Chapter 40, Problem 63E

(a)

To determine

To show: That the expression $T={\left[1+\frac{{\left(U_0\sinh \kappa L\right)}^2}{4E\left(U_0-E\right)}\right]}^{-1}$ approaches $16\frac{E}{U_0}\left(1-\frac{E}{U_0}\right)e^{-\frac{2\sqrt{2m\left(U_0-E\right)}}{\hslash }L}$ if $\kappa L>>1$.

(b)

To determine

Why the restriction $\kappa L>>1$ implies either that the barrier is relatively wide or that the energy $E$ is relatively low compared to $U_0$.

(c)

To determine

To show: That as the particles incident kinetic energy $E$ approaches the barrier height $U_0$, Then $T$ approaches ${\left[1+{\left(\frac{kL}{2}\right)}^2\right]}^{-1}$.