# Do bosons contradict basic probability laws?

• A
One shouldn't say "identical particles" when one means "indistinguishable particles"...
I agree. I would say identical particles are not necessarily indistinguishable. If they can be tracked they are distinguishable.

AndreasC
Gold Member
Well they can't be tracked in QM so...

Nugatory
Mentor
I would say identical particles are not necessarily indistinguishable. If they can be tracked they are distinguishable.
As far as the math of quantum mechanics (quantum field theory if you want to do it properly) is concerned, if they can be tracked they aren't identical.

mattt
vanhees71
Gold Member
I agree. I would say identical particles are not necessarily indistinguishable. If they can be tracked they are distinguishable.
"Identical particles" is just a often used misnomer. The expression is used synonymously with "indistinguishable particle", and indistinguishable particles are particles with all intrinsic properties (i.e., the quantum numbers needed to label an asymptotic free one-particle state for vanishing momentum, i.e., the particle at rest). These are within the Standard Model: spin, electric charge, color charge, flavor, and weak hypercharge.

mattt
One shouldn't say "identical particles" when one means "indistinguishable particles"...
"Identical particles" is just a often used misnomer. The expression is used synonymously with "indistinguishable particle", and indistinguishable particles are particles with all intrinsic properties (i.e., the quantum numbers needed to label an asymptotic free one-particle state for vanishing momentum, i.e., the particle at rest). These are within the Standard Model: spin, electric charge, color charge, flavor, and weak hypercharge.

Now you got me confused tbh. Are you arguing that one should just never use the term “identical particles”? Or do you actually have some situation in mind where “identical particles” would not be used synonymously with “indistinguishable particles”, and want to reserve “identical” for this?

vanhees71
Gold Member
As I said, I never use the expression "identical particles". I only wanted to say that you find this expression in many textbooks and papers meaning "indistinguishable particles".

In quantum mechanics the particles are really indistinguishable. It's a feature you cannot intuitively understand, because it's something we are not used to in our experience with macroscopic objects which obey to a very good approximation classical laws. Macroscopic objects can be individually followed. You just mark, e.g., a ball somehow and then you can distinguish it from other similar balls by this mark. Formally you can follow its trajectory from its initial position at some time ##t_0## and identify this individual object at any later time ##t##.

This you cannot in general anymore for an individual particle in a many-body system. The many-body quantum state of indistinguishable quantities must be either symmetric or antisymmetric under exchange of two particles, describing either bosons or fermions (where the bosons have necessarily integer and fermions necessarily half-integer spin).

Take two indistinguishable particles in non-relativistic quantum mechanics. Then you can describe a pure quantum state with a two-particle wave function ##\Psi(t,\vec{x}_1,\sigma_1, \vec{x}_2,\sigma_2)##, where ##(\vec{x}_j,\sigma_j)## are positions and spin-z components (##\sigma_j \in \{\pm s,\pm (s-1),\ldots \}##). The physical meaning is, according to Born's rule, given by the two-body probability distribution
$$w(t,\vec{x}_1,\sigma_1,\vec{x}_2,\sigma_2)=|\Psi(t,\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2)|^2,$$
where
$$\mathrm{d}^3 x_1 \mathrm{d}^3 x_2 w (t,\vec{x}_1,\sigma_1,\vec{x}_2,\sigma_2)$$
is the probability to find one particle with spin component ##\sigma_1## within a volume elment ##\mathrm{d}^3 x_1## around the position ##\vec{x}_1## and one particle with spin component ##\sigma_2## within a volume element ##\mathrm{d}^3 x_2## around the position ##\vec{x}_2##.
You can only say that much about indistinguishable particles: It doesn't make sense to say you find a specific particle around ##\vec{x}_1## and another specfic particle at ##\vec{x}_2##.

Indeed from ##\Psi(t,\vec{x}_2,\sigma_2;\vec{x}_1,\sigma_1)=\pm \Psi(t,\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2)## (upper sign bosons, lower sign fermions) you get
$$w(t,\vec{x}_2,\sigma_2;\vec{x}_1,\sigma_1)=w(t,\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2),$$
i.e., it's not observable which individual particle is which.

hutchphd and Dr.AbeNikIanEdL
PeroK
Homework Helper
Gold Member
2020 Award
As I said, I never use the expression "identical particles". I only wanted to say that you find this expression in many textbooks and papers meaning "indistinguishable particles".

You'll be pleased to learn that Chapter 5 of Griffiths' QM Book is entitled "Identical Particles"!

vanhees71
Gold Member
Well, I'm not in favor of this textbook anyway,... As I said, it's unfortunately common jargon in the physics literature. You cannot help it. One only has to carefully explain the meaning to the students.

