# What does it mean “not to have a definite trajectory”?

In a comment to my question someone stated the following:

"photons do not travel at some definite number of oscillations per second. In fact, they do not "travel" at all, no more than electrons or other quanta do, as by the Uncertainty Principle they don't have a definite speed and/or trajectory"

Nobody objected or denied it, can someone explain what that actually means?

• Does it mean that they do not have a definite/straight/regular trajectory and they wander erratically or that they have none at all? Can you try to graphically describe their motion ?

It is generally thought that QM describes weird things, laws and phenomena that are quite different from macroscopic world, can you be precise about one feature, please, i.e if it respects the basic old tenet "Natura non facit saltus":

• Does QM allow a particle to disappear from a point and reappear in another point that is not continuous to it? If so, what is the explanation?

A photon is a name given to a lump in an electromagnetic field that can cause a single electron to change from one energy level to another. The size of the lump in a given region tells you the probability that it will make an electron change its energy level. The first thing to note about such a lump is that it doesn't have a single location. Rather, it is spread over an extended region. Now, it can be the case that if you track the evolution of the field over time, a lump in some region R1 can give rise to another lump in some other region R2. But you can't pick a particular point x1 in R1 and say that the field at x1 gave rise to the field at a point x2 in R2. Rather, the lump in R1 gave rise to the lump in R2. If you change the shape of the lump in R1 away from x1 this will in general change the probability of observing something in a sub region of R2 around x2. So you can't say the photon travels along some trajectory from x1 to x2.

And what I have said above is only an approximation because in general you can't localise a field so that it only has a non-zero value in some bounded region. The best you can do is change the field so that you will have a higher probability of seeing a photon in some region.

The above discussion alone would mean that a photon doesn't have a trajectory, but in general the situation is even less trajectory friendly than that. Different photons with the same energy aren't distinguishable: all you can say is "there are so many lumps in the field in this region". If you have some region R3 at t2 and there are lumps in R1 and R2 at t1 both of which are within (t2-t1)/c of R3, then there is in general no fact of the matter about whether the lump at R3 corresponds to the lump at R1 or R2 since they both contribute to R3 and all you can measure is something like the number of lumps.

If you want to understand this issue properly you should read about quantum field theory. A good book about QFT is "Quantum field theory for the gifted amateur" by Lancaster and Blundell.

More explanation. The OP asks if the particle can be in two places at once. Suppose that the field in a particular region is such that you have a very high probability of only measuring one particle in some given period of time. In general you will not be able to explain the results of experiments in that region by saying the particle has gone down one particular trajectory. Changes in different places in that region will all change the final outcome of the experiment. You could say the particle is in more than one place at a time in that sense. The particle doesn't appear or disappear from one place or another in the region. Rather, the field changes gradually over time so that the particle changes its probability of being in different places.

• The electron in H ground state is everywhere at a given t? if not, if it does indeedmove, how does it move without a trajectory? can it be in two places at one time? can it disappear in one place and reappear in another place skipping, space-points in-between according to QM? This is the question – user104372 Feb 17 '16 at 12:14
• "Lump mechanics"? I like it! :-) – CuriousOne Feb 17 '16 at 12:25
• @user104 see the explanation in the paragraph added in the answer above – alanf Feb 17 '16 at 12:45
• So could we say that a laser forms a line of lumps, the first lumps forming right next to the device itself causing subsequent lumps to be formed next to the first lumps and so on? And the lumps propogate at the speed of light? And when the laser is switched off the lumps collapse starting at the laser end and the collapsing happens at the speed of light? – Todd Wilcox Feb 17 '16 at 17:30
• No. The laser forms a field, which when measured or interacted with in a certain way looks like lumps. To say the laser beam is made of lumps is a bit like looking at an object at an angle and saying that the object is made out of all the ways you can look at it at an angle. – alanf Feb 17 '16 at 23:19

It means that quanta are not "bodies" in the sense of classical mechanics. Let's review the necessary definitions:

A "body" is defined as an extended piece of matter.

A "particle" is the approximation that a body can be described by the motion of its center of mass, so that we don't have to care about either size, shape or composition of the body.

A "trajectory" is the time dependent position vector of that center of mass.

Quanta (like photons and electrons) are neither bodies nor particles in this sense (because they don't behave like either) and they do not have trajectories.

Now, one can analyze quantum mechanics in terms of "paths", but those are not the paths that you keep hearing about when people are poorly discussing things like the double slit experiment. Instead, one can reformulate the Schroedinger equation (and the equations of quantum field theory) into an integral formulation, where the propagator can be described by a so called "path integral". This path integral is the summation of the complex exponential of the classical action S over all possible geometric paths that connect the initial to the final state.

