Journal of Modern Physics, 2013, 4, 940-944
http://dx.doi.org/10.4236/jmp.2013.47126 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Solution of Dirac Equation with the Time-Dependent
Linear Potential in Non-Commutative Phase Space*
Xueling Jiang1, Chaoyun Long2#, Shuijie Qin2
1School of Mechanical Engineering, Guizhou University, Guiyang, China
2Laboratory for Photoelectric Technology and Application, Guizhou University, Guiyang, China
Email: #long.chaoyun@163.com
Received April 13, 2013; revised May 16, 2013; accepted June 15, 2013
Copyright © 2013 Xueling Jiang et al. 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
In this paper, the time-dependent invariant of the Dirac equation with time-dependent linear potential has been con-
structed in non-commutative phase space. The corresponding analytical solution of the Dirac equation is presented by
Lewis-Riesenfield invariant method.
Keywords: Non-Commutative; Dirac Equation; Time-Dependent Linear Potentials; Exact Wave Function
1. Introduction
It is well known that the quadratic Hamiltonian system
includes not only conservation system but also time-de-
pendent system [1,2]. The study of time-dependent
quantum-mechanical system has attracted considerable
interest over the past decades because their quantum
correspondence provides fundamental structure of basic
physics and interpretation of new physics in different
areas of physics, such as, gravitation [3], quantum optic
[4,5], the Paul trap [6-8] and spintronics [9]. In the past
few decades, an extensive effort has been made to obtain
exact solution of the Schrödinger equation with time-
dependent harmonic o scillator [1 0-14] making us e of dif-
ferent methods, for example path integral, second quan-
tization and dynamical invariant. Besides the time-de-
pendent harmonic oscillator, the time-dependent linear
potential model has also been employed to study several
problems in physics. For instance, Cocke and Reichl
have employed this model to study the effects of static
field on the stochastic layer of microwave-driven hydro-
gen [15] and on nonlinear quantum resonances and the
ionization spectrum of a simple bound particle [16] and
compute the spectrum of emitted radiation for a particle
in a triangular potential well and driven by an electro-
magnetic field [17]. Recently, Schrödinger equation [18-
21] and Dirac equation [22-24] of linear potential model
have also been investigated widely and the corresponding
exact solutions have been given, in one dimension.
In spite of a large variety of papers that has been pub-
lished concerning time-dependent Schrödinger equation
and Dirac equation in commutative space, no one has
reported the study of the time-dependent quantum prob-
lems in non-commutative space. While, there has been
much interest in the study of ph ysics in non-commutative
spaces (NCS) in recent year [25,26], not only because the
NCS is necessary when one studies the low energy effec-
tive of the D-brane with B field background, but also
because in the very tiny string scale or at very high en-
ergy situation, the effects of non-commutativity of space
may appear. Generally, the theory and methods of re-
searching non-commutative problems are mainly from
quantum field theory. But, it is fascinating to speculate
whether there might be some low-energy effects of the
fundamental quantum field theory. These effects might
arise as a non-commutative version of quantum mechan-
ics [27-29]. Under the frame of quantum mechanics, the
Quantum Hall effect [30], Aharonov-Casher effect [31],
gravitational quantum well [32] and the two-dimensional
anharmonic oscillator [33], have been studied extensively
in non-commutative space.
In this paper we are interested in non-perturbation in-
vestigation of Dirac equation with time-dependent linear
potential in non-commutative phase space [32,34,35].
The main purpose of the paper is to construct the time-
dependent invariant of the Dirac equation with time-de-
pendent lin ear potential and to obtain , throug h the Lewis-
*This work was supported by the Foundation of Guizhou province
master, China (Grant Nos.(2012)61) and Foundation of social devel-
opment of Guizhou province, China ( G rant Nos.(2012)3073).
#Corresponding author.
C
opyright © 2013 SciRes. JMP
X. L. JIANG ET AL. 941
Riesenfield invariant method [36], a corresponding ana-
lytical solution to th e Dirac equation in non-co mmutative
phase space.
2. Solution of the Dirac Equation with
Time-Dependent Linear Potential in
Non-Commutative Phase Space
It is well known that in the two-dimensional commuta-
tive space, the coordinates i
x
and momenta satisfy
the usual canonical commutation relations: i
p

,,0,
,,1,2
ij ij
xx
ij




,i
,0
ij
ij
xp
pp




(1)
However, the recent study results on the NCS show
that at very tiny scales, say string scale, the space may
not commute anymore. Let us denote the operators of
coordinates and momenta in non-commutative space as
ˆi
x
and (12 1
ˆi
p,,
x
x
xx yp p and 2
y
pp
) re-
spectively, then in the two-dimensional non-commuta-
tive phase space [32,34,35], the ˆi
x
and satisfy the
following commutation relations: ˆi
p

