Journal of Modern Physics, 2013, 4, 940944 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 TimeDependent Linear Potential in NonCommutative 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 timedependent invariant of the Dirac equation with timedependent linear potential has been con structed in noncommutative phase space. The corresponding analytical solution of the Dirac equation is presented by LewisRiesenfield invariant method. Keywords: NonCommutative; Dirac Equation; TimeDependent Linear Potentials; Exact Wave Function 1. Introduction It is well known that the quadratic Hamiltonian system includes not only conservation system but also timede pendent system [1,2]. The study of timedependent quantummechanical 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 [68] 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 [1 014] making us e of dif ferent methods, for example path integral, second quan tization and dynamical invariant. Besides the timede pendent harmonic oscillator, the timedependent 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 microwavedriven 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 [2224] 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 timedependent Schrödinger equation and Dirac equation in commutative space, no one has reported the study of the timedependent quantum prob lems in noncommutative space. While, there has been much interest in the study of ph ysics in noncommutative spaces (NCS) in recent year [25,26], not only because the NCS is necessary when one studies the low energy effec tive of the Dbrane with B field background, but also because in the very tiny string scale or at very high en ergy situation, the effects of noncommutativity of space may appear. Generally, the theory and methods of re searching noncommutative problems are mainly from quantum field theory. But, it is fascinating to speculate whether there might be some lowenergy effects of the fundamental quantum field theory. These effects might arise as a noncommutative version of quantum mechan ics [2729]. Under the frame of quantum mechanics, the Quantum Hall effect [30], AharonovCasher effect [31], gravitational quantum well [32] and the twodimensional anharmonic oscillator [33], have been studied extensively in noncommutative space. In this paper we are interested in nonperturbation in vestigation of Dirac equation with timedependent linear potential in noncommutative phase space [32,34,35]. The main purpose of the paper is to construct the time dependent invariant of the Dirac equation with timede 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 nonco mmutative phase space. 2. Solution of the Dirac Equation with TimeDependent Linear Potential in NonCommutative Phase Space It is well known that in the twodimensional commuta tive space, the coordinates i 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 noncommutative space as ˆi and (12 1 ˆi p,, xx yp p and 2 pp ) re spectively, then in the twodimensional noncommuta tive phase space [32,34,35], the ˆi 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 noncommutative parameters of the noncommutative phase space. Particularly, when 0, 0 the noncommutative phase space reduces to commutative space. One possible way of implement ing algebra Equation (2) is to construct the noncommu tative var iables ˆˆˆˆ ,, , y yp p ,, , from the commutative vari ables y 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 twodimensional noncommutative phase space where the timedependent linear potential: ˆˆ ,,Vxyt f 12 ˆ ˆ txf ty (12 tft 2 12 ˆˆ ˆˆ yy are arbitrary function of time). Then Hamiltonian of the system is given by: ˆxx 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 noncommutative 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 tx pftypm ff pp txftym In commutative space the timedependent 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 timedependent Dirac equ a tion [22,23] in commutative space. In this paper, we used the socalled LewisRiesenfield invariant method. Now let us suppose that there exists a quantummechanical invariant 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 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 t to be 112 2 y tAtpBtxAtp BtyCt (8) ,,, 112 2 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 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 (10a10d), we h a ve Aaa (11a) 234 , Aaa (11b) 112 , Bbb (11c) 234 , 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 and is given by 12 3 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 ,. 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 ap ap ax ay af aft (18) It is easy to see that the eigenfunction of t 22 4 ty 4t is of the form: 123 ,, exp xyt tx ty tx ,,ttt (19) where 12 3 and are arbitrary timedependent functions. From Equation (7), it is obvious that if is a solu tion of the timedependent Dirac equation, then the any function defined by will also be. In this way, if we take , we have that is an eigenfunction of . This suggests that the solutions of the timede pendent Dirac equation have the form of trial function ,, ,, yt tx yt (20) where t 21is a timedependent 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 t and 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 timedependent linear potentials in noncommutative phase space. In particular, when momentummomentum 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 timedependent linear potentials has been studied in noncommutative phase space. By LewisRiesenfield 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 spacespace and momen tummomentum 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. 36653668. 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. 527529. doi:10.1103/PhysRevLett.66.527 Copyright © 2013 SciRes. JMP
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