According to DeBroglie's model of an atom, an electron wave travels as a standing wave in orbit around the nucleus of an atom. My question is, do DeBroglie electron waves behave as standing waves when they are free electrons as well or only when they orbit an atom's nucleus? For example, an electron gun that fires single electrons through a slit. Would these electrons behave as standing waves?
> According to DeBroglie's model of an atom
First of all, DeBroglie did not have a model of the atom. I think you're confusing him with Neils Bohr, who published his concept for the atom in 1913.
DeBroglie published his ideas on electrons having a wavelength equal to h/p in 1924. Sometime after that, it was noted (strangely enough, I've never been able to find out when this was noted, or who noted it) that the circumference of a circle with the Bohr Radius (.529 A) was equal to the DeBroglie Wavelength of an electron with the momentum ('p') that Bohr stated it would have. It was also noted that doubling the momentum -- thus allowing a standing DeBroglie Wave twice the Bohr Circumference -- was the same as having an electron at one state above ground state. Some thought this was a better explanation of a higher energy state than saying the electron was farther from the nucleus than the Bohr Radius.
> do DeBroglie electron waves behave as standing waves when they are free electrons as well
> or only when they orbit an atom's nucleus?
The problem with all this discussion of standing DeBroglie Waves for an electron in orbit around a nucleus is that the entire idea is WRONG -- and we've known this for over 75 years. The (relatively) simplistic ideas of Bohr
have been completely superseded by the quantum mechanics of Schroendinger, Heisenberg, and Born -- with a lot of help from Bohr himself. We now know that it is quite misleading -- if not just outright wrong -- to speak of an electron orbiting a nucleus at a specific radius. We now state that the electron is at a certain energy level and, when it is at that level, it has a certain probability of being at a certain place. If the electron is at its ground state, it is MOST LIKELY to be one Bohr Radius from the nucleus, but it is WRONG to say it IS at that distance. Thus, any discussion of DeBroglie Waves for electrons in an atom is meaningless, other than in a historical setting.
Nevertheless, it is USEFUL to speak of the DeBroglie Wavelength when discussing free electrons. The reason is that it becomes very easy to understand electron interference patterns as no different than the interference patterns we find in light, just with a different wavelength. This approach is also too simplistic for the reality of electron scattering, but it is also much easier to comprehend.
So, in answer to your question, DeBroglie Waves are MORE useful, and less misleading, when discussing a free electron than when discussing an electron around a nucleus. However, it is BEST if we don't discuss them at all.