The path integral is a pretty hard to use mathematical formalism if one attacks it directly, but it can be expressed as a perturbation series... those are the Feynman diagrams that you may have seen.

So what are quanta then, you may ask?

A quantum is the smallest unit of measurement on a quantum field. No matter what we do to that field, it will never interact in any other way than by exchanging a quantum with us.

Practically this means that we can initialize a quantum field with a finite number of quanta at the beginning of an experiment and we can measure a finite number of quanta at the end of it. The initial state is a configuration of quanta and the final state is a configuration of quanta. The dynamical theory of the field tells us what the probability distributions for final configurations of quanta will be when we keep repeating the same experiment over and over, again. Whatever we do, though, we can not assume that the quanta we put in have travelled on some paths from the initial state to the quanta of the final state.

Why is that so? Because in quantum field theory the number of quanta is simply not a constant and even if it was, quanta are not distinguishable. We can't label them Q1, Q2, Q3 and expect to see three labeled quanta to come out at the end. Instead nature can make some of them disappear or add some. More importantly, though, all the propagation of identical quanta will follow either the symmetry of bosons (i.e. only wave functions that are fully symmetric in all bosons appear) or fermions, which means that all wave functions have to be antisymmetric in the fermionic quanta.

So why do we see all this talk about "particles" in quantum mechanics? Because (beginners level) quantum mechanics is really a non-relativistic single quantum theory. It avoids all the mathematical problems with relativistic quantum field theory and can make reasonable statements about system with low energy bound states and low energy scattering. We don't have to worry about ever seeing more or less quanta than we have put in and it pretends to have a simple interpretation in terms of "particles". That, unfortunately, is somewhat of a mirage and it actually pays not to waste any effort on trying to reach the promised oasis of an interpretation of quantum theory where "particles" more on well defined "paths". That is (and always was) a nonsensical misunderstanding/misstatement of the theory that stems from its beginnings in the 1900s-1920s. By 1930 physicists had understood that single particle quantum mechanics was not sufficient to describe nature and by the late 1940s quantum field theory was blooming. At that point, latest, one should have stopped using the wrong concepts even in non-relativistic quantum mechanics. For whatever reason (probably because it seems to be easier to teach it that way) inertia has won out and students are still too often being forced to learn one version of QM first, before they basically have to re-learn the same thing, again, this time with the proper concepts. Those who never reach the level of even rudimentary understanding of relativistic fields are stuck with the wrong mental picture about quanta and particles.

To answer your questions directly: quanta don't have trajectories and it does not make sense to try to define any for them. "Particles" don't disappear and they do not reappear. What high energy physicists call "particles" are high momentum states that are being subjected to weak position measurements in particle detectors. It can be shown that under these circumstances reasonably straight "tracks" (not paths!) will appear in detectors. QM textbooks may glance over the phenomenological difference by showing particle tracks without explaining that a particle detector never even comes close to probing the quantum regime for the momentum/position uncertainty. What these detectors are built for are charge, momentum/energy and mass measurements (occasionally also for spin), but they are essentially classical devices.

• Could I get some feedback from the downvoter what is technically false about this description? I am really curious. – CuriousOne Feb 17 '16 at 11:31
• In the usual interpretation, an electron/photon is not a quantum. The measured quantities characterizing a system - i.e. the observables - may (but not always) have quantized (discrete) numerical values. The "fundamental" (in the right scale) constituents of a quantum system may be described mathematically by some suitable set of observables and/or fields (that are observables as well), but the interpretation of them is still of (more or less) tangible objects. – yuggib Feb 17 '16 at 11:36
• As far as I know, the word quanta have been introduced to describe the discreteness of the spectrum of some observables (already in non-relativistic QM), and not to refer to the fundamental constituents of the system under examination. – yuggib Feb 17 '16 at 11:37
• @yuggib: If by "usual interpretation" you mean the technically incorrect explanation of quantum mechanics that you can find in so many textbooks, then you are correct. That shouldn't stop us from giving the OP the correct explanation, instead. Falsehoods have no protection because of their popularity. The only way for quantum systems to interact (and to exist in the first place) is by means of quantum fields. There is nothing else. – CuriousOne Feb 17 '16 at 11:38
• It is just a matter of terminology, but an important one. I agree that you need to introduce the quantum fields to describe relativistic quantum mechanics. But "quantum" means a definite discrete quantity (as opposed to the continuum), and it is related to the spectral behavior of some observables rather than to particles. "Particle" come from the latin word for tiny object, and so is more suitable to define the fundamental constituent of a physical system. Again, I agree that in relativistic quantum mechanics this constituent is described by an operator (the field) rather than... – yuggib Feb 17 '16 at 11:43

tl,dr: This is really just a verbous synthesis of what has already been said by alanf and CuriousOne with a more basic experiment and theory explanation approach and a smattering of my own limited knowledge.

