BREMSSTRAHLUNG IN NONCOMMUTATIVE QUANTUM MECHANICS

SEVDA HEYDARNEZHAD

Payame Noor Univeristy (PNU), Orumieh, IranE-mail: sevda.heydarnezhad@live.com

Received September 30, 2013

The intensity cross section for a non-relativistic bremsstrahlung is calculated innoncommutative space. It is shown that for a soft bremsstrahlung the cross section isincreased by a factor which is second order in noncommutativity parameter.

Key words: Soft bremsstrahlung, noncommutative quantum mechanics.

1. INTRODUCTION

In recent years, many aspects of the noncommutative quantum mechanics, asthe low energy limit of the noncommutative quantum field theory, has been studied bythe several authors [1-10]. The theory of noncommutative fields deals with the fieldsdefined in space-time with noncommutating coordinates xi (i= 1,2,3) satisfying

[xi, xj ] = iγij , γij =−γji. (1)

where γij is an antisymmetric 3× 3 matrix. In a noncommutative space one mustreplace the product of the fields with Moyal product. For the fields φ1(x) and φ2(x)it is defined as

φ1(x)?φ2(x)≡ φ1(x)exp( i

2γij←−∂ i−→∂ j

)φ2(x). (2)

Using the above definition, one can easily show that [11]

eik1 ·x ?eik2 ·x = ei(k1+k2)·xe−ik1∧k2 , (3)

andeik1 ·x ?eik2 ·x ?eik3 ·x = ei(k1+k2+k3)·xe−ik1∧k2e−ik1∧k3e−ik2∧k3 , (4)

where k∧k′ = 12γ

ijkik′j .

In this note, we will derive the radiation cross section in a non-relativistic bremsstrahlungin noncommutative space. The correction on the radiation cross section, due to non-commutativity of space, is calculated to second order in γ. In noncommutative space,we find an increase in the intensity cross section. In this work, we will use the naturalunits, i.e. ~ = c= 1.

RJP 59(Nos. 5-6), 500–503 (2014) (c) 2014-2014Rom. Journ. Phys., Vol. 59, Nos. 5-6, P. 500–503, Bucharest, 2014

2 Bremsstrahlung in Noncommutative Quantum Mechanics 501

2. NON-RELATIVISTIC BREMSSTRAHLUNG

Let us consider an incoming particle with momentum p = mv and charge e,scattered by a static source with charge −Ze. The bremsstrahlung cross section isgiven by [12-14]

dσp′ =4e2

3ωv|〈p′|x|p〉|2 d

3p′

(2π)3=

4mp′e2

3ωv|〈p′|x|p〉|2

dωdΩp′

(2π)3. (5)

Here |p〉 and |p′〉 denote the exact wave functions of the incoming and outgoingparticles with momentum p and p′, respectively. The outgoing particle has the mo-mentum p′ = mv′ and the energy of emitted photon is ω = p2

2m −p′2

2m . By engagingthe classical equation of motion

mx = Ze2∇ 1

|x|, (6)

(5) takes the form

dσp′ =4mp′e2

3ωv

(Ze2m

)2|〈p′|∇ 1

|x||p〉|2

dωdΩp′

(2π)3. (7)

In Born approximation, one replaces the exact wave function with the plane wave i.e.φp ≈ eip·x. Thus, by taking into account

〈p′|∇ 1

|x||p〉=

∫d3xe−ip

′·x∇ 1

|x|eip·x = 4πi

p′−p|p′−p|2

, (8)

and integrating over the solid angle dΩp′ = dφdθ sinθ one arrives at

dσω =16

3

Z2e6

mv2log(v+v′

v−v′)dωω, (9)

where θ denotes the angle between the incoming and outgoing particles. The inten-sity of emitted photon is dI = ωdσω. So, for a soft bremsstrahlung, i.e. ω 1 andp′ ≈ p, the intensity per frequency is found to be [14]

dI

dω=

16

3

Z2e6

mv2log(2mv2

ω

). (10)

3. BREMSSTRAHLUNG IN NONCOMMUTATIVE SPACE

Now, we consider the bremsstrahlung cross section in noncommutative space:

dσp′ =4mp′e2

3ωv

(Ze22m

)2|〈p′|∇ 1

|x||p〉?|2

dωdΩp′

(2π)3, (11)

RJP 59(Nos. 5-6), 500–503 (2014) (c) 2014-2014

502 Sevda Heydarnezhad 3

where the usual product is replaced by the Moyal product, namely

〈p′|∇ 1

|x||p〉? =

∫d3xe−ip

′·x ?∇( 1

|x|

)?eip·x

= 4πip−p′

|p−p′|2eip∧p′

.

(12)

Also, we have invoked

∇ 1

|x|= i

∫d3k

(2π)3keik·x

|k|2. (13)

Now, we expand the exponent in the second line of (12) to first order in noncommu-tativity parameter

eip∧p′ ≈ 1 +i

4γpp′ sinθ. (14)

Thus, one arrives at

|〈p′|∇ 1

|x||p〉?|2 ≈ |〈p′|∇

1

|x||p〉|2 +

π2p2p′2 sin2 θ

|p−p′|2γ2. (15)

On substituting the above expression in (11) and using∫dΩp′

p2p′2 sin2 θ

|p−p′|2= π(p2 +p′2)− 2πm2ω2

pp′ln

(p+p′)2

2mω, (16)

and noting that for a soft bremsstrahlung p' p′, we obtain the emission intensity perfrequency in noncommutative space to second order in γ as

dIθdω

=dI

dω+γ2

dI ′

dω

=16

3

Z2e6

mv2log(2mv2

ω

)+Z2e6

3

[m2v2

ω− ω

v2log(2mv2

ω

)]γ2.

(17)

Therefore, we observe that the intensity in usual space, i.e. equation (10), is correctedby the term arising from the noncommutativity of space implying an increase in theintensity cross section. In γ → 0 limit, i.e. when the noncommutativity of spacedisappears, equation (17) reduces to (10) and one recovers the standard textbookresult.

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