Introduction: The Impossible Becomes Possible
Quantum tunneling is perhaps the most striking quantum effect: a particle can pass through a barrier even when it doesn't have enough energy to go over. Classically forbidden; quantum mechanically probable.
The Barrier Potential
\[V(x) = \begin{cases} 0 & x < 0 \\ V_0 & 0 \leq x \leq a \\ 0 & x > a \end{cases}\]For \(E < V_0\): classically, particle is reflected. Quantum mechanically, there's transmission!
Inside the Barrier
In the classically forbidden region, \(\psi'' = \kappa^2\psi\) where \(\kappa = \sqrt{2m(V_0 - E)}/\hbar\)
Solution: \(\psi = Ce^{-\kappa x} + De^{\kappa x}\)
The wavefunction doesn't oscillate—it decays exponentially.
Physical Picture
The wavefunction "leaks" into the barrier. If the barrier is thin enough, the wavefunction doesn't decay to zero before reaching the other side. The remaining amplitude propagates as a transmitted wave.
The Quantum Connection
Tunneling explains alpha decay (nucleus tunneling through nuclear barrier), scanning tunneling microscopy, fusion in stars, and flash memory. It's not that particles "borrow" energy—they genuinely traverse classically forbidden regions because quantum mechanics isn't classical mechanics.