2-particle Wavefunction and the Pauli Exclusion Principle

[Isaac0427]

Say you have two particles a and b with respective positions $x_a$ and $x_b$. Particle a is in the state $\psi_a$, and particle b is in the state $\psi_b$. If they are distinguishable, the wavefunction is
$$\psi=\psi_a(x_a)\psi_b(x_b)$$
However, if they are identical fermions, the wavefunction is
$$\psi(a,b)=A\left(\psi_a(x_a)\psi_b(x_b)-\psi_a(x_b)\psi_b(x_a)\right)$$
If $\psi_a=\psi_b$ then $\psi=0$. But, if $\psi_a$ is arbitrarily close to $\psi_b$ (i.e. $\psi_a=\psi_1$ and $\psi_b=\sqrt{.99}\psi_1+.1\psi_2$ where 1 and 2 denote the 1st and 2nd principal quantum numbers of some potential) then the wavefunction is not zero, and can be normalized if A is arbitrarily large. So, it would seem as though you could have an infinite number of identical fermions in just two energy levels. I know, however, that is not the case. So, I came up with a thought on why this is, but it had a problem. Could tell me if I'm on the right track, and where I am going wrong?

The constant A in the identical fermion equation must be $\frac{1}{\sqrt{2}}$ (for N particles it would be $\frac{1}{\sqrt{N!}}$). This would require that $\psi_a$ and $\psi_b$ (and any other wavefunctions corresponding to more identical fermions) be orthogonal in order to keep $\psi$ normalized. But, this has two issues. First, as a and b get farther apart, the second identical fermions start to act distinguishable. So, $\psi_a(x_b)$ and $\psi_b(x_a)$ go to zero, and $\psi$ goes to $\frac{1}{\sqrt{2}}\psi_a(x_a)\psi_b(x_b)$, which is not normalized (as well as not the same as the distinguishable particle equation).

I also had a thought on that; my thinking was that even as a and b get arbitrarily far apart, in theory they are still identical, and particle a being in in state $\psi_a$ and particle b being in $\psi_a$ are the same things, as the fermions are indistinguishable. Thus, $\psi_a(x_b)\psi_b(x_a)$ does not go to zero. But, since $\psi_a$ and $\psi_b$ do not overlap, they are essentially orthogonal. It appears as though this would make the identical fermion equation yield about the same practical result as the distinguishable equation.

Again, this is just my thinking, I have no idea how far off I am.

Thanks in advance!

 

[mikeyork]

Trying to discuss the spin-statistics theorem without including spin dependence in your wave functions is a sure recipe for misunderstanding and error.

Then once you have spin in the system you need to be very careful how you define the frames of reference in which you measure the z-component as a $2\pi$ rotation on just one of the particles can change the sign of a fermion wave function.

Until you get that clear, any attempt to answer your question will create more confusion.

 

[Isaac0427]

Trying to discuss the spin-statistics theorem without including spin dependence in your wave functions is a sure recipe for misunderstanding and error.

I understand that spin dependence is a factor too—but it still yields the same orthogonality question.

 

[mikeyork]

I understand that spin dependence is a factor too—but it still yields the same orthogonality question.

If you rely on an anti-symmetrization rule you need to be clear about where it comes from and exactly how it applies.

BTW you have actually hit on a really important question which is how do we move without a sudden jump from distinguishable states to indistinguishable states.

 

[Isaac0427]

If you rely on an anti-symmetrization rule you need to be clear about where it comes from and exactly how it applies.

Okay, I know that fermions, by definition, have antisymetric wave functions under exchange. They also happen to have half-integer spin.

BTW you have actually hit on a really important question which is how do we move without a sudden jump from distinguishable states to indistinguishable states.

The answer to this may also help me with my first question. Is there a good explanation for this somewhere?

 

[mikeyork]

Okay, I know that fermions, by definition, have antisymetric wave functions under exchange. They also happen to have half-integer spin.

