Identical and indistinguishable particles

[Stephen Tashi]

<Moderator's note: Thread split off from https://www.physicsforums.com/threa...e-particles-obey-boltzmann-statistics.939086/>

Two electrons are identical to each other. They have exactly the same properties and characteristics.

This raises the question of how you know that there are two electrons in the first place. It seems to me that counting two things implies some way of telling them apart.

In classical mechanics , someone who talks about "two identical spherical masses" means certain properties of the two objects are identical, but not that they are identical with respect to their position coordinates. So there is a tacit understanding that the use of phrases like "identical" or "same properties" does not actually refer to all possible properties.

But these two electrons become INDISTINGUISHABLE if they are so close to one another that their wavefuctions significantly overlap, so much so that you can no longer distinguish which is which.

My interpretation is: Suppose electron A and electron B are initially distinguishable. After they get close enough, we can't perform any process to separate them again and tell which is electron A and which is electron B. I understand that interpretation.

 

[ZapperZ]

This raises the question of how you know that there are two electrons in the first place. It seems to me that counting two things implies some way of telling them apart.

In classical mechanics , someone who talks about "two identical spherical masses" means certain properties of the two objects are identical, but not that they are identical with respect to their position coordinates. So there is a tacit understanding that the use of phrases like "identical" or "same properties" does not actually refer to all possible properties.

Then you are simply nitpicking unnecessarily here. You might as well argue that there is no such thing as an infinite square well and thus, the physics doesn't work.

The idealized description does not diminish the validity, even if it is "experimentally difficult". Those are two different things.

My interpretation is: Suppose electron A and electron B are initially distinguishable. After they get close enough, we can't perform any process to separate them again and tell which is electron A and which is electron B. I understand that interpretation.

You may understand it, but it isn't valid. A and B are not "indistinguishable" in the Drude Model, because I used the Boltzmann distribution to get Ohm's Law. THAT is an experimentally-verified observation.

And I can definitely separate out two electrons that have been entangled! EPR-type experiments on electrons do that all the time. Furthermore, electrons in the valence band of a semiconductor obeys the FD statistics, and yet, I can easily pull one out to become free. Now I've taken one electron which I couldn't pick out while it was in the Fermi sea, and have made it a free particle that I can distinguish and follow. So I do NOT understand this last claim.

Zz.

 

[Philip Koeck]

This raises the question of how you know that there are two electrons in the first place. It seems to me that counting two things implies some way of telling them apart.

It's actually quite easy to count indistinguishable particles. For electrons, for example, you can measure the total charge. For many systems you can measure the mass, etc.

 

[bhobba]

This raises the question of how you know that there are two electrons in the first place.

Well its not talked about a lot but there are in fact quite a few different formulations of QM:
https://www.colorado.edu/physics/phys5260/phys5260_sp16/lectureNotes/NineFormulations.pdf

They really all are equivalent. Have a look at the second quantization formulation. Its not a common one, but actually should be a bit more widely known than it is. In that formulation, as you can see, the number of particles is a key observable coming from the creation and annihilation operators. Like any other observable it can, at least in principle, be measured. That's how we know the number of electrons or whatever in the same way we know any other quantum observable. If there are say two electrons then the number operator applied to the state always gives 2. Its also very important in understanding how particles emerge from fields in QFT. The reason they are indistinguishable is as I said before exchanging particles gives exactly the same wavefunction with just maybe a change in sign - but there is this axiom of QM that if you multiply a state (alright strictly speaking a pure state) by a phase factor, and -1 is a phase factor, you get the same state. It means all quantum particles that are part of the same system are indistinguishable - part of the same system meaning their wave-functions overlap so it has to be considered as a whole.

Thanks
Bill

 

[bhobba]

It means all quantum particles that are part of the same system are indistinguishable - part of the same system meaning their wave-functions overlap so it has to be considered as a whole.

Actually it occurs to me that may be one way to get Maxwell's statistics - if the 'gas' of particles is so dilute none of the wave-functions overlap so are all distinguishable.

That in turn raises an interesting question I don't know the answer to. Theoretically exactly how much overlap is required to consider it Maxwell describable ie the particles distinguishable - is even a smidgen enough - can't say I have ever com across that one before - someone else may know.

Thanks
Bill

 

[Stephen Tashi]

Then you are simply nitpicking unnecessarily here. You might as well argue that there is no such thing as an infinite square well and thus, the physics doesn't work.

