In Classical Mechanics we describe the evolution of a particle giving its trajectory. This is quite natural because it seems a particle must be somewhere and must have some state of motion. In Quantum Mechanics, on the other hand, we describe the evolution of a particle with its wave function $\Psi(x,t)$ which is a function such that $|\Psi(x,t)|^2$ is a probability density function for the position random variable.

In that case, solving the equations of the theory instead of giving the trajectory of the particle gives just statistical information about it. Up to there it is fine, these are just mathematical models. The model from Classical Mechanics has been confirmed with experiments in some situations and the Quantum Mechanics model has been confirmed with experiments in situations Classical Mechanics failed.

What is really troubling me is: does the fact that the Quantum Mechanics model has been so amply confirmed implies a particle has no trajectory? I know some people argue that a particle is really nowhere and that observation is what makes it take a stand. But, to be sincere, I don't swallow that idea. It always seemed to me that it just reflects the fact that we don't really know what is going on.

So, Quantum Mechanics implies that a particle has no trajectory whatsoever or particles do have well defined trajectories but the theory is unable to give any more information about then than just probabilities?

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    $\begingroup$ All you will ever get from quantum mechanics is probabilities, that's exactly what makes it different from classical mechanics. $\endgroup$ – user81619 May 26 '15 at 23:03
  • $\begingroup$ @AcidJazz, I know, that is the mathematical model. But does it imply that the particle really doesn't have a trajectory? Or is just a limitation of the model itself? $\endgroup$ – user1620696 May 26 '15 at 23:06
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    $\begingroup$ There are no particles in classical mechanics, either. The particle picture is simply a complete misrepresentation of the fact that one can simplify the equations of motion of extended pieces of matter to the motion of the coordinates of the center of mass in cases where rotational and other internal degrees of freedom do not matter. At some point people went from that simplification to a completely abstract and 100% false image of "particles". $\endgroup$ – CuriousOne May 26 '15 at 23:09
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    $\begingroup$ There are at least eight variations of what quantum mechanics "means" that I know of, and that I spent a summer reading about. All of them had pros and cons but none had a way that you could prove it was correct and the others were wrong. It's an opinion thing, in my opinion :) $\endgroup$ – user81619 May 26 '15 at 23:12
  • $\begingroup$ related: en.wikipedia.org/wiki/Path_integral_formulation $\endgroup$ – Phoenix87 May 26 '15 at 23:22

Quantum systems do not have a position. This is intuitively hard to grasp, but it is fundamental to a proper understanding of quantum mechanics. QM has a position operator that you can apply to the wavefunction to return a number, but the number you get back is randomly distributed with a probability density given by $|\Psi |^2$.

I can't emphasise this enough. What we instinctively think of as a position is an emergent property of quantum systems in the classical limit. Quantum systems do not have a position, so asking for (for example) the position of an electron in an atom is a nonsensical question. Given that there is no position, obviously asking for the evolution of that position with time, i.e. the trajectory, is also nonsensical.

You say:

I don't swallow that idea. It always seemed to me that it just reflects the fact that we don't really know what is going on.

and you are far from alone in this as indeed his Albertness himself would have agreed with you. The idea that we don't know what is going on is generically referred to as a hidden variable theory, however we now have experimental evidence that local hidden variable theories cannot exist.

  • $\begingroup$ I got your point. However please explain those precise trajectories 'particles' draw at the Bubble Chambers in 'Particle Colliders'. May I expect to find some 'particle flashes' outside the main trajectories as QM suggest? I read 'anna v' answer below but I would like to know your comments. $\endgroup$ – fante Oct 21 '15 at 14:55
  • $\begingroup$ @fante: that's quite a long explanation and it really warrants its own question. $\endgroup$ – John Rennie Oct 21 '15 at 15:02
  • $\begingroup$ Or please provide some links where I can get some info about. That would be fine for me :-) $\endgroup$ – fante Oct 21 '15 at 15:06
  • $\begingroup$ Are you sure there is experimental evidence that local hidden variable theories cannot exist? Isn't The actual claim that hidden variables are not consistent with quantum mechanics and both theories cannot be correct at the same time. As far as I know it doesn't say one or the other is wrong. $\endgroup$ – Bill Alsept Jan 12 '17 at 21:29
  • $\begingroup$ +1 you got me with "his Albertness". $\endgroup$ – QuantumBrick Feb 4 '19 at 17:31

So, Quantum Mechanics implies that a particle has no trajectory whatsoever

It depends what "whatsoever" means and what "particle" means and what "trajectory" means. All these words in physics depend on the framework. For distances larger than nanometers and energies larger than some kilo electron volts or so, the classical framework is what defines these words. A particle has a fixed center of mass that given a momentum describes a trajectory according to the classical mechanics theories.

or particles do have well defined trajectories but the theory is unable to give any more information about them than just probabilities?

