Transmission coefficient limit

[TheCanadian]

I've attached the equation for the transmission coefficient of a particle going through a potential barrier and E < V. I was simply wondering in the limit V --> E, why does T --> 0 (i.e. the V-E term --> 0 and thus the denominator would approach infinity, making T --> 0)? Shouldn't it be approaching 1?

 

[Samy_A]

I've attached the equation for the transmission coefficient of a particle going through a potential barrier and E < V. I was simply wondering in the limit V --> E, why does T --> 0 (i.e. the V-E term --> 0 and thus the denominator would approach infinity, making T --> 0)? Shouldn't it be approaching 1?

There is also the $sinh²(k_1 a)$ to take into account, since $V_0 \rightarrow E$ implies $k_1 \rightarrow 0$.

Purely mathematically, you will get as limit $1+\frac{mEa²}{2 {\hbar}^2}$ in the denominator.
(Physically maybe $E \rightarrow V_0$ makes more sense, so in the limit, replace $E$ by $V_0$.)