## Homework Statement

This isn't actually a homework question/problem, but a conceptual problem that i've been having regarding tunneling.
Can someone please tell me what will happen to a particle's energy if a particle tunnels through some potential barrier, given that the particles energy is less than that of the potential barrier. I would assume that the particles kinetic energy would deffinitely decrease, but how could you find by how much it decreases? its not simply (V - Uo), is it? (where V is the particles KE and Uo is the potential of the barrier).

Also, if the potential barrier has lower energy than the energy of the particle, how does the energy of the particle change once it crosses the barrier?

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Nevermind, found an explanation online: The energy of the particle doesn't change after tunneling -- but, the wavefunction amplitude decreases, which makes sense.

Yes, energy is conserved even in quantum mechanics. You're right, the wavefunction amplitude should decrease because the particle is less likely to be found in the region after the tunneling barrier. What about the wavelength of the particle?

The wavelength of the particle should be the same on either side of the barrier since E and lambda are proportional to one another. However, if the particle has more initial energy V than the potential barrier Uo, its energy in the region of the barrier will be (V-Uo), right? Then the wavelength will increase because the particles energy will be decreased,but only in that region. Is that last part right? Thanks for the help!

Right, for the particular case of tunneling the wavelength stays the same because the kinetic energy is the same. However, think about a potential step (where V<E). Since E=K+V then K=E-V, and so if a particle sees a step up or down then the wavelength will change. Are you more likely to see a particle in a certain region if it moves faster through the region or slower? Slower obviously, and so we can see that the kinetic energy is related to the probability, which then relates back to the wavefunction. A big kinetic energy means a small wavelength, and also a small probability.

These same arguments extend into all kinds of situations, which is why I brought it up, so that you can tell what the solution to the Schrödinger equation will look like even before you solve it. For example, you could figure out the stationary states of the harmonic oscillator (a U shaped potential) just by looking at the energies and wavelengths, and how they relate to the stationary state's wavefunction. Knowing how to see the solutions before you start in will either save you calculations altogether, or it will help you check what calculations you do make.

Hope I made things clearer rather than muckier.

thank you so much for helping me out with my question and going beyond. You definitely cleared things up.