NON COMMUTATING SPACE

   In quantum theory, we use two important equations to handle the dynamics of the particles. These are the Schrodinger equation and the Dirac equation [1-5] .In the non relativistic case (v<<c) we use the well known Schrodinger equation to solve the different potential problems such as harmonic oscillator, particle in a box potential step barrier coulomb potential etc. However in the relativistic case (v>>c) we use the Dirac equation different types of scaler and vector potentials. Hence therefore we briefly see the results for central and hydrogen atom potential cases .

          Initially the Dirac’s free particle equation was originated in an attempt to express linearly the relativistic quadratic relation between energy and momentum. We introduce a Dirac equation which, besides the momentum, is also linear in the coordinates. We call it the Dirac oscillator because in the non-relativistic limit it becomes a harmonic oscillator with a very strong spin-orbit coupling term. The eigen states and eigen values of the Dirac oscillator can be obtained in an elementary fashion, with the degeneracy of the latter being quite different from that of the ordinary oscillator. We briefly mention the symmetry Lie algebra responsible for this degeneracy and the generalisation of the problem to many particle system.

        Then therefore in this regard we give an introduction to non-commutative space and find the behaviour of Dirac oscillator corresponding to the relativistic non-commutative space case.

2. RELATIVISTIC QUANTUM MECHANICS

In the case of non-relativistic quantum mechanics case [6] the well-known Schrödinger Equation to study the dynamics of the particle. Similarly the relativistic case (v≅c) we us the Klein-Gordon equation and Dirac equation. In this chapter we will obtain two equation which are useful in relativistic quantum mechanics.

2.1 KLEIN-GORDON EQUATION

In this chapter we will be discussing Klein-Gordon equation which is useful in relativistic quantum mechanics. In relativistic quantum mechanics particle with spin zero is described by the Klein-Gordon equation.

       In non-relativistic case the energy of a particle is

E=p^2/2m

        For mass m and momentum p. Now in quantum mechanics we replace

E→ⅈℏ∂/∂t, p → -ⅈℏ∇

         And the Schrödinger equation for the free particle

Hψ=Eψ

ℏ^2/2m ∇^2 ψ=ⅈℏ∂ψ/∂t

         We want to introduce the relativistic effect then we must use relativistic quantum mechanics

E^2=p^2 c^2+m^2 c^4 [2.1.1]

〖(ⅈℏ ∂/∂t)〗^2=〖(-ⅈℏ ∇)〗^2 c^2+ m^2 c^4

ⅈ^2 ℏ^2 ∂^2/(∂t^2 )=ⅈℏc^2 ∇^2+m^2 c^4

-ℏ∂^2/(∂t^2 )=-ℏ^2 c^2 ∇^2+m^2 c^4

         Multiplying both sides by ψ, we get

-ℏ^2 (∂^2 ψ)/(∂t^2 )=-ℏ^2 c^2 ∇^2 ψ + m^2 c^4 ψ [2.1.2]

         This Equation is known as Klein-Gordon equation for free particle. Dividing eqn. [2.1.2] by ℏ^2 c^2 both sides we get

-ℏ^2/(ℏ^2 c^2 ) (∂^2 ψ)/(∂t^2 )=-(ℏ^2 c^2 ∇^2 ψ)/(ℏ^2 c^2 )+(m^2 c^4 ψ)/(ℏ^2 c^2 )

-1/c^2 (∂^2 ψ)/(∂t^2 )=-∇^2 ψ+(m^2 c^4 ψ)/(ℏ^2 c^2 )

∇^2 ψ-1/c^2 (∂^2 ψ)/(∂t^2 )=(m^2 c^4 ψ)/(ℏ^2 c^2 )

(∇^2-1/c^2 ∂^2/(∂t^2 ))ψ= (m^2 c^4 ψ)/(ℏ^2 c^2 )

□^2 ψ= (m^2 c^4 ψ)/(ℏ^2 c^2 ) [2.1.3]

       Where □^2= (∇^2-1/c^2 ∂^2/(∂t^2 )) is called D’ Alembertian operator. Now the equation has a plane wave solution is given by

Ψ=e^(ⅈ(k ⃗.r ⃗-ωt)) [2.1.4]

     Again differentiation with respect to t two times we get

(∂^2 ψ)/(∂t^2 )=-ω^2 e^(ⅈ(k ⃗.r ⃗-ωt)) [2.1.6]

    Now we have to find ∇^2 ψ

∇^2 ψ=-k^2 e^(ⅈ(k ⃗.r ⃗-ωt)) [2.1.7]

     Now putting the values of eqn. [2.1.4], [2.1.4], [2.1.7] in eqn. [2.1.2] we get

-ℏ^2 [-ω^2 e^(ⅈ(k ⃗.r ⃗-ωt))]= -ℏ^2 c^2 [〖-k〗^2 e^(ⅈ(k ⃗.r ⃗-ωt))]+m^2 c^4 e^(ⅈ(k ⃗.r ⃗-ωt))

ℏ^2 ω^2= ℏ^2 c^2 k^2+ m^2 c^4

E= ℏψ= ∓√(ℏ^2 c^2 k^2+ m^2 c^4 )

     Thus, a free particle in a relativistic quantum mechanics will have positive as well as negative energy Eigen value.

∂x/∂t