Imagine a particle with energy $E$ heading towards a potential step with height $V_0$ where $E<V_0$. The particle's wave function is oscillatory before the boundary and exponentially decaying in and after the boundary. However, there is still a small probability the particle can be found in the 'forbidden' region. Is this quantum tunnelling? I read that for a potential step like this, there is a 100% reflection so I'm getting getting the indication that there is no tunnelling.
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It is quantum tunneling if the step potential has finite length. In this case, the continuity conditions on both sides of the step potential will show you that there is a nonvanishing amplitude behind the potential. If the potential has infinite length, there is no space to tunnel into.
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