+ αx2. 2. −x∂x − 1. 2 x∂x + 1. 2. −β∂2 x. 2. − βx2. 2. ) where α, β are two constants, α,β ... The wave function ψ(x, t) describe the probability distribution in time.
ACTA UNIVERSITATIS APULENSIS
No 13/2007
THE NONCOMMUTATIVE DIRAC EQUATION FOR SOME WAVE FUNCTIONS
Pis¸coran Laurian Abstract. In this paper we will analyzed the noncommutative Dirac equation which was introduced in the paper [7], for some wave functions. For instance, we will study the movement of an electron inside of an atom using the noncommutative Dirac equation. 2005 Mathematics Subject Classification: 34L40, 58B99. Keywords and phrases: noncommutative Dirac equation, wave functions. 1. Introduction In the paper [7], we construct the ”noncommutative Dirac equation”, which was: ∂ ψ(t, x) = Q(x, ∂x )ψ(t, x) (1) ∂t Here Q(x, ∂x ) represent the noncommutative harmonic oscilator which is the second-order ordinary differential operator: i~
Q(x, ∂x ) =
α 0 0 β
2 ∂x x2 1 0 −1 − + + x∂x + = 1 0 2 2 2 2 − α∂2 x
+ x∂x + 12
2 α x2
1 2
−x∂x − 2 2 −β ∂2x − β x2
!
where α, β are two constants, α, β > 0. ψ(x, t) represent the wave function and ~ represent the Planck constant. The wave function ψ(x, t) describe the probability distribution in time and space of an particle. 71
L. Pi¸scoran - The noncommutative Dirac equation for some wave functions
For instance, the wave function ψ(x, t) for an electron which is moving inside of an atom in the fundamental state, defined for n = 1 in the equa4 1 tion E = − 8εme 2 ~2 n2 , have the following wave function (see [4]): ψ(x, t) = 0 q ~2 ε0 2 − ar e cos ωt where a = πme 2 and r represent the radius of the orbit of an πa3 electon. 2. Main Result q r Theorem 2.1 For the wave function ψ(x, t) = πa2 3 e− a cos ωt , which represent the wave function of an electron which is moving inside of an atom, the noncommutative Dirac equation is: ~ω tan ωt = x∂x + 21 . Proof.Starting with the noncommutative harmonic oscilator : ! 2 2 − α∂2 x + α x2 −x∂x − 21 Q(x, ∂x ) = 2 2 x∂x + 12 −β ∂2x − β x2 and using the noncommutative Dirac equation, one obtain : ! 2 2 ∂ − α∂2 x + α x2 −x∂x − 21 ψ(t, x) i~ ψ(t, x) = 2 2 ∂t x∂x + 12 −β ∂2x − β x2 In this equation we take r ψ(t, x) = and we get: r − ar
i~ωe
2 (− sin ωt) = πa3
2 −r e a cos ωt πa3
2 − α∂2 x
+ x∂x + 12
αx2 2
−x∂x − 2 − β∂2 x −
1 2 βx2 2
!r
2 −r e a cos ωt πa3
Using the matricial representation for the complex numbers, i =
0 1 −1 0
,
one obtain : ~ ∂ψ(t,x) ∂t
0 −~ ∂ψ(t,x) 0 ∂t
!
=
2 − α∂2 x
x∂x +
2 α x2
+ 1 2
72
ψ(t, x)
ψ(t, x)
1 2
−x∂x − ψ(t, x) 2 2 −β ∂2x − β x2 ψ(t, x)
L. Pi¸scoran - The noncommutative Dirac equation for some wave functions
Identifying, one obtain the following equation: r r 2 2 −r 1 − ar ~ωe sin ωt = x∂x + e a cos ωt 3 πa 2 πa3 Finally, one obtain: ~ω tan ωt = −x∂x + 21 , so the theorem is proved. In the paper [7], we obtain the noncommutative Dirac equation for a free particle as: 1 ∂ ψ(x, t) (2) ~ ψ(x, t) = − x∂x + ∂t 2 iEt
Lemma 2.2. If we take the deBroglie wave function ψ(t, x) = Ae ~ + − iEt Be ~ , (where E represent the total energy), using the noncommutative Dirac equation (1), one obtain the total energy: 1 E=− i
1 x∂x + 2
Ae
iEt ~
+ Be−
iEt ~
Ae
iEt ~
+ Be−
iEt ~
.
Proof. Applying equation (1) to the de Broglie wave function (2) , one obtain: iEt iE iEt iE − iEt 1 IEt ~ A e ~ − B e ~ = − x∂x + Ae ~ + Be− ~ ⇒ ~ ~ 2 iEt iEt 1 − IEt − iEt ~ ~ ~ ~ = − x∂x + 2 Ae + Be iE Ae − Be Finally, one obtain: 1 E=− i
1 x∂x + 2
Ae
iEt ~
+ Be−
iEt ~
Ae
iEt ~
− Be−
iEt ~
so, the lemma is proved. Corollary 2.3. If we consider a plane wave function: ψ(t, x) = i iEBt − ce( ~ Apx− ~ ) , where A, B are constants, using noncommutative Dirac equation one obtain the total energy: 1 ~ px + . E=− ~B 2i 73
L. Pi¸scoran - The noncommutative Dirac equation for some wave functions
Proof. ipAx iEBt ∂ ( ipAx 1 − iEBt ) ~ ~ ~ = − x∂x + ce ce( ~ − ~ ) ⇒ ∂t 2 ipAx ipAx iEBt 1 iEB − iEBt ( ) ~ ~ ~ce − = − x∂x + ce( ~ − ~ ) . ~ 2 So, finally, we obtain: iEB = x∂x + 1 E= iB
1 x∂x + 2
1 = ~B
1 ⇒ 2
i~ ~ − 2 x∂x + i 2i
=
1 ~ (px + ). ~B 2i
References [1] Thaller B., The Dirac equation, Springer-Verlag,Berlin, Heildelberg, New-York, 1992. [2] J. G. Bondia, J. Varilly, Elements of Noncommutative Geometry, Birkhauser, Berlin, 2001. [3] J. Varilly, Introduction in Noncommutative Geometry, Summer School in Lisbon, 1997. [4] Halliday David, R. Resnick, Physics- part two, Ed. Did. si Pedag., Bucuresti 1975 [5] P. Almeida, Noncommutative Geometry, Lisbon, 2001. [6] Pi¸scoran Laurian, Finsler Geometry and Noncommutative geometry with application, Presa Univ. Clujeana, Cluj-Napoca 2006 [7] Pi¸scoran Laurian, The Dirac equation and the noncommutative harmonic oscilator, Studia Mathematica, 2006 (to appear) Author: Pi¸scoran Laurian Department of Mathematics and Computer Science North University of Baia Mare,Str. Victoriei nr. 76 Baia Mare, Romania. 74
