Tunneling: wave function in different regions

Quantum tunneling is the passing of a particle through a potential barrier whose height $V_0$ is greater than the particle's energy $E$. In classical mechanics this is impossible, but in quantum mechanics the wave function does not become zero inside the barrier. It decays, so a small part of it comes out on the other side and there is a finite chance of finding the particle there.

Barrier and regions

 V(x)
  ^
  |            +----------+   V = V0
  |            |          |
  |  - - - - - |- - - - - |- - - - -  E < V0
  |  Region I  | Region II| Region III
  |  V = 0     | V = V0   | V = 0
--+------------+----------+-----------> x
               0          a

The time-independent Schrödinger equation is

$$\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E - V)\psi = 0$$

Region I ($x < 0$, $V = 0$)

$$\psi_1 = A e^{jkx} + B e^{-jkx}, \qquad k = \frac{\sqrt{2mE}}{\hbar}$$

$A e^{jkx}$ is the incident wave and $B e^{-jkx}$ the reflected wave. The wave function is oscillatory.

Region II ($0 \le x \le a$, $V = V_0 > E$)

$$\frac{d^2\psi_2}{dx^2} - \alpha^2\psi_2 = 0, \qquad \alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$$ $$\psi_2 = C e^{\alpha x} + D e^{-\alpha x}$$

The solution is exponential, not oscillatory. For a wide barrier the decaying term dominates, so $\psi$ falls off roughly as $e^{-\alpha x}$ inside the barrier.

Region III ($x > a$, $V = 0$)

$$\psi_3 = F e^{jkx}$$

Only a transmitted wave travels to the right (there is nothing to reflect it back). It is oscillatory with the same $k$ (same energy) but a much smaller amplitude.

Boundary conditions

$\psi$ and $d\psi/dx$ must be continuous at $x = 0$ and $x = a$. These four conditions fix $B, C, D, F$ in terms of $A$.

Transmission probability

$$T = \frac{|F|^2}{|A|^2} \approx T_0\, e^{-2\alpha a}, \qquad T_0 = 16\frac{E}{V_0}\left(1 - \frac{E}{V_0}\right)$$

(valid when $\alpha a \gg 1$). $T$ falls very quickly as the barrier width $a$, the mass $m$ or $(V_0 - E)$ increases.

 psi(x)
     Region I       Region II     Region III
  /\    /\    /\ |\            |
 /  \  /  \  /  \| \           |
/    \/    \/    |  \__        |/\/\/\/\/\/\
                 |     ``--..__|  (small amp.)
                 0             a

Examples

Alpha decay of nuclei, the tunnel diode, the scanning tunneling microscope (STM), and conduction through thin oxide layers between metal contacts all rely on tunneling.