This is what happens when a quantum wave-function hits a wall that it does not have enough energy to cross. Part of it goes through anyway. This is the famous "tunnel-effect". A particle without quantum properties cannot do that.

The tunnel effect is real and has been measured many times. It plays a big role in many chemical reactions and even in biochemistry. It's also what an electron tunnel microscope works with.

A peculiar aspect of the tunnel-effect is that it seems to break the speed of light limit. This is because the peak of the wave-function which goes through (which could be photons) actually moves faster than the peak of the wave-function would have moved without the wall.

You can see this here, on top is the case with the wall, at the bottom the case without. See how the peak of the part that goes through seems to move faster?

The standard explanation for this is that it's not correct that anything moves faster than light, it's that some photons at the front of the wave-packet made it through. Fact is, one can't send information faster than light with it, sorry.

Those animations are not drawings, they're actually visualization of solutions of the Schrödinger equation with a Gaussian wave-packet hitting a square potential.

Here's another illustration of this phenomenon, demonstrating the wave function's evolution in phase space using the Wigner function.

It’s also critical to the rate of fusion in the core of a star.

Yes, that's right, should have mentioned this!

what is a "wall" in this case?

it's a potential barrier