If you are up to a trick question, consider the following. There are no forces inside the pipe, so the particle has to keep moving until it hits an end of the pipe, then reflect backward until it hits the other side and so on. So, it has to cross the center of the pipe regularly. But in the energy eigenstate
Almost every word in the above story is a gross misstatement of what nature really is like when examined on quantum scales. A particle does not have a position, so phrases like
hits an end”, are truly meaningless. On macroscopic scales a particle may have an relatively precisely defined position, but that is only because there is uncertainty in energy. If you could bring a macroscopic particle truly into a single energy eigenstate, it too would have no position. And the smallest thing you might do to figure out where it is would kick it out of that single energy state.
reflect backward, and “keep moving