When send in a small opening particle acquire the spread of impulse p. E.g. there can be sometimes a huge value of p, which the particle doesn't have initially. So is with Energy too. The question is where do these surplus energy and momentum come from? I never encountered that in books. I think there is no other place where it can come from except the atoms of the opening. But in that case follows the question what if the opening is cooled down to near -273? Take into account that one can have very slow particles so they don't deliver much energy but must take away. So they will cool down the opening and the whole barrier but not under -273,16.
Subatomic particles are (or are associated with) wave packets (of finite size). Fourier analysis tells us that if the packet is very short, the frequency of the wave cannot be accurately determined. I believe this is the source of the uncertainty principle. I am not aware of any situation where a particle can gain energy by passing through a small opening. It gains energy by interacting with other particles or fields. A particle confined to the volume of a nucleus (by the strong nuclear force) would have at any instant a very uncertain energy.
The Heisenberg principle is related to uncertainties of measurements. The particle is in a QM-state, described by a wave function. If we make some measurement that tells us that it is confined in a very small location, there is a limitation in my max. accuracy of knowledge about its momentum.
For example, it is necessary a electronic microscope to be sure about where things are, for small distances. But that means that electrons collide on whatever is being observed, and the device manages to form an image on the screen from the scattered electrons. Electrons are used instead of photons exactly because they have smaller wave lengths, what also means bigger momentum.
So, we can say that the source of the momentum of the observed particles is the momentum of the incoming ones, that must be greater and greater for smaller accuracy of the location.