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In the case of a potential step when $E<V$, the transmission coefficient of the particle is zero. However, there is also an exponentially decaying wavefunction of the particle in that classically forbidden region. The existence of a finite valued wavefunction implies non-zero probability in that region.

So what exactly is the case? Is the probability of finding the particle on the other side of potential step exactly zero (as we see from transmission coefficient) or is it finite non-zero but very small (as is suggested by the non zero wavefunction)?

Here is a similar question which does not answer my query Potential step and tunnelling

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  • $\begingroup$ If the step is infinitely long, how can there be an other side? $\endgroup$ – lalala Oct 19 '19 at 10:04
  • $\begingroup$ but the probability is non zero. that's what is bugging me $\endgroup$ – user185887 Oct 19 '19 at 10:05
  • $\begingroup$ If you like this question you may also enjoy reading this Phys.SE post. $\endgroup$ – Qmechanic Oct 19 '19 at 10:15
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You are correct that for a finite potential the probability will not be zero. Inside the 'forbidden zone' the amplitude of the wave function decays with a length-scale $\xi$ given by $$ \xi^{-1} = \sqrt{2m(V-E)}/\hbar $$ so if $V\gg E$ it is very short and the wave-function decays very fast. At the limit of $V\to\infty$ or if the width of the barrier is very long (as is implied in the title of your question), the probability to find the particle at the other side will be zero. However, for finite barriers the particle can pass through the barrier.

In fact, a very famous phenomenon in physics - the $\alpha$-decay, was explained by Gamow exactly as a process of tunneling through a finite potential. You can read about it here

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