Talk:Indistinguishable particles/Archive 1

Archive 1

Removed paragraph

This has profound implications for thermodynamics. Suppose you have two coins which are not identical. The probability of finding the coins in heads-tails is one in two, while the probability of finding the coints in heads-heads or tails-tails in one in four. If the coins are identical, then the probability of finding the coins in heads-heads, tails-tails, or heads-tails is one in three. Since the thermodynamic properties of a material are determined by the probability of particles being at a certain energy, the assertion of identical particles has some very noticable impacts.

I removed this paragraph, as I have doubts about it. Could whoever wrote it please explain in greater detail? I do not find it useful, because coins (like all macroscopic objects) are not identical. Also, I do not understand what you mean by "flipping" identical coins.

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I'm not the author of the omitted paragraph, but I think it comes down to the distinguishability between the two coins allowing for allowing a difference between "heads/tails" and "tails/head" for the pair. If the coins are identical, there's no way to distinguish between {H,T} and {T,H}, so the number of available combined states is three, as the original author states. If they can be distinuished, then the total number of possible pair states is four. Perhaps this plays into the equipartition theorem...

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— Preceding unsigned comment added by 69.174.136.201 (talk) 02:21, 23 September 2022 (UTC)

Rewrote the paragraph. I'm basically trying to paraphrase the discussion of quantum statistics in Kittel and Kromer's *Thermal Physics*. The basic point is that the concept of identical particles has some profound macroscopic consequences. The discussion is for bosons. It would be nice to include a section of the implications of antisymmetry on fermions.

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The Pauli exclusion principle doesn't always work, according to my quantum textbook. See my discussion in talk:Pauli exclusion principle. -- Tim Starling

I think you're misunderstanding your textbook. Even when the particles are far apart, the wavefunction is still antisymmetric, but this contributes very little. Let me explain with some math:

Define the two-electron wavefunction as follows (note that it's antisymmetric):

$\psi (x,y)=\psi _{1}(x)\psi _{2}(y)-\psi _{1}(y)\psi _{2}(x)$

Here, $\psi _{1}$ and $\psi _{2}$ are two single-electron wavefunctions. So, $|\psi (x,y)|^{2}$ is the probability (really probability density) of finding one electron at position x, and the other at position y.

Now suppose that the two electrons are very far apart. To be specific, let's say that $\psi _{1}$ is a Gaussian centered at some position $x_{0}$, and $\psi _{2}$ is a Gaussian centered at some position $y_{0}$, with $x_{0}$ and $y_{0}$ far apart.

The value of the Gaussian function is small at positions which are far from the center. So the amplitude of $\psi (x,y)$ is small except if you have either

CASE 1: $x$ is close to $x_{0}$ and $y$ close to $y_{0}$

OR

CASE 2: $x$ is close to $y_{0}$ and $y$ close to $x_{0}$

In Case 1 the second term in our definition of $\psi (x,y)$ is basically zero, and we're left with:

$\psi (x,y)=\psi _{1}(x)\psi _{2}(y)$

In Case 2 the first term in our definition of $\psi (x,y)$ is basically zero, and we're left with:

$\psi (x,y)=-\psi _{1}(y)\psi _{2}(x)$

An overall minus sign in our wavefunction doesn't change any observable physics (only relative signs between different terms matter), so in any case we just have:

$\psi (x_{1},x_{2})=\psi _{1}(x_{1})\psi _{2}(x_{2})$

where $x_{1}$ is the position of particle 1, and $x_{2}$ is the position of particle 2. This is what we would have had if we hadn't made our wavefunction antisymmetric in the first place.

To summarize my point: Even when two identical (fermionic) particles are far apart, the wavefunction is still antisymmetric. But when they're far apart, the antisymmetric wavefunction is approximately equal to what you would have had if you didn't bother making it antisymmetric. So the Pauli exclusion principle always applies, but it only matters when the particles are close together (i.e., close enough together that the two single-particle wavefunctions overlap significantly). -- Tim314 16:04, 2 May 2007 (UTC)

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Is there any experimental evidence for the existence of plektons (which if I'm not mistaken are anyon-like solutions for non-Abelian gauge fields)? If not, the article should mention that they are hypothetical particles, whereas the existence of anyons is pretty well-confirmed by experiments on the fractional quantum Hall effect. -- CYD

I honestly don't know if plektons have been experimentally observed so far (anyone knows?), but they certainly are a very important theoretical possibility. Phys 12:16, 22 Aug 2003 (UTC)

question

this sounds really stupid but i dont understand how fermions can be identical, like electrons can differ in their momentum uncertainty and spin, that make them difference, doesnt it?