I don't consider it nitpicking to use precise language. Various definitions of "equal" are defined with respect to particular equivalence relations. "Identical" should be a clearly defined equivalence relation.

A fundamental issue in this thread is the definitions used in counting things. One way to clarify how to count is to say what it means for "two" things to be "the same thing" - i.e. be a set with cardinality of 1 - versus being two "identical" or "indistinguishable" things that form a set of cardinality 2.

One definition of "microstate" advocated in this thread is that two microstates that have the same number of "identical" particles and the same values of certain statistical measurements of these particles form a set of cardinality 1. Another view is that two microstates that are "identical" in that respect may form a set of cardinality 2 due to distinctions we can make among "identical" particles.

It wouldn't hurt to be explicit about what is meant by "identical" and "indistinguishable" in particular contexts.

You may understand it, but it isn't valid.

Ok, I'll look at @bhobba 's link.

 

[Stephen Tashi]

It's actually quite easy to count indistinguishable particles. For electrons, for example, you can measure the total charge. For many systems you can measure the mass, etc.

As a follow-up, can we clarify the distinction between "identical particles" and "indistinguishable particles"? Posts #17 and #20 agree that two "identical particles" can be distinguished.

A perhaps obsolete approach to classical thermodynamics asks the student to think of a phase space of many dimension where a point is a long vector giving the position and velocity components for each particle in a gas. In such a vector we have data for the k-th particle in some place in that vector. So, it seems to me that this description distinguishes one particle from another - just by virtue of the fact that each particle's data is assigned certain components of the vector. Data for a path in this phase space gives data to track each particle.

We can define micro-states in this phase space by partitioning this phase space into subsets. Where things get hazy in my mind is when we attempt to derive probabilities from counts of points in the micro-states. If are actually counting discrete points we get can get two different points by taking one point and then swapping particle k with particle j, in the sense of writing particle k's data in the places were particle j's data used to be, and vice-versa. If we are not counting discrete points then we may be counting things that consist of a set points. So 1 thing of this sort might include a point plus all possible points that can be generated from by permuting the particles.

From this viewpoint, the crux of the matter is not whether the particles themselves are distinguishable. (They are.) The crucial question is: What is the definition of the things we are counting? For example in @NFuller 's insight https://www.physicsforums.com/insights/statistical-mechanics-part-equilibrium-systems/ it is said that we count "subsystems" and that we consider positions and momenta discretized. In the above description of phase space, what is 1 subsystem? The arguments in favor of considering all the particles "indistinguishable" actually seem to be arguments in favor of lumping together data from distinguishable particles in a set and then counting that set as 1 thing.

From the current Wikipedia article https://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics)

Correcting overcounting in phase space
Typically, the phase space contains duplicates of the same physical state in multiple distinct locations. This is a consequence of the way that a physical state is encoded into mathematical coordinates; the simplest choice of coordinate system often allows a state to be encoded in multiple ways. An example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system. It is important in statistical mechanics (a theory about physical states) to recognize that the phase space is just a mathematical construction, and to not naively overcount actual physical states when integrating over phase space. Overcounting can cause serious problems:

That passage is written as if "actual physical state" had been defined previously and the overcount is deduced from that definition However, a more coherent view is that "physical state" is defined by defining what set of points in phase space we intend to consider 1 physical state.

 

[bhobba]

The crucial question is: What is the definition of the things we are counting?

Its a probability question - see page 146 - Ross - Introduction to Probability Models for the details.

His remark is as follows (page 150):
Physicists studying how particles distribute themselves observed the behavior of particles such as photons and atoms containing an even number of elementary particles. However when they studied the resulting data they were amazed to discover the observed frequencies did not follow the multinomial distribution but rather seemed to follow the Bose-Einstein distribution. They were amazed because they could not imagine a physical model for the distribution of particles which would result in all possible outcomes being equally likely.

It's a simply an exercise probability models and outcomes being equally likely.

Thanks
Bill

 

[PeterDonis]

This raises the question of how you know that there are two electrons in the first place. It seems to me that counting two things implies some way of telling them apart.

The QFT answer would be that you don't say "there are two electrons". You say "the electron field is in an eigenstate of the number operator with eigenvalue two". This does not commit you to any statement about "which electron is which" or how you distinguish one "electron" from the other. It just says what it says.

 

[vanhees71]

I can only underline that the sloppy expression "identical particles" is misleading and should not be used. Among all features of QT, besides the phenomenon of entanglement, the indistinguishability of particles with identical intrinsic quantum numbers is the most mind-boggling one. You cannot comprehend it other than with the precise language of mathematics.

