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 : 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: and we can define the transmission and reflection coefficients: 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: 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. 14 T = | 1 − 2 µ | 2 ∣ $_{∣}$M B (1 − 2 µ ) e 2 µaL + N $^{[}$B (2 − 2 µ ) e ( − 2 µ +1) aL − D (2 µ + 1) e − 2 µaL ] ∣ 2 (44a) R = E + k E − k | A [ M (1 − 2 µ ) e 2 µaL + N (2 − 2 µ ) e − $^{aL}$] − C [ N (1 + 2 µ ) e − 2 $^{µaL}$] | 2 | B [ M (1 − 2 µ ) e 2 µaL + N (2 − 2 µ ) e − $^{aL}$] − D [ N (1 + 2 µ ) e − 2 $^{µaL}$] | 2 (44b) T = J$_{trans}$ J$_{inc}$ = | d 2 $_{1}$| | H | 2 (42) R = J$_{refl}$ J$_{inc}$ = ( E + k ) ( E − k ) | G | 2 | H | 2 (43) J$_{L}$ = J$_{inc}$ − J$_{refl}$ = | H | 2 k ( E − k ) m 2 0 − | G | 2 k ( E + k ) m 2 0 , (40) J$_{R}$ = J$_{trans}$ = | d$_{1}$ | 2 k ( E − k ) m 2 0 , (41) J ( x ) = ¯ ψ ( x ) γ$_{x}$ψ ( x ) = i [ u ∗ $_{1}$( x ) u$_{2}$ ( x ) − u ∗ $_{2}$( x ) u$_{1}$ ( x )] J ( x ) = 1 2 $^{[}$| φ ( x ) | 2 − | χ ( x ) | 2 ]