According to special relativity if an object is in motion relative to your frame of reference it contracts in the direction of its momentum. Thus, if a wave-particle duality, such as an electron, appeared to be moving from your frame of reference it is my understanding that said particle's wave form would appear contracted from your frame of reference.
This contraction would seemingly result in a particle's location being determinable to a greater degree of certainty with no cost to the degree of certainty with which an observer could determine the particle's momentum which would contradict the Heisenberg uncertainty principle and the limit it applies to the information an observer could collect about said particle.
So, what is the explanation for this? Is the Heisenberg uncertainty principle only applicable to particles at rest relative to the observer? Is Planck's constant relative (does it dilate proportionally with the momentum of said particle)? Or, alternatively, have I misinterpreted either special relativity or the uncertainty principle?
Best Answer
The uncertainty principle arises because the relationship between the position states of a particle-like system1 and the momentum states of that same system is a Fourier transformation.
Even in classical optics or electronics there exist a theorem linking the spread of a signal in the time-domain and in frequency-domain. The uncertainty principle is exactly the same math.
Notable the relationship between the two sets of states being that of a Fourier transform is not dependent on the object having any particular momentum distribution, so the Heisenberg principle is likewise insensitive to the value of a particle momentum.
1 By which I mean a quantum system that will exhibit particle-like properties if you probe it properly.