As stressed by @bhobba, the most natural way to deal with many-particle systems, particularly of those consisting of indistinguishable particles, is the formulation of QT in terms of quantum field theory, and it's not so uncommon as he might think, but to the contrary it's the most common way to formulate QT in the communities of condensed-matter physicists (who most often can rely on non-relativistic QFT, because at the energy scales in involved in the physics of atoms, molecules, etc. you usually are in the relativistic realm; of course, if it comes to large atoms, you also have to deal with relativistic effects), high-energy-particle and nuclear physicists who have to deal more often with relativistic QFT (the nuclear physicists not only with vacuum but also with many-body QFT). In the relativistic case it's the only formulation that makes sense since particle numbers are not conserved (but only net charges like electric charge, baryon number, etc.).

One calls two particles indistinguishable if all relevant intrinsic quantum numbers are equal. Intrinsic quantum numbers are defined by all observables that need to be specified to completely characterize a single-particle state at rest. As a short-hand notation you simply call the particles by their name. E.g., an electron has the intrinsic quantum numbers: mass $m=m_{e} \simeq 0.511 \; \mathrm{MeV}$, spin $s=1/2$, electric charge $Q=-1$, color charge=0, etc.

Then you can define a basis of single-particle states by defining a complete set of compaticle observables. For an electron you can choose, e.g., momentum and spin (or more precisely the Wigner basis in the relativistic case, where the corresponding state is defined as the state from the electron at rest with specified spin-z component boosted by a rotation-free Lorentz boost to the given momentum value). Then you can build the full many-electron space by the completely antisymmetrized single-particle product states (or occupation-number states). Here, the antisymmetrization rule for fermions applies due to the spin-statistics theorem which tells you that all half-integer-spin particles are necessarily fermions. This antisymmetrization (fermions) or the corresponding symmetrization prescriptions implements or rather defines what's meant by the indistinguishability of particles. The symmetrization or antisymmetrization is most conveniently implemented using the QFT (or "2nd quantization") approach, introducing annihilation and creation operators wrt. the chosen single-particle basis, in our example $\hat{a}(\vec{p},\sigma_z)$, obeying the canonical (anti-)symmetrization rules,
$$[\hat{a}(\vec{p},\sigma_z),\hat{a}(\vec{p}',\sigma_z')]_{\pm}=0, \quad [\hat{a}(\vec{p},\sigma_z),\hat{a}^{\dagger}(\vec{p}',\sigma_z')]_{\pm}=\delta^{(3)}(\vec{p}-\vec{p}') \delta_{\sigma_z,\sigma_z'},$$
where the upper sign is for fermions and the lower one for bosons.

 

[PeroK]

<Moderator's note: Thread split off from https://www.physicsforums.com/threa...e-particles-obey-boltzmann-statistics.939086/>This raises the question of how you know that there are two electrons in the first place. It seems to me that counting two things implies some way of telling them apart.

.

I don't understand this statement. You could count red snooker balls, say, without making use of the ability to tell them apart. Apart from indirect techniques like weighing them, you simply move them from one place to another one at a time.

I don't think, therefore, it's the counting of snooker balls and electrons that is fundamentally different. It's the theoretical and practical ability to label them that is different.

 

[Stephen Tashi]

It's a simply an exercise probability models and outcomes being equally likely.

Well, you could say any physics involving probability models is simply an exercise in them.

As to the meaning and significance of "distinguishable" vs "indistinguishable" in physics, my current (today! ) opinion is that these adjectives do not actually refer to the ability to make distinctions about objects - or the lack of that ability. Instead they refer to what type of probability model describes statistical properties of the objects. So if I have a set of objects that I can distinguish and some statistical property of them is best described by the combinatorics of indistinguishable objects then, for the purposes of a physical theory about that property, my objects are "indistinguishable".

Perhaps that's what you're saying, but the quotation from Ross doesn't clearly say it.

My interpretation (insofar as I have one) of what @PeterDonis and @vanhees71 said is consistent with that outlook - i.e. that "distinguishable" vs "indistinguishable" are terms we apply after we know what physical models describe properties of the objects. We don't actually begin with the question of whether some objects can or cannot be distinguished and proceed by logical deduction to their statistical properties. Instead, we look at the statistical properties and decide what type of combinatorics explains them. Then we call the objects "distinguishable" or "indistinguishable" according what kind of combinatorics works.