They are anti-symmetric only with a specific choice of spin reference frames. See

``Symmetrizing The Symmetrization Postulate'', AIP Proceedings 545 (2000) ``Spin Statistics Connection And Commutation Relations'', ed. Hilborn and Tino, pp 104-110. (http://arxiv.org/pdf/quant-ph/0006101.pdf)

and

http://arxiv.org/pdf/quant-ph/9908078v2.pdf

where you will also find references to similar papers by other authors.

The answer to this may also help me with my first question. Is there a good explanation for this somewhere?

Yes, see the papers above. The point is that the conventional antisymmetry is due to a hidden permutation-dependent geometrical asymmetry that introduces a $2\pi$ rotation on one particle's spin frame when they are permuted. When that asymmetry is removed, the wavefunctions are symmetric for fermions as well as bosons and so the problem disappears. If the particle states are defined in a way that is completely symmetrical (permutation-independent) then the transition from distinguishable states to indistinguishable states is smooth.

To my knowledge, you won't find a reasonable answer to this question of a discontinuity in going from distinguishable to indistinguishable states anywhere else.

 

[PeterDonis]

as a and b get farther apart, the second identical fermions start to act distinguishable

What makes you think this? By which I mean, what experimental evidence?

 

[PeterDonis]

If they are distinguishable, the wavefunction is

...

However, if they are identical fermions, the wavefunction is

...

As @mikeyork has implicitly pointed out, these wave functions are not correct. You are assuming that the position part of the wave function must be antisymmetric, but that is not the case. Only the whole wave function has to be antisymmetric; but the wave function includes a spin part as well as a position part. So there are two possibilities: position antisymmetric and spin symmetric, or position symmetric and spin antisymmetric. (A good sidebar question: why are these the two possibilities?) It might be helpful to write out these two possibilities explicitly.

 

[Swamp Thing]

mikeyork, are you M J York, the author of the papers you referred to?

 

[Isaac0427]

What makes you think this? By which I mean, what experimental evidence?

Many textbooks state it, see Griffiths chapter 5.

As @mikeyork has implicitly pointed out, these wave functions are not correct. You are assuming that the position part of the wave function must be antisymmetric, but that is not the case. Only the whole wave function has to be antisymmetric; but the wave function includes a spin part as well as a position part. So there are two possibilities: position antisymmetric and spin symmetric, or position symmetric and spin antisymmetric. (A good sidebar question: why are these the two possibilities?) It might be helpful to write out these two possibilities explicitly.

If $\psi_a$ and $\psi_b$ are spinors, wouldn’t that issue be fixed?

If so, am I still correct about the one-over-root-2 term forcing the two states to be orthogonal?

Also, if I am getting this correct, separated identical fermions are always indestinguushable, just at a point thinking of them as distinguishable is a valid approximation. Is this correct or off-the-mark?

 

[bhobba]

Many textbooks state it, see Griffiths chapter 5.

I do have a copy of Griffiths and I could scan over chapter 5, but I can assure you are misinterpreting what he is saying. That being the case best to re-read it and post his EXACT wording if you are still in any doubts.

If so, am I still correct about the one-over-root-2 term forcing the two states to be orthogonal?

It's the difference between orthogonal and orthonormal. I suspect you haven't studied Linear algebra yet - that should be corrected. Once you do that the answer is obvious. Of course I could tell you - but then I would hvve to kill you . Seriously you learn and understand better what you sort out yourself.

Thanks
Bill

 

[Isaac0427]

It's the difference between orthogonal and orthonormal. I suspect you haven't studied Linear algebra yet - that should be corrected. Once you do that the answer is obvious. Of course I could tell you - but then I would hvve to kill you . Seriously you learn and understand better what you sort out yourself.

I apologize-I worded that question poorly. I understand a reasonable amount of linear algebra. And I understand that the one-over-root-2 keeps a superposition normalized if the super posed wavefunctions are orthogonal (I am pretty sure). My question was if the one-over-root-2 term has to be in the equation for identical fermions, which would mean that identical fermions would have to have orthogonal wavefunctions.

 

[PeterDonis]

best to re-read it and post his EXACT wording if you are still in any doubts.

