Barrier Tunneling — Question 4

Question 4

Mass changes tunneling exponentially even at the same incident kinetic energy

Two species have masses m and $2m$. They encounter the same rectangular barrier with the same energy $0<E<V_0$ and width a. Both are in the opaque regime. The lighter species has $\kappa_m a=4.00$. Use $T\simeq[16E(V_0-E)/V_0^2]e^{-2\kappa a}$.

1. Derive the ratio of decay constants and transmissions.

2. Calculate $T_{2m}/T_m$ and the inverse enhancement factor.

Question 4 - Solution

Mass changes tunneling exponentially even at the same incident kinetic energy

(a) Method and calculation

$$\kappa=\sqrt{2m(V_0-E)}/\hbar\quad\Rightarrow\quad\boxed{\kappa_{2m}=\sqrt2\,\kappa_m}.$$$$\boxed{\frac{T_{2m}}{T_m}\simeq\exp[-2\kappa_m a(\sqrt2-1)]}.$$

(b) Interpretation and further calculation

$$\boxed{T_{2m}/T_m=e^{-8(\sqrt2-1)}=.0363810}.$$$$\boxed{T_m/T_{2m}=27.4869}.$$

Physical check

At fixed energy and barrier height the prefactor is mass-independent in this approximation. The strong mass dependence comes from the exponential action.