Introduction: The Wavefunction Penetrates but Nothing Transmits
For \(E < V_0\) at a step, we have \(R = 1\) (total reflection). Yet the wavefunction extends into the forbidden region. How can probability flow into the step but nothing come out?
The Evanescent Wave
In the classically forbidden region:
\[\psi(x) = Ce^{-\kappa x}\]This isn't zero—probability density \(|\psi|^2 = |C|^2e^{-2\kappa x}\) exists inside the step!
Resolution: Probability Current
For a real decaying exponential:
\[j = \frac{\hbar}{m}\text{Im}(\psi^*\psi') = \frac{\hbar}{m}\text{Im}(|C|^2(-\kappa)e^{-2\kappa x}) = 0\]No probability current flows in the evanescent region!
Physical Picture
The particle "samples" the forbidden region but doesn't travel through it. It's like a wave that partly enters a medium where it can't propagate—a standing-wave-like penetration with no transport. The decay length \(1/\kappa\) sets how far the particle can "feel" into the barrier.
The Quantum Connection
This explains why tunneling requires a finite barrier. With a step (infinite forbidden region), the evanescent wave decays to zero before there's anything on the other side. With a finite barrier, the evanescent wave can "reach" the far edge and couple to a propagating wave—that's tunneling.