The upshot is the same: trajectories make little sense in nonrelativistic QM and if you factor in QFT, you can't even speak about particles.

The problem here is that our intuition is ill-defined when it comes to quantum mechanics. In particular, a "particle" is ill-defined. A second problem is that there are two sides to this question: theory and experiments - and another problem is that quantum experiments are much more difficult on a fundamental level.

Experimental Trajectories

Let's start with experiments (one should always start there, right?) and suppose we know what a "particle" is. Now, for classical physics, we can all agree that a body has a nice trajectory like a football shot on a goal. When you ask people whether quantum particles have trajectories, they will point to bubble chamber trajectories and violà, there are clear particle trajectories. Or they might point you to particle accelerators, where particle trajectories are visualised. As mentioned in another answer, the trouble is that the quantum particles here have very high energy. If the amount of energy involved is very high, the particle, albeit tiny, will behave classically where we are concerned about.

Thus, in order to see the "true" quantum world, this won't do. You'll have to think of low energy particles - maybe electrons in an atom or such. When you want to look at the "trajectory" of such a particle, you'll already be in trouble: you can't just shine light at it and watch it move - any photon with a moderately high energy will just ionize the atom and completely change whatever "trajectory" you wanted to look at. And even if you don't ionize the atom, you'll still heavily influence the path. If you want to "watch" a photon, you could use photo-detectors to measure the exact place where the photon is - but afterwards, it'll be destroyed, so that's also no good to measuring a trajectory.

First lesson: What you can do is already limited to one measurement of position or momentum. Everything after that just won't make much sense any more.

From Experiment to Theory

Now you might be clever and think: I can just set up an experiment, where I sent an electron time and time again and if I keep everything else fixed, every electron will have the exact same trajectory (this is what we expect from classical physics) and then I can just measure the trajectory by measuring speed and/or position at different places.

This is where the Heisenberg uncertainty jumps in and tells you that the result will never be a single trajectory. However hard you try, there will be a fundamental distribution of the results. If you measure "one" trajectory (meaning you prepare an electron, measure its position after time $t_1$, then reprepare, measure after $t_2$ and so on), you'd get an eratic behaviour. If you repeat the measurements at every point, you'd get a distribution of results. This is one way to say "particles don't have a trajectory".

Second Lesson: Repeating an experiment won't help - the outcome will be a probabilistic trajectory, not something you'd normally call a "trajectory".

However, what you can observe is how motion is built up: The probability of where to find the particle changes over time as does the speed. If you think you are sending a particle along some linear trajectory, what you will observe if done right is that the particle will "probably" move along this trajectory.

Nonrelativistic Quantum Mechanics

You can even go further: If you look into quantum mechanics (which can perfectly describe anything I have just talked about), in the scenario of an electron moving along some path* we have Ehrenfest's theorem, which tells us that the expectation value of position and momentum behave like a classical particle in classical mechanics: the expectation value has a classical trajectory! Note however that since the probability distributions are not very sharp, we cannot say the same for a single particle.

In nonrelativistic quantum mechanics, people tend to interpret this (and some other results together) as meaning that no particle really has a well-defined position and/or momentum. For large momenta (i.e. large energies) or large particles this doesn't really matter because the probability distributions become very sharp, but for our electron at low speed it does. This doesn't mean that the position is just a bit uncertain, it means that asking for a position of a particle when you don't measure it doesn't make sense. Note that this is a theoretical statement. If you don't measure, you can't really say anything, but using measurements you can rule out simple theories where a position and momentum are well-defined.

Third lesson: Mostly, people interpret quantum theory to imply that drawing the movement of a particle doesn't make sense. The particle moves neither along an eratic, nor a straight trajectory and it also doesn't magically disappear at a point and reappear at another. The question "How does the movement look like" just doesn't make sense (and you will always fail when you answer it experimentally).

Can we get around this?

Yes. Bohmian mechanics, a different yet equivalent interpretation of nonrelativistic quantum mechanics assigns each particle a position and a trajectory "pilot wave". This trajectory uses hidden variables (i.e. variables which we have no experimental access to). Note that the predictions of Bohmian mechanics and canonical quantum mechanics are the same, it is (so far) only a reinterpretation of the theory.

However, let me warn you: Bohmian mechanics has many problems of its own, not least that (as of now) it cannot really make sense out of QFT.

More theory and philosophy

I have always talked about particles and nonrelativistic quantum field theory. As pointed out by other answers here, the story gets even more complicated with QFT (from the classical intuition - it actually gets simpler in a certain way).

