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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 $0$.

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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.

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  • $\begingroup$ As said, If the wave after say x>0 is not propagating, but we see the probability of finding the particle, i cant just get this thing in my head physically. I know for real wave function current density is always 0, but if you can justify my doubt would be much more clearer. $\endgroup$ May 27, 2021 at 12:00
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    $\begingroup$ If the wave function is real, the momentum is zero - there is no flux, i.e., the wave is not propagating. One could think of it as a stationary wave. $\endgroup$
    – Roger V.
    May 27, 2021 at 12:02
  • $\begingroup$ So doesn't that means particle never crossed the barrier. And wave not propagating means that T=0 means, no wave is present or T=0 means no particle is present. Also how can we correlate to stationary wave. Also Stationary wave concept how are using it here can you explain a bit more. $\endgroup$ May 27, 2021 at 12:09
  • $\begingroup$ To say that particle crossed the barrier, one needs to know its position and momentum - it is not possible for classical particle. If we compare with theelectromagnetic field, then within the barrier we have an evanescent wave - the amplitude is non-zero, but the wave is not propagating (evanescent wave is actually more precise term than stationary wave) $\endgroup$
    – Roger V.
    May 27, 2021 at 12:14

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