DrClaude
Mentor
You'll be pleased to learn that Chapter 5 of Griffiths' QM Book is entitled "Identical Particles"!
So does Sakurai.

I think that "indistinguishable" comes from classical statistical physics, and is independent from identical particles. It just turns out that QM showed us that identical particles are fundamentally indistinguishable!

PeroK
vanhees71
Gold Member
Yes, I know. Even Weinberg uses "identical particles". I don't say that any text book that uses this phrase is bad. Then there'd be almost no textbook left, I guess.

PeroK
In quantum mechanics the particles are really indistinguishable. It's a feature you cannot intuitively understand, because it's something we are not used to in our experience with macroscopic objects which obey to a very good approximation classical laws. Macroscopic objects can be individually followed. You just mark, e.g., a ball somehow and then you can distinguish it from other similar balls by this mark. Formally you can follow its trajectory from its initial position at some time ##t_0## and identify this individual object at any later time ##t##.
This marking, does it have to be an actual change to the objects or could it just be in an image of the objects? For example if one could follow the trajectories of large molecules or clusters (for example in a solution) using some high speed, high resolution, imaging technique and then just mark the objects in the computer and track them, would that make normally indistinguishable particles distinguishable?

Lord Jestocost
Gold Member
In Walter Greiner's "QUANTUM MECHANICS An Introduction" (Fourth Edition) one reads:

"One characteristic of quantum mechanics is the indistinguishability of identical particles in the subatomic region. We designate as identical particles those particles that have the same mass, charge, spin etc. and behave in the same manner under equal physical conditions. Therefore, in contrast with macroscopic objects, it is not possible to distinguish between particles like electrons (protons, pions, α particles) on the basis of their characteristics or their trajectory. The spreading of the wave packets that describe the particles leads to an overlapping of the probability densities in time (Fig. 15.1); thus we will not be able to establish later on whether particle no. 1 or no. 2 or another particle can be found at the point in space r. Because of the possible interaction (momentum exchange etc.), dynamical properties cannot be used to distinguish between them, either." [Italics in original, LJ]

DrClaude, mattt, vanhees71 and 1 other person
AndreasC
Gold Member
Imo Shankar explained it pretty well. At least I feel like I understood the difference.

Imo Shankar explained it pretty well. At least I feel like I understood the difference.
What difference?

vanhees71
DrClaude
Mentor
This marking, does it have to be an actual change to the objects or could it just be in an image of the objects? For example if one could follow the trajectories of large molecules or clusters (for example in a solution) using some high speed, high resolution, imaging technique and then just mark the objects in the computer and track them, would that make normally indistinguishable particles distinguishable?
You can of course follow individual particles and make a distinction between identical particles in different locations. For instance, when researchers trap an electron in a Penning trap and keep it there for days on end, they know it is always the same electron until it escapes the trap.

Likewise, if I follow a single carbon dioxide molecule that I expire, I can differentiate it from one you just breathed in.

It is only when these identical particles are part of the same system or can somehow interact that the indistinguishability plays a role.

I recommend reading Feynman on the subject: https://www.feynmanlectures.caltech.edu/III_04.html

PeterDonis, Lord Jestocost and Philip Koeck
AndreasC
Gold Member
What difference?
Between identical particles and indistinguishable particles.

vanhees71
Gold Member
There is no difference. Both expressions are used synonymously in the literature.

AndreasC
Gold Member
There is no difference. Both expressions are used synonymously in the literature.
In classical mechanics you can call two particles "identical" (in terms of intrinsic features) but they can still be distinguishable, unlike quantum mechanics where they are indeed used synonymously. Although as Dr Claude said, even in QM you can sometimes distinguish between them.

vanhees71
In classical mechanics you can call two particles "identical" (in terms of intrinsic features) but they can still be distinguishable, unlike quantum mechanics where they are indeed used synonymously. Although as Dr Claude said, even in QM you can sometimes distinguish between them.
I wonder if you've noticed what you have written. You say that in QM "identical" and "indistinguishable" are synonymous, and that identical particles can be distinguishable even in QM, according to what Dr Claude wrote.
Essential you've stated that in QM indistinguishable particles can be distinguishable.

DrClaude
Mentor
Let me clarify some things before this turns into a Brian Cox moment.