Exactly. (Pun sort of intended. )

 

[Isaac0427]

Well, I do see where I got confused. Griffiths did say that you could pretend that the particles are distinguishable for practical matters. I’m assuming then that it is not really the limit of the identical fermion equation, it is just another way look at the situation that yields the same practical result.

 

[mikeyork]

mikeyork, are you M J York, the author of the papers you referred to?

Yes.

 

[mikeyork]

Also, if I am getting this correct, separated identical fermions are always indestinguushable, just at a point thinking of them as distinguishable is a valid approximation. Is this correct or off-the-mark?

Identical particles, by themselves, are intrinsically indistinguishable by definition. However, their space-time or energy-momentum states may be distinguishable.

 

[Isaac0427]

Identical particles, by themselves, are intrinsically indistinguishable by definition. However, their space-time or energy-momentum states may

Right, but theoretically for any case saying that particle a is in the state $\psi_a$ is essentially the same as saying that particle b is in that state, as particles a and b are identical (i.e. it doesn’t make sense to talk about particle a or particle b as you need to talk about particle a and particle b because they are identical). Is that correct?

However, if $\psi_a$ and $\psi_b$ do not overlap, you can say that one particle is in the state $\psi_a$ and one particle is in the state $\psi_b$, and get the same practical results as if you had made the particles identical fermions. Is this correct as well?

 

[PeterDonis]

theoretically for any case saying that particle a is in the state $\psi_a$ is essentially the same as saying that particle b is in that state, as particles a and b are identical

No, that's not what "identical particles" means. What it means is that you can't say "particle a is in state $\psi_a$" at all (or particle b). All you can say is "a particle is in state $\psi_a$ and a particle is in state $\psi_b$". Or, more precisely, that you have a quantum system consisting of two identical particles, with one in state $\psi_a$ and one in state $\psi_b$. Labeling the particles as "particle a" and "particle b" is an artifact of the way we model such systems mathematically; it doesn't correspond to anything in the system itself.

If the two identical particles are fermions, you can also say something else: that it is impossible to have a state of the two-particle quantum system in which both particles are in state $\psi_a$. (Note that the state here has to include all the degrees of freedom of the system, both position and spin, and in the general case there might be others as well.) Mathematically, this implies that the wave function of the system must be antisymmetric.

 

[Isaac0427]

No, that's not what "identical particles" means. What it means is that you can't say "particle a is in state ψaψa\psi_a" at all (or particle b). All you can say is "a particle is in state ψaψa\psi_a and a particle is in state ψbψb\psi_b". Or, more precisely, that you have a quantum system consisting of two identical particles, with one in state ψaψa\psi_a and one in state ψbψb\psi_b. Labeling the particles as "particle a" and "particle b" is an artifact of the way we model such systems mathematically; it doesn't correspond to anything in the system itself.

I guess that is sort-of what I meant (though using very incorrect wording). I meant that you can't say anything about particle a being in a state or particle b being in a state as you can only talk about "some particle," and you cannot draw distinctions between a and b.

If the two identical particles are fermions, you can also say something else: that it is impossible to have a state of the two-particle quantum system in which both particles are in state ψaψa\psi_a. (Note that the state here has to include all the degrees of freedom of the system, both position and spin, and in the general case there might be others as well.) Mathematically, this implies that the wave function of the system must be antisymmetric.

Again, though, do the states just have to be different or orthogonal? If they have to be orthogonal, why would that be, mathematically. If they don't, then couldn't an infinite number of identical fermions all be in the ground energy level with slightly different spin-up and spin-down superpositions?

 

[PeterDonis]

do the states just have to be different or orthogonal?

The short answer is "just different". But I think it's worth expanding on that.

First, there are no "states" of the individual particles, strictly speaking. There is only the state of the 2-particle system. The Hilbert space of states for the system will have a basis of orthogonal vectors, but those are state vectors for the 2-particle system, not for either particle in isolation.

One way to construct the Hilbert space of the 2-particle system (for fermions) is to start with the Hilbert space for a one-particle system and then form an antisymmetric tensor product of two copies of that Hilbert space. (For bosons, you would form the symmetric tensor product.) Similarly, you can construct a basis for the Hilbert space of the 2-particle system by starting with a basis for the one-particle system (which would consist of a set of mutually orthogonal vectors in that Hilbert space) and forming antisymmetric combinations of those vectors. That's basically what you were getting at in the OP of this thread.

