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?
Best Answer
The reason is just conservation of energy. You have to keep two things separate here.
There's the amplitude of the wave function $|\psi(x)|^2$ which tells you how likely it is to find the particle at the position $x$ and there is the energy of $E$ the particle, these two have nothing to do with each other. Since your looking at at solutions of the stationary Schrödinger equation this has to be constant.
For the description of tunneling it is misleading to picture the electron as a particle that hit's the wall and may or may not bounce back.