For distances larger than nanometers and energies larger than a few keV particles have well defined trajectories.( The sizes depend on the Heisenberg Uncertainty Principle and the very small value of h_bar).

Here is an electron trajectory, the electron is a particle


and the width of its trajectory is smaller than a micron. There is no ambiguity to its particleness , and the trajectory can be computed classically, given the magnetic field which is perpendicular to the plane of the photo.

What brings quantum mechanics in by force is if one accumulates a lot of electron-on-proton scatterings and tries to model mathematically what happens when an electron hits a proton . Classical mechanics fails and the theory of quantum mechanics has been very successful in describing the data at the microscopic level of an electron hitting a proton. The result is that the classical trajectory idea falls down in these small distances. One has instead of a particle meeting a particle, a quantum mechanical entity meeting a quantum mechanical entity and their interaction implies that there is a probability distribution controlling what is happening.

So "whatsoever" is defined as "trajectories exist in the macroscopic dimensions, quantum mechanical probabilities reign at the micro system." The classical trajectory emerges smoothly from the underlying quantum mechanical level.

P.S. There do exist macroscopic manifestations of quantum mechanics , like superconductivity , but that is another story.

  • $\begingroup$ Now I get a clue about those 'particle trajectories' challenging the 'wave' nature of them. Thanx a lot!!! $\endgroup$ – fante Oct 20 '15 at 16:10
  • $\begingroup$ This topic is hard to grasp. Thinking about it, do you know what the outcome would be if the 'double slit' experiment is done in a 'bubble chamber'? That would be pretty interesting... $\endgroup$ – fante Oct 21 '15 at 15:42
  • $\begingroup$ @fante it is one of the few questions I asked here. It seems it is very difficult to fulfill the conditions for such an experiment, physics.stackexchange.com/questions/193364/… $\endgroup$ – anna v Oct 21 '15 at 16:11
  • $\begingroup$ Thanx a lot for the comment. I just took a look at the link with your question and now have a stronger curiosity for your second question there: "@PyRulez yes, it would be one path I would bet on it, but would the collection of tracks show the interference pattern or not?". I think that is the billon dollars answer to be solved!!!. Because as far as my limited layman knowledge tell me, "The wave collapse" interpretation followers will say that "It should not be any interference pattern" just 2 peaks in front of each slit. But what if not?. It sounds interesting enough!!! Or not? $\endgroup$ – fante Oct 21 '15 at 19:29
  • $\begingroup$ @fante well, the whole point of asking it was whether seeing the tracks would destroy the interference.From the numbers in the answer I checked I do not think the accuracy is enough in a bubble chamber. $\endgroup$ – anna v Oct 22 '15 at 4:34

Being strict, quantum mechanics doesn't rule out trajectories. In the first place quantum particles can follow classical trajectories under special settings. Anna provided the image of the classical trajectory followed by an electron in a cloud chamber.

Then we have quantum trajectories. Those are radically different from classical trajectories and produce all the 'exotic' behaviors we observer at quantum scales. The next quantum trajectories are plotted for the double slit experiment

Double slit trajectories

The literature on quantum trajectories is very broad, because not only them are used for raising foundational questions about quantum mechanics, but to solve practical problems such as getting rates of chemical reactions, modeling of classical-quantum couplings, and so on. A good introductory book that cites relevant articles is the book by the one of the pioneers on computational quantum trajectory methods

Quantum Dynamics with Trajectories


If you have a conserved probability (such as in nonrelativstic quantum mechanics), then you get a conserved current, the probability current.

In almost all situations it will do for you what you want a velocity to do. Just don't try to get it to do more than you want by expecting it to be too classical. For instance the expectation value of momentum could be zero in an energy eigenstate, even if the probability current is zero everywhere.

Furthermore, the probability current is not unique. If you add the curl of an arbitrary vector field to your probability current it will accomplish just as much. So you don't want to read too much into it.

But it might be what you need if you are looking for something.


Quantum physics has momentum. It's just not something fundamental. Saying that quantum physics doesn't have momentum because it's just the waveform evolving is like saying that an airplane doesn't have wings because it's really just a bunch of atoms.


“In quantum mechanics, due to Heisenberg's uncertainty principle, the notion of a particle trajectory does no longer make sense. QT is probabilistic not deterministic.. ...no definite trajectory.

  • $\begingroup$ Is it due to the HUP, or is the HUP a consequence of the lack of position in QM (as John states)? $\endgroup$ – Kyle Kanos Jan 12 '17 at 21:31

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