2
ˆ
, i,
,,1,2
1, .
4
xy
effij
p p
ij
ˆˆˆ
,i,
ˆˆ
,i
ij
eff
xy
xp










(2)
where
and
are non-commutative parameters of
the non-commutative phase space. Particularly, when
0, 0

the non-commutative phase space reduces
to commutative space. One possible way of implement-
ing algebra Equation (2) is to construct the non-commu-
tative var iables
ˆˆˆˆ
,, ,
x
y
x
yp p
,, ,
from the commutative vari-
ables
x
y
x
yp p by the following means of linear
transformations:
11
,,
2
11
,.
22
yx
yy
y p
p x

ˆˆ
2
ˆˆ
xx
xx py
pp yp


 
,
 

(3)
Now let us consider a Driac particle moving in the
two-dimensional non-commutative phase space where
the time-dependent linear potential:

ˆˆ
,,Vxyt f
 
12
ˆ ˆ
txf ty (12
f
tft

2
12
ˆˆ
ˆˆ
yy
are arbitrary
function of time). Then Hamiltonian of the system is
given by:


ˆxx
H
tcp
ftx



cp cm
f ty
1 0.
0 1
 
 
 
(4)
with the Pauli matrices:
010 i
,,
10i 0
xy

 

 
 
To find the solution of the Dirac equation of system
considered here, two approaches are possible. One can
directly work in the non-commutative space variables or
use the phase space transformations to reduce the prob-
lem on the usual commutative space. In following dis-
cussion, we work in the second approach.
Substituting Equation (3) into (4), we have corre-
sponding Hamiltonian expressed by the commutative
variables

 
 
12
21
12
,, ,1
22
22
22
.
22
xy
xx yy
yx
xxyy
yx
xyp pc
Htpyp x
f
tx pftypm
ff
pp
f
txftym





 






 



 




 




In commutative space the time-dependent Dirac equa-
tion is written as:
 
,1.iHt
t
(5)
So the present task is to solve Equation (5). There are
also several methods to study time-dependent Dirac equ a-
tion [22,23] in commutative space. In this paper, we used
the so-called Lewis-Riesenfield invariant method. Now
let us suppose that there exists a quantum-mechanical
invariant
I
t that satisfies the equation:
 
d1,0,1.
d
It I
It Ht
ti t
 


(6)
By applying Equation (6) on
and after some minor
algebra, we obtain

iI
H
tI
t
(7)
which implies that the action of the invariant operator on
a Dirac wave function produces another solution of the
Dirac equation. This result is valid for any invariant if the
latter involves the operation of time differentiation. For
the model considered here let the linear invariant
I
t
to be

 
112
2
I
x
y
tAtpBtxAtp
BtyCt

 (8)

,,,
112 2
A
where tBtAtBt

Ct
, and are matrices.
The substitution of Equations (8) and (4) into Equation (6)
we have
Copyright © 2013 SciRes. JMP
X. L. JIANG ET AL.
942











22
12
2
21
22
11
22
21
21
,,
,,
2
,,i
,,
2
,,i
2
,,
,,
2
xx y y
x
2
1
1
,
2
,i
i
y
x
x
x
AmAp
p
Bx
By





yy
y
x
xy
xy
x
x
ApApB
ByC
CAmA
CBm
CBm
AApp
BA

 













 




 



















12
21
11
22
21
12
,,
2
,,
22
,,
2
,,
2
ii
22
i
x
y
yy
yx
y
xx
x
yy
xy
yp
BA
BB
BA x
BAy
ff
BB

 






















































21
,,0,
xy
AA


xp
xy
p
p

112 2
12
i
22
i,i ,i
yx
xy
Af Af
BBC
 



 





 


, 0.Cm
(9)
A solution of the above relation is obtained by
1,0,
x
A
2,0,
y
A


(10a)
(10b)
1,0,
2
y
B




(10c)
2,0,
2x
B




(10d)
11
i0,mA
,,
x
CA

(10e)
22
i0,mA
,,CA

y

 (10f)

11
,i0,mB
 




,
2
y
CB


 (10g)

22
i0,mB
,,
2x
CB



 (10h)


(10i)
21
,,0,
2x
x
BA




(10j)
12
,, 0,
2
y
y
BA







 

(10k)
21
,,0,
22
yx
BB
 








(10l)
11
,, 0,
2
y
x
BA








(10m)
22
,,0,
2x
y
BA






(10n)


21
1211
221 2
iii
222
ii,i,i
2
,0.
y
xy
x
xy
ff
BBAf
AfBBC
Cm


 


 




 
 
 


112
,
(10o)
From commutation rela t i ons (10a-10d), we h a ve
x
Aaa (11a)
234
,
y
Aaa (11b)
112
,
y
Bbb (11c)
234
,
x
Bbb
a