It's interesting to ask if there can be a mathematical or physical theorem that would let us do the deduction in the order that is usually presented - i.e. The usual order is : We don't have the ability to distinguish the particles. Therefore the statistics of their such-and-such property must be given by the combinatorics of indistinguishable things.

A indirect proof of such a theorem could follow the pattern: Assume there exists a property of the particles whose statistics are not described by the combinatorics of indistinguishable objects. We shall show this implies there is a way to distinguish one particle from another.

 

[PeroK]

@Stephen Tashi fundamentally, physics is an empirical science. It's no surprise that physics in general works by experimentally discovering or verifying something and then defining it mathematically so that it has the observed properties.

Note that all particles are indistinguishable from those of their own kind. An electron is distinguishable from a proton, but not from another electron.

 

[Stephen Tashi]

Note that all particles are indistinguishable from those of their own kind.

Yes, keeping in mind that the statement is essentially a proof-by-definition. When can't distinguish among group of things, we can define them to be "of their own kind".

 

[PeroK]

Yes, keeping in mind that the statement is essentially a proof-by-definition. When can't distinguish among group of things, we can define them to be "of their own kind".

Not quite. Because it ties "indistinguishability" of measureable properties - mass, charge, spin etc. - with certain statistical behaviour.

There is no "tashitron" that has a set of defining quantum properties, but behaves statistically as though one tashitron could be distinguished from another.

 

[PeterDonis]

"distinguishable" vs "indistinguishable" are terms we apply after we know what physical models describe properties of the objects

Not just what physical models, but what experimental conditions. For example, suppose I am running some experiment whose initial conditions are that I have prepared an electron at spacetime point A and an electron at spacetime point B, and the final conditions are that I detect an electron at spacetime point C and an electron at spacetime point D. If those are the only measurements made, then the amplitudes for the two possibilities (A -> C, B -> D; and A -> D, B -> C) have to be added (and there is also a sign flip because electrons are fermions, whereas there wouldn't be if we were doing the experiment with, say, photons) before I can square the amplitude to obtain a probability for this event. That is what is usually being referred to when the term "indistinguishable" is used.

By contrast, if the experimental conditions also include, say, the fact that there is a tube with impermeable walls running between spacetime points A and C, and another one between B and D, then the amplitudes no longer add, because there is no way for the electron that starts at A to get to D; it can only go to C (and similarly the electron that starts at B can only go to D, not C). So this experiment really has a different set of measurements than the one described above; and it's the different set of measurements that makes the two electrons "distinguishable" in this experiment, whereas they were indistinguishable in the one described above. And, of course, the probabilities reflect that, since the amplitudes no longer add so there are no interference terms.

 

[bhobba]

Perhaps that's what you're saying, but the quotation from Ross doesn't clearly say it.

No - my quote doesn't say it - but prior to the quote the mathematical analysis does - there is nothing strange - probability model wise going on.

Why is it like that? As I alluded to and Vanhees elaborated on the second quantisation formulation emphasizes you have creation and annihilation operators. A creation operator does not create a particle in the sense you can tell apart from the others to derive the multinomial distribution which requires you to be able to label them in some way - the creation operator does not create a particle labelled a, then when you apply it again it is labelled b and so on - that's what you need to derive that distribution. It simply creates a particle - that's it. In that case you derive the Bose-Einstein (for Bosons) statistics. The derivations are all found in Ross. Its not really that hard.

Why don't we see that behavior here in the macro world when dealing with usual gasses? Here the energy is high enough, and considering the particles as 'wave-packets' you have the wave-packet of such small width there is no overlap so its not a single system and they act as individually labelled particles. That's the answer you will find in texts/papers on quantum statistics - the concept used being the thermal wavelength:
https://itp.uni-frankfurt.de/~valenti/WS13-14/all_1314_chap11.pdf

What I personally have difficulty with is theoretically there should be still some overlap even if just a smidgen so small you think you can ignore it - but from the theory as I understand it even that smidgen should be enough for it to be a single system. There is obviousy something I am missing - but right now I do not see it.

Its similar to the debate over when Brian Cox talked about heating a diamond affecting minutely all the energy levels throughout the entire universe. I thought about that one quite a bit then realized - how silly - between any two real numbers is another real number - all that would happen is the energy levels of the atoms in the diamond will simply fill the gaps available. But here the answer to whatever is going on doesn't seem so easily won - of course I am missing something - but exactly what isn't clear to me.

Thanks
Bill