Tunneling Through a Barrier

Classically, a particle with energy $E$ cannot enter a region where the potential energy $V_0 > E$ — it simply bounces back. Quantum mechanics is different: because the wavefunction must remain continuous and smooth, a non-zero amplitude penetrates into the barrier and emerges on the other side. This phenomenon is quantum tunneling.

Setting up the rectangular barrier

Consider a particle of mass $m$ and energy $E$ moving in one dimension. For $0 \leq x \leq a$ a constant potential $V_0 \gt E$ is present; outside the barrier $V = 0$ . The time-independent Schrödinger equation in each region reads

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\,\psi = E\,\psi.

Region I ( $x \lt 0$ ): free propagation, so $\psi = A e^{ikx} + B e^{-ikx}$ with $k = \sqrt{2mE}/\hbar$ .

Region II ( $0 \leq x \leq a$ ): the kinetic energy is negative, so instead of oscillating the wavefunction grows and decays exponentially. The solution is $\psi = C e^{\kappa x} + D e^{-\kappa x}$ where

\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}.

Region III ( $x \gt a$ ): free propagation again, $\psi = F e^{ikx}$ .

Matching $\psi$ and $d\psi/dx$ at both boundaries ( $x = 0$ and $x = a$ ) determines the coefficients. The transmission coefficient is $T = |F|^2 / |A|^2$ .

The thick-barrier approximation

Carrying out the full boundary-matching algebra yields an exact expression for $T$ . When $\kappa a \gg 1$ (a barrier that is both wide and tall compared with the de Broglie wavelength inside), the growing exponential term is negligible and the expression simplifies to

T \approx 16 \frac{E}{V_0}\!\left(1 - \frac{E}{V_0}\right) e^{-2\kappa a}.

The prefactor $16(E/V_0)(1-E/V_0)$ is of order 1 for $E$ not too close to $0$ or $V_0$ , so the dominant factor is the exponential suppression $e^{-2\kappa a}$ . The tunneling probability falls off exponentially with barrier width $a$ and with the square root of the energy deficit $V_0 - E$ (because $\kappa \propto \sqrt{V_0 - E}$ ).

A minimal estimate — accurate to within a factor of a few for thick barriers — simply drops the prefactor:

T \approx e^{-2\kappa a}.

A concrete example

Take an electron ( $m = 9.109 \times 10^{-31}\,\text{kg}$ ) with energy $1\,\text{eV}$ below the barrier top, so $V_0 - E = 1\,\text{eV} = 1.602 \times 10^{-19}\,\text{J}$ , and a barrier width $a = 0.5\,\text{nm}$ .

First, compute $\kappa$ :

\kappa = \frac{\sqrt{2 \times 9.109 \times 10^{-31} \times 1.602 \times 10^{-19}}}{1.0546 \times 10^{-34}} \approx 5.12 \times 10^{9}\,\text{m}^{-1}.

Then the exponent is

2\kappa a = 2 \times 5.12 \times 10^{9}\,\text{m}^{-1} \times 0.5 \times 10^{-9}\,\text{m} \approx 5.12,

and the thick-barrier estimate gives

T \approx e^{-5.12} \approx 0.006.

So roughly six electrons in a thousand pass through — a small but physically real probability that underlies technologies from alpha decay to modern semiconductor tunnel junctions.

Try it

This is a numerical exercise — return a number. Using the thick-barrier approximation $T \approx e^{-2\kappa a}$ , compute the tunneling probability for an electron with an energy deficit of $1.0\,\text{eV}$ and a barrier width of $0.5\,\text{nm}$ .

Run your code to see the quantum state.