Quantum Tunnelling: Where Does Energy Go?

[mrcotton]

Homework Statement

The waves amplitude appears to decrease as it tunnels through the barrier.
If we think of the amplitude as energy like in a sound wave etc, where does this energy go?

The Attempt at a Solution

I think it has something to do with the fact that for a matter wave the amplitude has no interpretation and that it amplitude squared is the probability of finding it in a location. So this will decrease through the barrier until it exits on the other side at that point it represents the probabilty of it being at that point?

Incidentally the electron does not move through the barrier?

Thanks for any clarification or further mystery

 

[BruceW]

yeah, the amplitude of the wave is only telling us about the probability of finding the particle in that position. But you are also correct that we can interpret this amplitude as related to the energy of a beam of particles. Think about the case where we have a bunch of similar, but non-interacting particles, all with this same wavefunction. Why will we expect a detector to receive more energy in places with a large amplitude for the wavefunction?

 

[mrcotton]

would this be because if there is a higher probability of finding more particles in a particular location then there must be more energy concentrated there. Would this be there kinetic energy detected by the detector?

Thanks for responding

 

[BruceW]

yeah, that's it. It just means that on average, there is going to be a lot more particles there, so if there are N particles, each with energy E, then the detector will pick up a total energy of N*E (well, the detector might not necessarily pick up all the energy of each particle, but this does not change the main idea of the argument).

 

[Nickie]

I can provide some insight into the concept of quantum tunnelling and where the energy goes in this phenomenon. Quantum tunnelling is a quantum mechanical process in which a particle, such as an electron, can pass through a potential barrier even though it does not have enough energy to overcome the barrier. This is due to the probabilistic nature of quantum mechanics, where particles can exist in multiple states at the same time.

In the case of quantum tunnelling, the particle's wave function extends beyond the barrier, allowing it to "tunnel" through the barrier even though it does not have enough energy to overcome it. The decrease in amplitude of the wave function as it passes through the barrier can be explained by the fact that the wave function represents the probability of finding the particle at a particular location. As the particle tunnels through the barrier, the probability of finding it in that location decreases, hence the decrease in wave amplitude.

Now, as for the question of where the energy goes, it is important to note that energy is a conserved quantity in quantum mechanics. This means that the total energy of the system remains constant, even during quantum tunnelling. The energy of the particle is not lost or dissipated, but rather it is redistributed throughout the system. This can be seen by considering the potential energy of the particle before and after it tunnels through the barrier. Before tunnelling, the particle has a certain potential energy due to the barrier. After tunnelling, the particle has a lower potential energy due to being on the other side of the barrier. This decrease in potential energy is balanced by an increase in kinetic energy of the particle, allowing it to continue moving through the barrier.

In conclusion, the energy of a particle undergoing quantum tunnelling is not lost or dissipated, but rather it is redistributed throughout the system. This phenomenon is a result of the probabilistic nature of quantum mechanics and is a fascinating concept in the study of subatomic particles.