Well sure, you can distinguish electrons by what state they're in (e.g., spin up vs. spin down), in the sense that you can say "the electron in state 1" and "the electron in state 2" (assuming state 1 and state 2 are different). But the point is you can't distinguish them beyond that. If I had a grey dog and a brown dog, I could point at them and say "that one is the grey dog" and "this one is the brown dog". But if I have a spin up and a spin down electron, the best I can say is "I'm more likely to find the spin up electron over there" and "I'm more likely to find the spin down electron over here." And who is to say the one that was spin up a second ago isn't the one that's spin down now?

Consider two electrons in a head-on collision. Say you have an electron with spin up approaching from the left, and an electron with spin down approaching from the right. Then after the collision you have an electron with spin up leaving to the left, and an electron with spin down leaving to the right. You could say "The electron with spin up bounced off and went back the way it came, and so did the one with spin down." Or you could say "The two electrons passed right by each other, but they each flipped their spins on the way." But the point is that in quantum mechanics you can't say which of these processes "really happened", because there's no difference between the two.

Of course, this is pretending the electrons have a definite position and velocity, when in fact the probability of finding them to have any particular position or velocity is part of the state. So really you might have started with "an electron in spin up that's highly likely to be approaching from the left", and "an electron in spin down that's highly likely to be approaching from the right" -- but there's a small probability of finding the spin up electron over by where the spin down one was, or vice versa. (Of course you can also find them somewhere else altogether.) Another way to say this: We have two terms in our antisymmetric wave function, but when the electrons are far apart then for a certain pair of positions (call them x1 and x2) the amplitude of one term (spin up at x1 and spin down at x2) is much greater than the amplitude of the other term (spin up at x2 and spin down at x1).

However, as the electrons get near each other, the probability of finding the spin up electron where the spin down electron used to be, or vice versa, becomes as great as the probability of finding them where you thought they were. In other words, the amplitudes of the two terms in our antisymmetric function (for whatever x1 and x2 we want to look at) both contribute significantly. At that point, we can't say "we have the spin up particle over here", and "we have the spin down particle over there", because we're just as likely to measure it the other way around. This can be understood as a consequence of the uncertainty principle.

Hopefully you can see what I'm getting at. If we have two electrons in two different states, such as spin up and spin down, we can talk about the electron with spin up and the electron with spin down. But as they get close together, you're just as likely to find the spin up one where you thought the spin down one was, or vice versa -- so how do you know it's the same electron in a given spin state from one moment to the next? You don't, really. So while you can talk about "the electron in state 1" and "the electron in state 2", really you just have two electrons and two internal states distributed among them.

Incidentally, I don't think it's a stupid question at all. I hope my effort to explain things didn't just make it more confusing. --Tim314 17:41, 1 May 2007 (UTC)

"ket" was not a mistake

The article originally read: "Note that if n1 and n2 are the same, our equation for the antisymmetric state gives the zero ket" On Sept 24 2006, a user with IP address 71.106.164.227 changed it to read "gives the zero set". I guess they thought "ket" was a typo. Actually, it's a reference to bra-ket notation. Anyway, I changed it to "which gives zero", which I think will be clearer to the lay reader. Of course if you subtract two equal vectors you get the zero vector, but I think that should be clear anyway. -- Tim314 16:43, 1 May 2007 (UTC)

Helium-4 nuclei are bosons?

I don't doubt the accuracy of this statement, but I don't understand how a combination of fermions could result in a boson. Certainly quarks are fermions and their various combinations, in the form of hadrons, are fermions as well. How can you combine two neutrons and two protons to form a boson? And why is it that helium-3 nuclei are still fermions?

If there's no way to explain this other than hardcore math, then don't bother, since there's not a chance I would understand any of it. If there's a simpler, more general explanation, however, could somebody please help me out? Eebster the Great 01:05, 4 December 2007 (UTC)

Simply put, fermion has spin 0.5, boson has spin 1.0. When you put two fermions together, you get a boson. 0.5+0.5=1.0 --Cubbi 03:34, 4 December 2007 (UTC)

Thanks, I guess I didn't realize the spins would add like that. Does this mean larger nuclei have very large spins? Also, I guess this means multiple alpha particles can occupy the same space (quantum state) even though traditionally they are considered matter? Eebster the Great (talk) 00:47, 5 December 2007 (UTC)

LOL you wanted the simplest possible explanation. No, it is not so simple with large nuclei. They have weird spins, like 7/2 or 5 or even zero, or whatnot. Also, quantum state is not space, it's a kind of a measure of energy. When bosons occupy the same quantum state, you get Bose–Einstein condensate, such as liquid helium-4. --Cubbi (talk) 01:07, 5 December 2007 (UTC)

Single Particle Approximation.

