Wave Function for a Step Potential

Question

If we sole the TISWE, and if energy or the particle lies between $0<E<V$. 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 $0$.

Answer 1

The transmission coefficient is zero, because the wave within the barrier is not propagating, even though the probability of finding particle within the barriers is non-zero.

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.