Consider an experiment where two electrons are in different traps in different sections of a lab. Technically, the wave function describing the two electrons should have a definite symmetry under the exchange of the two electrons, meaning that both electrons have to be in a superposition of being in both traps, and we cannot distinguish one from the other, as they are fundamentally indistinguishable. However, just like Schrödinger's cat is never really in a superposition of alive and dead, in practice the two electrons are not in a superposition of being in both traps.

If we take the electrons and send them hurtling towards one another, we will still talk of the electron coming in from the left and the electron coming in from the right. After the collision, however, we cannot distinguish between the two electrons passing by each other or bumping and reversing direction.

Fundamentally, identical particles are indistinguishable. In practice, there are plenty of cases where we can distinguish them, if they are not part of what we would consider the same system. In other words, even if the electrons are in a superposition of being in both traps, if we measure electron L in the left trap and electron R in the right trap (at ##t=0##), for well-separated traps the later (##t>0##) probability of now measuring electron R as being in the left trap and electron L in the right trap is so low that it can be neglected FAPP.

Lord Jestocost, PeroK and Dr.AbeNikIanEdL
You say that in QM "identical" and "indistinguishable" are synonymous

No, he said that the terms are used synonymously [by many sources]. As @vanhees71 pointed out, this is probably not a good idea in the first place. Obviously, in the special cases where particles with identical properties can approximately be distinguished one has to be particularly careful with one’s nomenclatur.

Philip Koeck
Lord Jestocost
Gold Member
"Equation (462) shows that the symmetry requirement on the total wavefunction of two identical bosons forces the particles to be, on average, closer together than two similar distinguishable particles. Conversely, Eq. (465) shows that the symmetry requirement on the total wavefunction of two identical fermions forces the particles to be, on average, further apart than two similar distinguishable particles. However, the strength of this effect depends on square of the magnitude of ## \left< x \right>_{ab}## , which measures the overlap between the wavefunctions ##\psi(x,E_a)## and ##\psi(x,E_b)##. It is evident, then, that if these two wavefunctions do not overlap to any great extent then identical bosons or fermions will act very much like distinguishable particles."

From: Quantum Mechanics by Richard Fitzpatrick
http://farside.ph.utexas.edu/teaching/qmech/Quantum/node60.html

Last edited:
Philip Koeck and PeroK
vanhees71
Gold Member
No, he said that the terms are used synonymously [by many sources]. As @vanhees71 pointed out, this is probably not a good idea in the first place. Obviously, in the special cases where particles with identical properties can approximately be distinguished one has to be particularly careful with one’s nomenclatur.
I have always given the definition: Two particles are called indistinguishable or identical if they have the same INTRINSIC properties. Intrinsic are all properties that are defined for a particle at rest, i.e., with vanishing momentum. So you have spin and various charge-like (electric charge, color, flavor) quantum numbers.

Of course you can always dinstinguish such particles by their momentum or position and their polarization/spin-##z## component. For fermions the ##N##-particle Hilbert space is spanned by the totally antisymmetrized product states ("Slater determinants").

In the above mentioned example of two electron, each trapped at different locations, you have a wave function like
$$\Psi(t,\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2)=\frac{1}{\sqrt{2}} [\psi_1(t,\vec{x}_1,\sigma_1) \psi_2(t,\vec{x}_2,\sigma_2) - \psi_2(t,\vec{x}_1,\sigma_1) \psi_1(t,\vec{x}_2,\sigma_2)].$$
If the single-particle wave functions have negligible spatial overlap you can distinguish the electrons by their position.

Fundamentally, identical particles are indistinguishable. In practice, there are plenty of cases where we can distinguish them, if they are not part of what we would consider the same system. In other words, even if the electrons are in a superposition of being in both traps, if we measure electron L in the left trap and electron R in the right trap (at ##t=0##), for well-separated traps the later (##t>0##) probability of now measuring electron R as being in the left trap and electron L in the right trap is so low that it can be neglected FAPP.
I quote Dr. Claude's post above, but I'm also referring to the following posts with similar content.

I'm wondering whether it is possible to make identical particles distinguishable in such a way that they still form a system in the thermodynamic sense, specifically that they have a distribution of energies that's described by, e.g., the BE-distribution?

As a possible example I could imagine a gas of carbon-60 clusters. If one follows these clusters using some high-speed and high resolution imaging method one would be measuring the position of each cluster at regular time-points. If I understand correctly this would make the wave function of the system collapse every time an image is taken.

Would this make the clusters distinguishable?

What would one actually see in the video? Would the clusters appear to be following classical paths or would they change position randomly?

In case the observed paths appear like classical paths, would the kinetic energies follow the BE-distribution or the MB-distribution?

vanhees71