Question #c4601

No, they don't have to, and they generally don't.

If they did, then we would have to wait an infinite amount of time for tunnelling to occur. We should expect a distribution of energies in the ensemble of particles that tunnel, as long as those particles follow the Heisenberg Uncertainty Principle.

Quantum tunnelling of electrons is a consequence of the two versions of the Heisenberg Uncertainty Principle:

#DeltaxDeltap_x >= ℏ/2# #" "bb((1))#,

#DeltaEDeltat >= ℏ/2# #" "bb((2))#,

where #Deltax# is uncertainty in the #x# position, #Deltap_x# is uncertainty in the #x# component of momentum, and #ℏ = h/(2pi)# is the reduced Planck's constant.

#DeltaE# is the uncertainty in the particle energy, and #Deltat# is the spread of time in which the particle is observed.

Conceptually, we know that if the uncertainty in one variable is high, the other is low, and vice versa.

Consider the following cases.

#"Case I"#

• If its position is known to 100% certainty, such as if we knew the electron did not tunnel yet (or did!), then from #(1)#, its momentum is known with infinite uncertainty and thus it has a large energy spread.

That also means from #(2)# that this situation only exists for a short amount of time. In other words, there is an instantaneous spike in the energy for a distribution of electrons, and in that instant, the tunnelling occurs.

Thus, we can't force multiple electrons to have a few discrete energies at the same time, but we can have a set of electrons with a distribution of energies.

#"Case II"#

• If its momentum is known to 100% certainty, then from #(1)#, the uncertainty in the position is infinite.

This situation doesn't really happen that often, since knowing a precise momentum indicates a single precise energy observed, and that indicates (from #(2)#) a large time spread, i.e. we'd have to wait a long time for this to occur.

In Case #"I"#, we found that a large energy spread corresponds to well-known positions.

If we know that tunneling occurred, it implies that the electrons that tunneled have a spread of energies, i.e. they don't all have a single, same discrete energy.