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q - deformation for quantum group and physical system has been one of remarkable and interesting issue of studies such as conformal quantum mechanics @xcite nuclear and high energy physics@xcite cosmic string and black holes @xcite fractional quantum hall effect @xcite .
applications of q - deformation were emerged in physics and chemistry after introducing q - deformed harmonic oscillator @xcite such as investigation of electronic conductance in disordered metals and doped semiconductors @xcite , analyzing of the phonon spectrum in @xmath0he @xcite , expressing of the oscillatory - rotational spectra of diatomic and multi - atomic molecules @xcite .
basically , q - calculus was established for the first time by jackson @xcite then arik and coon used it .
arik and coon by studied generalized coherent states that are associated with a generalization of the harmonic oscillator commutation relation @xcite .
they utilized @xmath1 = { a^\dag } , \quad \left [ { n , a } \right ] =   - a\end{aligned}\ ] ] where relation between number operator and step operators ig given by @xmath2_q},\end{aligned}\ ] ] where a q - number is defined as @xmath3_q } = \frac{{1 - { q^x}}}{{1 - q}}.\end{aligned}\ ] ]    another q - deformation exists that has been introduced by tsallis @xcite and has a different algebraic structure from jackson s .
for the tsallis s case the q - derivative and q - integral was given by borges @xcite .
recently chung and hassanabadi @xcite used the q - derivative emerging in the non - extensive statistical physics to formulate the q - deformed quantum mechanics .    in
what follows : sec . 2
is devoted to introduction to the kind of q - deformation of quantum mechanics which will be used in the next sections . in sec . 3 , q - deformed version of dirac equation is derived . as first relativistic scattering problem in q - deformed version of relativistic quantum mechanics , scattering from a dirac delta potential
is done in sec .
4 . sec . 5
is devoted to extended form of problem in sec .
4 , scattering problem from a double dirac delta potential . in the last ramsauer - townsend effect
is studied in considered formalism of quantum mechanics .
in this section , we want to introduce postulates of q - deformed quantum mechanics to use for the next sections . in this formalism of quantum mechanics we deal with :    1 .   in this formalism of quantum mechanics like the ordinary one , time - dependent form of schrdinger equation in q - deformed quantum mechanics
is written in form of @xmath4 in which we deal with the operators as @xmath5 where @xmath6 is a positive constant and the wave function @xmath7 .
2 .   inner product of hilbert space in one - dimensional q - deformed quantum mechanics can be written as @xmath8 3 .
expectation value of an operator @xmath9 regarding the wave function @xmath7 is given by @xmath10 also we have hermitian definition for the operator if we get @xmath11    remarking this point is suitable that we have been considering deformation only for coordinate part , then the time part has no deformation .
this point can be checked in the first postulate .    in this formalism of quantum mechanics , commutation relation between coordinate and its momentum should be deformed in form of @xmath12 = i\hbar \left ( { 1 + q \hat x ^2 } \right).\end{aligned}\ ] ] considering operator form of coordinate and momentum @xmath13 we can rewrite eq . in terms of the operators @xmath14 and to obtain time - independent form of shcrdinger equation in this formalism , we set @xmath15 then we have @xmath16    using , we can easily find continuity relation in this formalism of quantum mechanics as @xmath17 where @xmath18    by these considerations , we are in a position to study relativistic scattering of fermions in q - deformed relativistic quantum mechanics .
in this section , we want to study scattering of fermions in q - deformed formalism of relativistic quantum mechanics .
study of fermions can be done by dirac equation .
this can be written as @xmath19
@xcite @xmath20 in which the matrices are @xmath21 where @xmath22 stands for pauli matrices .
we have considered @xmath23 direction as interaction direction . to obtain stationary states
, we choose the wave function as @xmath24 also , we d like to consider @xmath25 for simplicity .
these assumptions give us a system of equation like @xmath26 from eq .
we find that @xmath27 if one substitutes eq . into eq .
, easily can derive @xmath28    in the next sections , we will study three important and famous scattering .
as first scattering study , we want to consider single dirac delta potential as @xmath29 where @xmath30 and @xmath31 are real constants .
this point can be derived that this potential produces a discontinuity for first derivative of wave function as @xmath32 we assume that particles come from @xmath33 then because of dirac delta existence they scatter . consequently some them reflected to region i @xmath34 and the other transmit to region ii @xmath35 . according to this assumption
, we can find wave functions of the regions as @xmath36 the coefficients @xmath37 and @xmath38 can be determined by using boundary condition of continuity and discontinuity of wave functions at @xmath39 .
these are @xmath40 where by solving them , we can find that @xmath41    on the other hand , current density of fermions can be derived by @xmath42 . because there is no sink or source , we have from current density @xmath43 we plot this equation considering @xmath44 and @xmath45 in figure [ fig:1 ] .    , @xmath46 and @xmath47 as energy varies . ]    for further study about effects of different parameters on the reflection and transmission coefficients , we have plot figure [ fig:2 ] in which readers can see that how different parameters in our system can affect of these confinements     and @xmath48 as parameters @xmath49 and @xmath30 vary , have been plotted .
