This shows how the wavefunction, behaves when meeting a barrier. The wave function is the solution of the appropriate quantum mechanical differential equation , with boundary conditions (the barrier).
Note that the energy of the tunneling particle is constant. What varies is the wavefunction whose complex conjugate squared gives the probability of finding the particle at a particular x in this one dimensional example.
Superposition of wave functions, i.e. addition of different wave functions can form interference patterns which, when the summed function is complex congugate squared will show the wave nature of the solutions in interference patterns, i.e. the probability of the particle appearing at x,y,z. This MIT video shows the interference appearing after the superposition of two coherent laser beams.
and The Uncertainty Principle
Here is a definition:
The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. There is a minimum for the product of the uncertainties of these two measurements. There is likewise a minimum for the product of the uncertainties of the energy and time.
This is not a statement about the inaccuracy of measurement instruments, nor a reflection on the quality of experimental methods; it arises from the wave properties inherent in the quantum mechanical description of nature. Even with perfect instruments and technique, the uncertainty is inherent in the nature of things.
I hope the above is an adequate description of the Heiseberg uncertainty. Further in the link, it shows the connection with the wavefunction solutions
and explanations of how tunneling is based on the fact that due to certainty in position particles can therefore gain enormous amounts of momentum due to its uncertainty, which allow it to overcome the barrier.
The statement in italics is wrong. For tunneling there is no change in "momentum" because there is no change in energy levels. It is based on the Heisneberg uncertainty principle (HUP) only in so far that the uncertainty principle can be derived from the wavefunctions (the second page in the last link). It explains the wavepacket model of a particle.
When the wave packet hits the barrier, a boundary value solution, the top diagram of tunneling holds, and a probability of tunneling will add up from the superposition of each individual wavefunction that mathematically composes the wavepacket .
In conclusion, tunneling depends on probability, not energy and/or momentum, and probability depends on the boundary conditions. The HUP allows it because it is an envelope that can describe the probabilistic particle behavior in general. For details one needs solutions of the appropriate differential equations.