1, 2, 3, 4bi
C
(11d)
respectively, where i and i are time-
dependent arbitrary functions. Next, let us assume that
the form of writing in terms of
x
and
y
is given
by
12 3
x
y
Cc cc

,ccc
123342
0,0,0,0,0,0,aacaac
(12)
where and are arbitrary functions of time.
12 3
Substituting Equations (12) and (11a) into Equation
(10e); Equations (12) and (11b) into Equation (10f) re-
spectively, we find


32
0,cc
(13)
where the dot on variables denotes derivative with re-
spect to time.
Noting
1.Cc
then C in Equation (12) is changed
to (14)
Substituting Equations (14) and (11c) into Equation
(10g), Equations (14) and (11d) into Equation (10h) re-
spectively, we get
2143
0, 0,0,0.bbbb

(15)
Making use of Equations (13) and (15), the Equation
Copyright © 2013 SciRes. JMP
X. L. JIANG ET AL. 943
(11) can be rewritten in the following form:
1123
,,
1123
,.
A
aA aB bB b (16)
Substituting Equation (16) into Equation (10o) leads to
13
,
2
ba
(17a)
31
,
2
ba
 (17b)

1132
d.afaft
1
1
4
c





(17c)
Finally, the invariant of the Dirac equation for time-
dependent linear potential can be written as


31
11 32
1d
.
13
2
4
xy
I
ap ap






ax ay
af aft
 
(18)
It is easy to see that the eigenfunction of
I
t


22
4
ty


4t
is of
the form:
  
123
,,
exp
xyt
tx ty tx


  
,,ttt

(19)
where 12 3
and are arbitrary
time-dependent functions.
From Equation (7), it is obvious that if
is a solu-
tion of the time-dependent Dirac equation, then the any
function defined by
I
will also be. In this way, if
we take

, we have that
is an eigenfunction
of
I
. This suggests that the solutions of the time-de-
pendent Dirac equation have the form of trial function
,, ,,
x
yt

tx yt
(20)
where
t
21is a time-dependent matrix:
 

1
2
.
t
tt




The substituting of Equation (20) into Equation (7)
gives
 
113
i
22
tft t

2
33
,
22
ft
tt





 
 




(21a)




2241
4
44
i
2
2
,
22
tft tft
tt
tt


 


4
2
i2
222



(21b)
 
  
21
12
ii ,
2
i,
2
x
y
ft
ttt
ft tt mt
 


 



 


(21c)
30,t
(21d)
40.t
(21e)
2
1
f
t and
f
Then, if t
0, 0
are specified, we can easily
solve Equation (2 1), and obtain the analytical wav e func-
tion for the Dirac equation with time-dependent linear
potentials in non-commutative phase space.
In particular, when momentum-momentum and space-
space are all commutative (namely,
), then
the solution (21) returns to that of general quantum me-
chanics.

113
i2,tft t

 
(22a)
22
i,tft

(22b)

 
1
2
,
i,
x
y
ttt
tt mt

 