You seem to believe that there are particles and a massive particle must of course have a place where you can find it. There is a rest frame for the particle, so I could just go and have a look at it. Sadly, that's not quite true. The fundamental objects in quantum field theory are not particles, they are fields and their excitations. The connection to experiments is the rule that a quantity in an experiment corresponds to a self-adjoint operator and the values that the experiment will give are elements of the spectrum of that operator. Quantum fields have a ground state and every other possible state is an excited state. Very often, the results of the measurement are discrete ("quanta") and therefore, so are the excitations. Often, people therefore call these excitations "particles". The excitation of the electromagnetic field is a "photon" and there are quantum fields where the excitation is an "electron". However, we have a problem: These "particles" are not really what we think of as particles. They might not be localised - i.e. they really don't need to have a "position", i.e. if you look at the values of the field everywhere, it might be that the excitation is very spread out and there is no place where you could really say "Here is the particle". They might be somewhat localised and look like "lumps" or they might not be. In a sense, the orbital of an atom is a visualisation of the electron-excitation. And looking at such an orbital, I wouldn't really call it a particle, because - well for one, because it doesn't have a well-defined position.

And this is a basic problem of quantum field theory: Excitations are rarely localised. Talking about particles as localised objects with definite properties will only ever (partially) make sense in the absence of interaction. There are mathematical results to back this up (see this entry in the Stanford encyclopedia to learn more about the ontology of QFT).

Last lesson: In other words: Since we can't even define particles (localised objects), we don't even have to care how they move from point to point or how their trajectory looks like - none of these questions has meaning. The only thing that we can talk about is the probability density corresponding to observables with respect to quantum fields. We can ask how they evolve (this is answered by QFT) and we can have a look at asymptotic behaviour and how classicality emerges.

Last but not least: In the LHC experiments alluded to earlier, we can now say a bit more: After the collision, we have a bunch of excitations with a very high energy. They are already in the asymptotic regime: the detectors interact only weakly with those particles and therefore they behave very much like particles. Since in addition their energy is quite high, their probability densities are very sharp so that you really do behave like a particle trajectory. The interesting part of course is the collision - where the interaction is certainly not small. At that point, we cannot make sense of the words "particle".

*(see how intuition dictates how I have to write this sentence? I'm trying to explain that the sentence doesn't quite make sense, yet have to write it down to explain the scenario...)

• It's implied but I think it should be made clear that at the point of collision is perhaps the only place where we can definitely say: here the particle was - interactions such as collisions collapse the probabilities to a definite point in space – slebetman Feb 17 '16 at 19:17

Yes, also check out Bell's Theorem which is brilliantly described in Brian Greene's "The Fabric of the Cosmos". In the case of a single stream of photons being beamed one at a time, they make a wave-like pattern. Each individual particle is a wave -- that's why they don't have a definite trajectory. In fact, you can't think of any single quanta (particle) as being in a specific location.... It's not because we can't know the location or because it might be in some other location... in fact, there is no specific location in space-time where a particle "is", until the effects of an interaction propegate which show the particle at one specific trajectory where that particle "could have been". It is in quotes because Bell's experiments demonstrate that the particle really was in an Eigenstate (a fancy way of saying it is in more than one place) -- right up until the "decoherence" when we can prove that it interacted with other particles from a specific location.

• My question refers to electrons, and they do have mass, so.. they must occupy space, and only one space at a time, and they should pass from one point to the next and only to a next point. They do progress like a wave, but a wave is indeed a trajectory , too – user104372 Feb 17 '16 at 15:29
• @user104: Think of it this way - until an electron interacts with another "thing" (usually this is called "observed") it doesn't occupy a definite space but rather a statistical/probabilistic range of space. That a single electron can cause interference with itself tells us that the probabilities does not mean we don't know (like the outcome of a dice), it means it actually occupy all the space according to the probabilities (like multiple virtual particles travelling together). – slebetman Feb 17 '16 at 19:10
• @user104 "x has mass, so x must occupy (behave as if it occupies) a particular single point in space and pass from one point to the next" is simply not true, on small scales our reality behaves in ways that contradict this. Electron "orbits" around atoms are a great example. In many cases, the concept of a trajectory is a very useful simplification but not always - the meaning of "x does not have a definite trajectory" is "as some cases show, the whole concept that 'stuff' or 'mass' has 'location' or 'trajectory' is conceptually wrong and not consistent with the physical reality we live in". – Peteris Feb 18 '16 at 1:55
• "they do have mass, so.. they must occupy space, and only one space at a time" - that makes sense but experiments show that it's wrong. – emery Feb 19 '16 at 15:04

## protected by Qmechanic♦Feb 17 '16 at 15:42

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