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.

Roger Vadim · Accepted Answer

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:
$$
mathbf{j}=frac{-ihbar}{2m}left(psi^*nablapsi - psinablapsi^*right)
$$
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.

Wave Function for a Step Potential

One Answer

Add your own answers!

Ask a Question