(22c)
30,t
(22d)
40.t
(22e)
3. Conclusion
In conclusion, the Dirac equation with time-dependent
linear potentials has been studied in non-commutative
phase space. By Lewis-Riesenfield invariant method, the
invariant of system has been constructed and the exact
wave function of the system has been obtained. In addi-
tion, under the condition that space-space and momen-
tum-momentum are all commutative (namely, η = 0, θ =
0) the results return to that of usual quantum mechanic.
REFERENCES
[1] K. H. Yeon, C. I. Um and T. F. Georgr, Physical Review
A, Vol. 68, 2003, Article ID: 052108.
doi:10.1103/PhysRevA.68.052108
[2] D. Y. Song, Physical Review A, Vol. 59, 1999, pp. 2616-
2623. doi:10.1103/PhysRevA.59.2616
[3] S. A. Fulling, “Aspects of Quantum Fields in Curved
Space,” Cambridge University Press, Cambridge, 1982.
[4] G. S. Agarwal and S. A. Kumar, Physical Review Letters,
Vol. 67, 1991, pp. 3665-3668.
doi:10.1103/PhysRevLett.67.3665
[5] H. P. Yuen, Physical Review A, Vol. 13, 1976, pp. 2226-
2243. doi:10.1103/PhysRevA.13.2226
[6] L. S. Brown, Physical Review Letters, Vol. 66, 1991, pp.
527-529. doi:10.1103/PhysRevLett.66.527
Copyright © 2013 SciRes. JMP
X. L. JIANG ET AL.
Copyright © 2013 SciRes. JMP
944
[7] W. Paul, Reviews of Modern Physics, Vol. 62, 1998, pp.
531-540. doi:10.1103/RevModPhys.62.531
[8] M. Feng, K. Wang, J. Wu and L. Shi, Physics Letters A,
Vol. 51, 1997, p. 230.
[9] X. F. Wang, P. Vasilopoulos and F. M. Peeters, Physical
Review B, Vol. 65, 2002, Article ID: 165217.
doi:10.1103/PhysRevB.65.165217
[10] M. Maamache and H. Bekkar, Journal of Physics A, Vol.
36, 2003, p. L359. doi:10.1088/0305-4470/36/23/105
[11] M. Maamache and H. Choutri, Journal of Physics A, Vol.
33, 2000, p. 6203. doi:10.1088/0305-4470/33/35/308
[12] H. G. Oh, H. R. Lee, T. F. George and C. I. Um, Physical
Review A, Vol. 39, 1989, pp. 5515-5522.
doi:10.1103/PhysRevA.39.5515
[13] H. R. Lewis Jr. and W. B. Rienesfeld, Journal of Mathe-
matical Physics, Vol. 10, 1969, p. 1458.
doi:10.1063/1.1664991
[14] I. Guedes, Physical Review A, Vol. 68, 2003, Article ID:
016102. doi:10.1103/PhysRevA.68.016102
[15] S. Cocke and L. E. Reichl, Physical Review A, Vol. 41,
1990, pp. 3733-3739. doi:10.1103/PhysRevA.41.3733
[16] S. Cocke and L. E. Reichl, Physical Review A, Vol. 52,
1995, pp. 4515-4522. doi:10.1103/PhysRevA.52.4515
[17] S. Cocke and L. E. Reichl, Physical Review A, Vol. 53,
1996, pp. 1746-1750. doi:10.1103/PhysRevA.53.1746
[18] J. Bauer, Physical Review A, Vol. 65, 2002, Article ID:
036101. doi:10.1103/PhysRevA.65.036101
[19] M. Feng, Physical Review A, Vol. 64, 2001, Article ID:
034101. doi:10.1103/PhysRevA.64.034101
[20] I. Guedes, Physical Review A, Vol. 63, 2001, Article ID:
034102. doi:10.1103/PhysRevA.63.034102
[21] H. Bekkar, F. Benamira and M. Maamache, Physical
Review A, Vol. 68, 2003, Article ID: 016101.
doi:10.1103/PhysRevA.68.016101
[22] M. Maamache and H. Lakehal, Europhysics Letters, Vol.
67, 2004, p. 695. doi:10.1209/epl/i2004-10109-6
[23] R. R. Landim and I. Guedes, Physical Review A, Vol. 61,
2000, Article ID: 054101.
doi:10.1103/PhysRevA.61.054101
[24] A. S. de Castro and A. de Souza Dutra, Physical Review
A, Vol. 67, 2003, Article ID: 054101.
doi:10.1103/PhysRevA.67.054101
[25] A. Smailagic, Physical Review D, Vol. 65, 2002, Article
ID: 107701. doi:10.1103/PhysRevD.65.107701
[26] E. M. F. Curado, M. A. Rego-Monteiro and H. N. Naza-
reno, Physical Review A, Vol. 64, 2001, Article ID: 012105.
doi:10.1103/PhysRevA.64.012105
[27] P.-M. Ho and H.-C. Kao, Physical Review Letters, Vol.
88, 2002, Article ID: 151602.
doi:10.1103/PhysRevLett.88.151602
[28] J.-Z. Zhang, Physics Letters B, Vol. 584, 2004, pp. 204-
209. doi:10.1016/j.physletb.2004.01.049
[29] J. Gamboa, M. Loewe and J. C. Rojas, Physical Review D,
Vol. 64, 2001, Article ID: 067901.
doi:10.1103/PhysRevD.64.067901
[30] H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C.
Rojas, Physical Review D, Vol. 66, 2002, Article ID:
045018. doi:10.1103/PhysRevD.66.045018
[31] S. C. Chu and P. M. Ho, Nuclear Physics B, Vol. 550,
1999, pp. 151-168. doi:10.1016/S0550-3213(99)00199-6
[32] O. Bertolami, J. G. Rosa, C. M. L. de Aragao, P. Casto-
rina and D. Zappala, Physical Review D, Vol. 72, 2005,
Article ID: 025010. doi:10.1103/PhysRevD.72.025010
[33] B. Muthukumar and P. Mitra, Physical Review D, Vol. 66,
2002, Article ID: 027701.
doi:10.1103/PhysRevD.66.027701
[34] R. Banerjee, B. D. Roy and S. Samanta, Physical Review
D, Vol. 74, 2006, Article ID: 045015.
doi:10.1103/PhysRevD.74.045015
[35] J.-Z. Zhang, Physical Review Letters, Vol. 93, 2004, Ar-
ticle ID: 043002. doi:10.1103/PhysRevLett.93.043002
[36] H. R. Lewis Jr. and W. B. Riesenfield, Journal of Mathe-
matical Physics, Vol. 10, 1969, p. 1458.
doi:10.1063/1.1664991