Barrier Tunneling — Question 9

Question 9

A sloped barrier changes the tunneling exponent through its action integral

Across the forbidden interval $0\le x\le a$, suppose $V(x)-E=\Delta(1-x/a)$ with $\Delta>0$. In a leading-exponent WKB model, $T\sim\exp[-2\int_0^a\kappa(x)dx]$, where $\kappa(x)=\sqrt{2m[V(x)-E]}/\hbar$. Define $\kappa_0=\sqrt{2m\Delta}/\hbar$ and omit all prefactors.

1. Evaluate the action integral and transmission exponent.

2. For $\kappa_0a=6$, compare with a rectangular barrier of the same width and height above E.

Question 9 - Solution

A sloped barrier changes the tunneling exponent through its action integral

(a) Method and calculation

$$\int_0^a\kappa(x)dx=\kappa_0a\int_0^1\sqrt{1-u}\,du=\frac23\kappa_0a.$$$$\boxed{T_{\mathrm{slope}}\sim e^{-4\kappa_0a/3}}.$$

(b) Interpretation and further calculation

$$\boxed{T_{\mathrm{slope}}\sim e^{-8}=3.35463\times10^{-4}},\quad\boxed{T_{\mathrm{rect}}\sim e^{-12}=6.14421\times10^{-6}}.$$$$\boxed{T_{\mathrm{slope}}/T_{\mathrm{rect}}\sim e^4=54.5982}.$$

Physical check

These are leading-exponent estimates, not exact probabilities. The action integrates the full forbidden profile; peak height and width alone do not determine tunneling.