The basis states of the 2-particle Hilbert space will be mutually orthogonal, as I said above. And if you express them the way I just described, they will each look like linear combinations of terms that are each products of two 1-particle basis vectors, and in each such product, the two 1-particle basis vectors will be different (since the combinations are antisymmetric), so they will be orthogonal. However, that is not the same as saying the two "one particle states" that make up the two-particle state (i.e., the two states that appear in the description "one particle is in state $\psi_a$ and one particle is in state $\psi_b$") must be orthogonal. That is not the case. They only need to be different.

The example you picked in the OP might obscure this because you only considered two possible positions. More generally, you were considering a discrete spectrum of states (which of course you have to if you are considering bound systems). For a discrete spectrum of states, there is no real difference, for the purposes of this discussion, between "different" and "orthogonal": heuristically, there are only a finite number of "buckets" into which you can put fermions. See further comments below.

couldn't an infinite number of identical fermions all be in the ground energy level with slightly different spin-up and spin-down superpositions?

No, because you are, again, talking about a discrete spectrum of states. Consider the difference between bound fermions (electrons in an atom, say) and free fermions (e.g., electrons moving through free space). In the latter case, you could indeed have an infinite number of identical fermions all with the same energy but slightly different superpositions of other degrees of freedom. (And of course there are also a continuous infinity of possible energies.)

It's also worth noting that so far we have not considered entangled states; considering them complicates all this even further. For example, here is a description of an entangled state: "there is a fermion at position $x_a$ and a fermion at position $x_b$, and their spins are opposite". You might want to think about how this state would be written down mathematically, as compared with a non-entangled state such as "there is a fermion at position $x_a$ and a fermion at position $x_b$" (with nothing specified about their spins).

 

[Isaac0427]

For a discrete spectrum of states, there is no real difference, for the purposes of this discussion, between "different" and "orthogonal": heuristically, there are only a finite number of "buckets" into which you can put fermions. See further comments below.

This I do not get. The spinors $\begin{pmatrix} 0\\ 1\\ \end{pmatrix}$ and $\begin{pmatrix} .1\\ \sqrt{.99}\\ \end{pmatrix}$ are different but not orthogonal. And if two identical fermions in the same ground state of the same atom can occupy those states, because they are different, then wouldn’t any number of electrons be allowed to be in that same ground state as long as their spin states are a little different?

 

[PeterDonis]

And if two identical fermions in the same ground state of the same atom can occupy those states

That's not what I said. The key thing you need to start with is that you do not have two identical fermions occupying one-particle states. You have one quantum system composed of two identical fermions, which has one state. If you want to write down such a state mathematically, you have to write it under the constraints I described for such states. Writing down a bunch of one-particle states and asking whether two identical fermions can be in such states is meaningless; that's not the quantum system we're talking about.

If you actually try to write down a valid state for a quantum system consisting of two identical fermions in a single energy level of a bound system, you will find that you cannot write down a state that could be described in words as "one particle is in the state $\begin{pmatrix} 0\\ 1\\ \end{pmatrix}$ and one particle is in the state $\begin{pmatrix} .1\\ \sqrt{.99}\\ \end{pmatrix}$". There is no such state that satisifies the constraints. By contrast, if you try to write down a valid state for a quantum system consisting of two identical fermions in free space that both have the same energy, you will find that you can write down such a state. But you actually have to go through the exercise of trying to write down actual states of the actual quantum system under discussion; just naming one-particle spin states won't do it.

 

[Isaac0427]

That's not what I said. The key thing you need to start with is that you do not have two identical fermions occupying one-particle states. You have one quantum system composed of two identical fermions, which has one state. If you want to write down such a state mathematically, you have to write it under the constraints I described for such states. Writing down a bunch of one-particle states and asking whether two identical fermions can be in such states is meaningless; that's not the quantum system we're talking about.

