# Wave Function for a Step Potential

If we sole the TISWE, and if energy or the particle lies between $$0. If we do the calculation, Transmission coefficient $$(T)$$ comes out to be zero. I get that part, but why then there exist a transmission probability of finding the particle if $$T=0$$. What does that signify.

Also if we compare it with the expression of $$J=\rho v$$, (here in this case, $$\rho$$ is our $$|\psi^2|$$ and $$v$$ is the velocity of particle) if $$|\psi^2|$$ is not zero, implies $$v=0$$, thus current density is zero. What does it means if velocity of particle being zero in classical forbidden region and current density thus finally becoming $$0$$.

This may become even clearer, if one calculates the current using the correct expression for the probability current: $$\mathbf{j}=\frac{-i\hbar}{2m}\left(\psi^*\nabla\psi - \psi\nabla\psi^*\right)$$
Remark: Expression $$j=\rho v$$ seems to me borrowed from classical physics, and applied here without justification - this expression is grounded in classical intuition and is not applicable without appropriate averaging.