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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?

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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.

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  • $\begingroup$ Given the potential energy remains constant, are there any circumstances in which the particle would lose energy? For example, the emission of thermal energy into its surroundings. $\endgroup$ – sausage Apr 25 '17 at 3:35
  • $\begingroup$ There are definitely systems that lose energy, in QM one just has to put a bit more effort in than for e.g. a block that's sliding on a surface and is slowed down by friction. One example I can think of is the interaction of atoms with electromagnetic fields, for simplicity let's take the atom as a 2 level system, one ground state and one excited state. It's an experimental fact, that the atom will eventually decay if you excite it. If you try to describe the system semiclassically you can only have oscillations between the two states (Rabi oscillations). $\endgroup$ – GaragePhys Apr 25 '17 at 6:25
  • $\begingroup$ The way to introduce energy loss (spontaneous emission into this system) goes under the name of Wigner Weisskopf theory, if you want to read up on it. There's a nice discussion in Scully 'Quantum optics' chapter 6. $\endgroup$ – GaragePhys Apr 25 '17 at 6:25

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