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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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  • $\begingroup$ You're looking at a stationary state, so you can't say the particle is "heading toward" or "tunneling" or doing anything. Once you make time-dependent wavepackets, you find that the particle 'bleeds into' the barrier a little when it hits it but then completely reflects off, so there's no tunneling. $\endgroup$
    – knzhou
    Jun 24, 2016 at 21:56
  • $\begingroup$ When there's genuine tunneling, the resulting wavepacket will have a component on the other side of the barrier for $t \to \infty$. $\endgroup$
    – knzhou
    Jun 24, 2016 at 21:56
  • $\begingroup$ So even a particle 'bleeding' even just a little does not count as tunnelling? I think what I have read was about formulating a plane wave. How do I construct the wavepackets then? $\endgroup$ Jun 24, 2016 at 22:01

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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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