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For a Finite Square Well where we have a wavefunction $\psi(x)$ which is an energy eigenfunction with eigenvalue $E = 2V_0$ in the following potential:

$V(x) = \begin{array}{ll} 6V_0 & x< 0 \\ 0 & 0\leq x\leq a \\ V_0 & x\gt a \\ \end{array}$

Since in the region $x<0$ the wavefunction will be in a classically forbidden region, does this imply that $R$, the reflection coefficient must be $1$ (no transmission; $T=0$)?

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Reflection, $R$ and transmission, $T$, probabilities are determined in terms of the current flux. For the case you describe, yes the reflection coefficient will be unity and the transmission coefficient will be zero. However there will still be a non-zero probability of finding the particle in the classically forbidden region, however this decays exponentially.

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