Through my classwork I have learned that an electron will not lose energy when it is tunneling. However, I am having a hard time understanding why.
From what I understand, as soon as the electron hits the barrier its wave function will be $\Psi(x)=Ae^{-\alpha x}$ where $\alpha = \sqrt{\frac{2m(U_0 -E)}{\hbar}}$. As the electron tunnels, I know its amplitude decreases and $x$ will increase as the particle moves past the barrier. Is this somehow related to the electron not losing energy? Or is there a different reason why?