we set the parameters in ( a ) and ( b ) @xmath50 , @xmath51 , ( c ) and ( d ) @xmath50 , @xmath52 , ( e ) and ( f ) @xmath51 , @xmath52 . ]    as it can be seen that there no fluctuations in the figures and each curve has a smooth treatment .
but in the next section we will deal with interesting . because kind of scatter potential makes interesting results resemble one of most important and famous effects in physics .
in this section , we suppose that particle are scattered from a double dirac delta potential in form of @xmath53 where @xmath54 and @xmath55 are real constants . from the previous section we know that this kind of potential makes discontinuity for derivative of wave functions at @xmath56 and @xmath57 .
they are @xmath58 similar the previous section , we suppose that particles come from region i @xmath59
. then they are scattered in region ii @xmath60 .
so some of them will be reflected into region i and the other will be transmitted into region iii @xmath61 . according to this assumption , we have wave functions @xmath62 where the coefficients are constant which can be determined from continuity and discontinuity condition at @xmath56 and @xmath57 . using these conditions
lead to the system of four equations    @xmath63    solving this system of equations for the constants of wave functions we obtain @xmath64    using the similar manner of previous section , we have found the constraint @xmath65 by plotting this equation using eq .
we can check validity of it .
this point can be seen in figure [ fig:3 ]    , @xmath66 and @xmath67 as energy varies . ]
it is instructive if we check treatments of reflection and transmission coefficients as different parameters in vary .
this one is done and plotted in figure [ fig:4 ] .     and @xmath68 as parameters @xmath69 and @xmath54 vary , have been plotted .
we set the parameters in ( a ) and ( b ) @xmath70 , @xmath51 , ( c ) and ( d ) @xmath70 , @xmath71 , ( e ) and ( f ) @xmath51 , @xmath71 . ]    the main difference between this section and the previous section is shown in the results . in this section by considering a double dirac potential as scatter potential , we find out some fluctuation in reflection and transmission coefficients , however in the previous section we dealt with a smooth treatment . on the other hand , such fluctuations
remember us one of most famous and important effect in physics , ramsauer - townsend effect . in the next section we will investigate this effect in q - deformed relativistic version .
ramsauer - townsend effect is a face of electron scattering .
this scattering due to a simple potential well .
actually when as electron which is moving through noble gas such as xenon with low energy ( like 0.1 @xmath72 ) something strange happens .
there an anomalously large transmission in this scattering@xcite .
importance of this simple scattering is that this effect can only be decried by quantum mechanics .
now , in considered formalism of relativistic quantum mechanics , we want to study . considering a potential well such @xmath73 and following previous assumptions , we can derived wave functions of our three regions in the problem as following @xmath74 in which the coefficients can be given by using continuity conditions of wave functions and their derivatives at @xmath75 and @xmath76 . using these conditions results in    @xmath77    from this system of equation
we can determine the coefficients as @xmath78 like previous sections , by using definition of current density we can find out that the constraint @xmath79 is governed here . by solving eq . and plotting eq . in figure [ fig:5 ]
we have    , @xmath80 and @xmath81 as energy varies . ]    which is similar to what we faced in the previous sections .
furthermore treatments of reflection and transmission coefficients in terms of different parameters are plotted in figure [ fig:6 ] .     and @xmath82 as parameters @xmath83 and @xmath84 vary , have been plotted .
we set the parameters in ( a ) and ( b ) @xmath85 , @xmath51 , ( c ) and ( d ) @xmath85 , @xmath86 , ( e ) and ( f ) @xmath51 , @xmath86 . ]
in this article , we introduced a q - deformation of quantum mechanics .
then in such formalism of quantum mechanics , we studied three important and famous scattering problems in relativistic region .
we first rewrote dirac equation in q - deformation then as first case , scattering due to a single dirac delta potential was studied . in this case
we dealt with smooth treatments if reflection and transmission coefficients . in the next case ,
a double dirac delta potential was considered . in this case
we saw that there were some fluctuation in reflection and transmission coefficients which were similar to ramsauer - townsend effect . to check this point
, we also investigated this effect in relativistic region . by plotting the coefficients we found out that effect of scattering from a potential well in q - deformed version of relativistic quantum mechanics
could be simulated by considering double dirac delta potential .
_ in this article , after introducing a kind of q - deformation in quantum mechanics , first , q - deformed form of dirac equation in relativistic quantum mechanics is derived . then 
 three important scat erring problem in physics are studied . 
 all results have satisfied what we had expected before . 
 furthermore , effects of all parameters in the problems on the reflection and transmission coefficients are calculated and shown graphically . _    _ * key words * : q - deformed quantum mechanics , dirac equation , dirac delta potential , ramsauer - townsend effect . _    _ * pacs * : 03.65.ta , 03.65.pm , 03.65.nk._