The tunnel barrier: Discussion

For energies of the particle $E<V$, we have a finite transmission coefficient $T$ that increases with $E$. In particular, the wave function below the barrier, i.e. in the interval $[-a,a]$, is non zero which means that there is a finite probability to find the particle below the barrier. This is a very important quantum mechanical phenomenon called the tunnel effect. Classically, a particle can not be in areas where the potential energy $V$ is larger than its total energy $E$.

For energies of the particle $E>V$, the transmission coefficient oscillated as a function of energy $E$. At particular values of $E$, the $\sin(2k_2a)$ in (2.67) vanishes, and $T$ exactly becomes unity. These peaks in $T$ are called transmission resonances. The condition for the resonance energies is

$$\sin(2k_2a)=0\leadsto 2k_2a=n\pi\leadsto k_2 = \frac{n\pi}{2a},\quad E_n= \frac{n^2\pi^2\hbar^2}{2m(2a)^2} + V.$$ (146)

We recognize that the energies $E_n$ are just the energy eigenvalues of the infinite potential well of width $2a$, shifted by the height $V$ of the potential!