The wavefunction of a free electrons is variously described as a plane wave or a wave packet. I am fairly happy with the wave packet, as it is localised.
But if we change to the electron's rest frame, then there is no direction of motion, and thus the wavefunction must be isotropic, presumably as a Gaussian wavepacket.
Yet the wavepacket supposedly expands over time in real space, permitting it to drop in energy in momentum space.
Does this mean that the free-electron wavefunction is not stable? In its own frame, will it simply expand ad infinitum until it interacts with another particle?
If the electron can continue to expand arbitrarily, then the probability that it retains its original momentum grows vanishingly small, which would deny conservation of momentum for any free particle with a significant path length.