University of Helsinki
11th.November.2023
When thinking about some scientific principle or equation many would imagine Einsteins E=mc2 or Newtons F=ma , but some of you might imagine the mysterious Heisenberg’s uncertainty principle (UP for short). Heisenberg’s famous relation
(1) ∆x∆p>h/2
changed the understanding of what the real world, deep inside, really is. However, it is many times poorly understood and even more often purely presented. What does it really mean that position and momentum cannot be known simultaneously? And more importantly, if we can’t determine position of particle precisely, can we describe its trajectory?
Quantum world is very different from ours. Many concepts can’t be understood through classical point of view. Young Heisenberg in the times when quantum physics was in crib, understood the strange nature of this world and proposed the principle, which in fact protects quantum physics from being inconsistent. In this blog, we will explore his argument and try to answer the question if the trajectories of particles are indeed real.
Heisenberg’s argument in 1927/1929
Heisenberg’s original argument wasn’t in the form we all know today. In fact, the famous equation (1) was shown in the same year by Kennard, but we can’t underestimate the importance of what Heisenberg argued.
Imagine we want to know the position and momentum of the particle simultaneously. This data would provide us with enough initial conditions, that we could predict where the particle will be in the future and what momentum it will have. Now take one electron which is in rest. If we try to measure its position to inaccuracy ∆q, we are in fact shining the γ-ray (photons) on electron. The resolution of this microscope will be proportional to inaccuracy ∆q of our measurement . What this really means is that we shine photon on electron, thanks to which we determine exact position up to order but at the same time we destroy the electron. Compton effects will take place and electron will be scattered. The scattered photon will come back to our eye as light which is the “seeing position of electron”.
It was known to Heisenberg that photon appears both as wave with a wavelength λ, and also as particle with momentum p=h/λ (thanks to de Broglie). This λ is the resolution of microscope, because λ is what we see in microscope. But it is related to
inaccuracy of position ∆q. If we equate the change of electron’s momentum with elastic transfer, then we can donate p as ∆p and our relation becomes,
∆q∆p~h
which is the original Heisenberg’s uncertainty relation. This means that when we more accurately measure the position we will have huge uncertainty in momentum ∆p thanks to kick of momentum from scattering. By observing we destroy the particle (state).
Trajectories
Thanks to this argument, he believed that the concept of position, velocity, or momentum must be redefined in quantum physics. There is no such thing as the exact location of particle or exact velocity with which the particle is moving. Rather there is some probabilistic gaussian distribution which determines how likely it is to be in certain position with certain momentum. This implies that the classical meaning of trajectories as set of points in space which particle takes and moves continuously is irrelevant.
With experiments we might more precisely determine one quantity, (position or momentum) but that would destroy the knowledge about the other. Heisenberg further associates the particles with wavefunctions. In the present the wavefunction formulation is the most widely accepted. Wave-packets form the particles, but they behave according to quantum rules, according to uncertainty principle. So, speaking about trajectories doesn’t really make sense for Heisenberg since we cannot determine trajectory since we can’t know the position and momentum exactly.
It might be the case that we can reconstruct the position and momentum of particles in our experiment. Meaning that uncertainty relation is broken for past. This, however, doesn’t seem like problem since anyway, even with this knowledge we can’t determine the future behavior of trajectory since there is uncertainty in momentum thanks to observation. Uncertainty relation is broken for past, but that is not what it is meant to describe. Therefore, if we don’t know the exact momentum p, which means exact velocity v (p=mv), we don’t know where the particle will be in future exactly, until we measure it. Speaking about trajectory here, doesn’t make sense. We might, but we can’t experimentally verify it, so it is a matter of taste as Heisenberg mentions. We shall think in terms of wave distributions and in that sense, trajectories are not well defined.
In conclusion, we cannot really describe in the classical sense the word “trajectory” of a particle, but nor are we absolutely sure what it really means. As quantum physics evolves, the meaning of what is real and what is not evolves with it and so we might expect that maybe one day we will be able to resolve the problem completely. What we know, however, is that uncertainty relation holds and played crucial role in development of modern physics in past and for years to come.
Michal Bires
S. Aristarhov (2023) , Heisenberg’s Uncertainty Principle and Particle Trajectories, Physics of Particles and Nuclei Vol. 54, No. 5, pp. 984–990.