[2] J. B. Bjorken and S. D. Drell, “Relativistic Quantum M

chanics,” McGraw-Hill, New York, 1964.

[3] J. J. Sakurai, “Advanced Quantum Mechan

Wesley, Menlopark, 1967.

[4] J. B. Marion and S. T. Thor

Particles and Systems,” 3rd Edition, Harcourt Brace Jo-

vanovich, New York, 1988.

[5] H. Goldstein, “Classical M

Publishing Company, Inc., Boston, 1974.

[6] M. Brack, “Virial Theorems for Relativis

Spin-0 Particles,” Physical Review D, Vol. 27, No. 8, 1983,

pp. 1950-1953. doi:10.1103/PhysRevD.27.1950

[7] W. Greiner, S. Schramm and E. Stein, “Quantum Chro-

modynamics,” 2nd Edition, Springer-Verlag Berlin Hei-

delberg, Berlin, Heidelberg, 2002.

doi:10.1007/978-3-662-04707-1

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