Quantum tunneling
Even without enough energy, a particle has a chance to pass through the wall.
According to classical mechanics, an electron whose energy E is below the barrier height V₀ must bounce back. In quantum mechanics, the electron's wave function does not drop to zero at once inside the barrier but decays exponentially; if the barrier is thin enough, a little of the wave remains on the other side, so the electron has a certain probability of getting through. The wider and higher the barrier, the faster the transmission probability falls. Scanning tunneling microscopes, flash memory and α decay all rely on the tunneling effect.
Transmission vs. electron energy (log)
For E < V₀ it rises roughly exponentially; for E > V₀ it oscillates near 1, with full transmission at certain energies (resonance).
Think about it
If the barrier width is doubled, roughly what does the transmission probability become?
Roughly the square of its original value (in order of magnitude). T ≈ e^(−2κa), so doubling a doubles the exponent: if T = 10⁻³ originally, it becomes about 10⁻⁶ after widening. Tunneling is therefore extremely sensitive to the barrier width, and this is exactly what scanning tunneling microscopes exploit to resolve height differences on the atomic scale.
Model assumptionsElectron incident on a one-dimensional square barrier · shown are the real part and |ψ| of the stationary-state wave function ψ(x)·e^(−iωt), with incident amplitude 1 · time runs slowed down on screen
Data log
Set the parameters and click “Record data”. Record several rows with different parameters to plot their relationship and fit a straight line; pause and record at different times to study how a reading changes with time.