Barrier Tunneling — Question 2

Question 2

The opaque-barrier approximation has a quantifiable relative error

For a rectangular barrier at $E=V_0/2$, the exact formula reduces to $T=[1+\sinh^2z]^{-1}$, with $z=\kappa a>0$.

1. Rewrite T in exponentials and derive its leading approximation at large z.

2. Find the relative overestimate of that approximation at $z=3$.

Question 2 - Solution

The opaque-barrier approximation has a quantifiable relative error

(a) Method and calculation

$$T=\operatorname{sech}^2z=\frac{4e^{-2z}}{(1+e^{-2z})^2}.$$$$z\gg1:\quad\boxed{T_{\mathrm{op}}=4e^{-2z}}.$$

(b) Interpretation and further calculation

$$\boxed{\frac{T_{\mathrm{op}}-T}{T}=2e^{-2z}+e^{-4z}}.$$$$z=3:\quad\boxed{T=.00986604,\quad T_{\mathrm{op}}=.00991501,\quad\text{relative error}=.00496365}.$$

Physical check

The exponential alone omits a boundary prefactor of four in this special case. The leading opaque expression can exceed one if used outside its valid regime.