Introduction: How Much Gets Through?
The transmission coefficient \(T\) gives the probability of tunneling through a barrier. It depends exponentially on the barrier width and height.
For a Rectangular Barrier
Exact formula (for \(E < V_0\)):
\[T = \frac{1}{1 + \frac{V_0^2\sinh^2(\kappa a)}{4E(V_0 - E)}}\]For thick barriers (\(\kappa a \gg 1\)):
\[T \approx 16\frac{E(V_0 - E)}{V_0^2}e^{-2\kappa a}\]The Exponential Dependence
The key factor is \(e^{-2\kappa a}\) where:
\[\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}\]Doubling the barrier width squares the suppression factor!
Worked Example
Electron (\(m = 9.1 \times 10^{-31}\) kg) with \(E = 1\) eV tunneling through \(V_0 = 2\) eV barrier of width \(a = 1\) nm:
\[\kappa = \frac{\sqrt{2 \times 9.1 \times 10^{-31} \times 1.6 \times 10^{-19}}}{\hbar} \approx 5.1 \times 10^9 \text{ m}^{-1}\] \[T \approx e^{-2 \times 5.1 \times 10^9 \times 10^{-9}} = e^{-10.2} \approx 4 \times 10^{-5}\]The Quantum Connection
The exponential dependence makes tunneling exquisitely sensitive to barrier parameters. Alpha decay half-lives vary from microseconds to billions of years based on tiny differences in nuclear barriers. The STM relies on tunneling current varying exponentially with tip-sample distance.