just reading through the section "Wavefunction representation", You guys haven't made it clear that everything done there is in the single particle approximation and is not in general true i.e. in general any wavefunction must be written as an infinite sum of the w.f. you have defined there. —Preceding unsigned comment added by 128.250.54.197 (talk) 03:22, 14 February 2008 (UTC)

N particles equation for totally symmetric state does not compute correctly

According to the equation, for 2 bosons with 1 distinct possible states, the state vector should be

${\begin{aligned}|1,\,1;A\rangle &={\frac {2!}{\sqrt {2!}}}(|1\rangle |1\rangle +|1\rangle |1\rangle )\\&={\sqrt {2}}|1\rangle |1\rangle \\\end{aligned}}\,\!$ 。

This is a factor of ${\sqrt {2}}\,\!$ off. Can someone find out what's the problem? --Thurth 13:43, 30 September 2008 (UTC)

Probably the author copied from the erroneous Landau-Lifshitz. The correct normalizing constant should rather be ${\sqrt {\frac {1}{N!\prod _{j}n_{j}!}}}$ than ${\sqrt {\frac {\prod _{j}n_{j}!}{N!}}}$, at least this is the result of my calculation. --TurionTzukosson (talk) 15:17, 22 February 2010 (UTC)

Statistical effects of indistinguishability

The claim that the (1/N!) prefactor in the definition of the entropy in equilibrium statistical physics finds its explanation in quantum mechanics is unfortunately very common, but nevertheless incorrect. Indeed, the identification between the thermodynamic entropy and the corresponding quantity in stat. mech. comes from a relation involving the differential of the entropy at fixed number of particles. So, from this definition, one can only determine the entropy up to a function of the number of particles. This is true both in classical and quantum statistical physics. If one wishes to have an extensive entropy, this has to be added as an extra assumption... All this is very clearly explained in Jaynes' well-known paper on the Gibbs paradox.--YVelenik 15:32, 10 February 2010 (UTC) —Preceding unsigned comment added by Velenik (talk • contribs)

Composite particles

I removed "which are in precisely the same quantum state" which was an error; note that fr.wiki speaks in this place about internal states: French: ayant le même état interne. If two particles were "precisely in the same quantum state", then they would be identical by definition, not as a phenomenon. Apparently, the author tried to make some remark about intrinsic degrees of freedom such as atomic/molecular energy levels, but this is useless without a proper clarification. If we consider e.g. a hydrogen-like atom at all its energy levels as a quantum particle, then we have to describe it with somewhat complicated Hilbert space, such as a wave function valued in an infinite dimensional Hilbert space or, equivalently, by a separate wave function for each possible combination of quantum numbers. Then, after careful symmetrization/alternation of the tensor power, we get "identical particles" even for all quantum numbers. But if we neglect transitions and consider as a particle, say, an atom at some given energy level, or at fixed principal quantum number, then we have a simpler Hilbert space, similar to one for an elementary particle. Incnis Mrsi (talk) 09:06, 30 August 2011 (UTC)

It's perhaps worth noting that this subtlety impacts even fundamental particles, such as electrons, since electrons also have internal state (spin). A spin-up electron is certainly a distinct and distinguishable kind of particle compared to a spin-down electron, much as an orthohydrogen_-1 molecule is distinguishable from an orthohydrogen₊₁ molecule, or a parahydrogen molecule. Of course in another sense all electrons are the same kind of particle and completely interchangeable -- as long as the internal state is interchanged. Nanite (talk) 11:11, 16 January 2014 (UTC)

I agree with your point that there are no specific clauses about composite particles that are not applicable to elementary ones. But I do not agree with interpretation of
e⁻
_↑ and
e⁻
_↓ as distinct particles: these are distinct states, but the particle is the same. Antisymmetrization by the spin quantum number is not different in any sense from antisymmetrization by the motion degrees of freedom. But if you put electron(s) to a very strong z-axis magnetic field, such that the gap between the two states becomes comparable with its rest energy, then yes, they’ll become “different particles”: you will not be able to apply Pauli equation (and Schrödingerian mechanics at all) to the “electron” because two states will have different rest masses and a transition from
e⁻
_↑ to
e⁻
_↓ will not be different from an ordinary particle decay. Note that these extreme
e⁻
_↑ and
e⁻
_↓ will not be “elementary” anymore:
e⁻
_↓ will be the ground state of an electron together with electromagnetic field, and
e⁻
_↑ will be a resonance. And from the point of view of Dirac’s theory nothing changed but a specific alignment of states: it is our interpretation that did change. Conclusion: there is no fundamental, firm boundary between different states and different flavours, but in certain problems we intuitively distinguish different states of the same particle from really different kinds of particles. Incnis Mrsi (talk) 23:50, 16 January 2014 (UTC)