content_image/342074.jpg content_image/342075.jpg content_image/342072.jpg content_image/342073.jpg content_image/342070.jpg content_image/342071.jpg content_image/342068.jpg content_image/342069.jpg
current such terms simply drop out as in the constant mass case. The probability current density is given by: 

which can be also given in terms of the auxiliary functions φ e χ : 

<img src='content_image/342074.jpg'>

<img src='content_image/342075.jpg'>

With the asymptotic form of the wave function we can compute the left ( x < 0) J$_{L}$ , and the right ( x > 0) J$_{R}$ current density: 

<img src='content_image/342072.jpg'>

and we can define the transmission and reflection coefficients: 

<img src='content_image/342073.jpg'>

<img src='content_image/342070.jpg'>

<img src='content_image/342071.jpg'>

and from the current conservation ∂$_{x}$J ( x ) = 0 it follows that ∫ + ∞ −∞ ∂$_{x}$J ( x ) = [ J ( x )] + ∞ −∞ = J$_{R}$ − J$_{L}$ = 0, and therefore J$_{L}$ = J$_{R}$ from which we have the unitarity condition R + T = 1. The transmission and reflection coefficients can then be written as: 

<img src='content_image/342068.jpg'>

<img src='content_image/342069.jpg'>

Fig. 2 shows the transmission coefficient in the constant mass case (left plots) and for the position dependent mass case (right plot) for two choices of the parameters ( a,L ) and in the so called Klein range m < E < W − m . We note that in the PDM case we still observe the transmission resonances found for constant mass. We observe that, while for m = m$_{0}$ when E → W − m , T → 0, in the PDM case T → 1. This is an important fact wortwhile to be pointed out. In Fig. 3 we plot the transmission coefficient as a function of the barrier height W. We note that as opposed to the case of constant mass where T = 0 for E − m < W < E + m , in the position dependent mass case, T ( W ) always oscillates and does not go to zero in the interval E − m < W < E + m . We note also that we have verified our numerical calculations of the constant mass case with those of ref. [24] finding complete agreement. Finally, we have numerically checked the validity of the unitarity condition R + T = 1. 