If you actually try to write down a valid state for a quantum system consisting of two identical fermions in a single energy level of a bound system, you will find that you cannot write down a state that could be described in words as "one particle is in the state $\begin{pmatrix} 0\\ 1\\ \end{pmatrix}$ and one particle is in the state $\begin{pmatrix} .1\\ \sqrt{.99}\\ \end{pmatrix}$". There is no such state that satisifies the constraints. By contrast, if you try to write down a valid state for a quantum system consisting of two identical fermions in free space that both have the same energy, you will find that you can write down such a state. But you actually have to go through the exercise of trying to write down actual states of the actual quantum system under discussion; just naming one-particle spin states won't do it.

Then I guess I’m completely lost here. Pretty much every source talks about the state of identical fermions as $$\psi (x_a,x_b)=A(\psi_a(x_a)\psi_b(x_b)-\psi_a(x_b)\psi_b(x_a))$$ In this equation $\psi_a$ and $\psi_b$ are single-particle states.

 

[PeroK]

Then I guess I’m completely lost here. Pretty much every source talks about the state of identical fermions as $$\psi (x_a,x_b)=A(\psi_a(x_a)\psi_b(x_b)-\psi_a(x_b)\psi_b(x_a))$$ In this equation $\psi_a$ and $\psi_b$ are single-particle states.

Maybe you can figure this out for yourself. Hint: In that two-particle state, ask yourself this question:

What is the state of particle a?

 

[Isaac0427]

Maybe you can figure this out for yourself. Hint: In that two-particle state, ask yourself this question:

What is the state of particle a?

It is in the two-particle state, right? I’m not entirely sure what you are asking though.

 

[PeroK]

It is in the two-particle state, right? I’m not entirely sure what you are asking though.

I might have answered that there is no particle a.

 

[Isaac0427]

I might have answered that there is no particle a.

Right. It’s just a fermion and cannot be identified among other identical fermions in the system. So if you ignore my poor wording, when I talked about the state of particle a I meant $\psi_a$ and when I talked about the state of particle b I meant $\psi_b$. I am still completely lost as to how these states do not exist.

 

[PeterDonis]

In this equation $\psi_a$ and $\psi_b$ are single-particle states

Yes, but they occur as factors in products, and the state as a whole is a sum of those products. So there are no actual single-particle states, physically, for this system; they are just artifacts of the mathematical way we represent the state. So are the labels $a$ and $b$. You can't understand what's going on if you focus on these artifacts of the math and don't think about the overall way the state is constructed. You have to actually start from a valid basis for the Hilbert space of the two-particle system, and then see what states you can write down as valid linear combinations of those basis states. So the first question you need to ask is, what is a valid basis for the Hilbert space of the two-particle system? (I described how to construct one in a previous post, but you should not take my description for granted; you should construct the basis yourself.)

 

[Isaac0427]

Yes, but they occur as factors in products, and the state as a whole is a sum of those products. So there are no actual single-particle states, physically, for this system; they are just artifacts of the mathematical way we represent the state. So are the labels $a$ and $b$. You can't understand what's going on if you focus on these artifacts of the math and don't think about the overall way the state is constructed. You have to actually start from a valid basis for the Hilbert space of the two-particle system, and then see what states you can write down as valid linear combinations of those basis states. So the first question you need to ask is, what is a valid basis for the Hilbert space of the two-particle system? (I described how to construct one in a previous post, but you should not take my description for granted; you should construct the basis yourself.)

I do not see how my example is invalid.

 

[PeterDonis]

I do not see how my example is invalid.

That's because you're not doing what I asked you to do. You are just waving your hands and writing down two one-particle states and assuming that there is some valid state in the Hilbert space of the two-particle system which can be described as one particle being in each of these states. But just waving your hands doesn't show that your assumption is correct. You have to actually show it, by constructing a valid basis for the Hilbert space of the two-particle system and then writing down the state you think you are talking about in that basis. You have not done that. And I think that until you try, you will continue to be confused.

 

[Swamp Thing]

The two "one particle states" that make up the two-particle state ... [don't need to be orthogonal]. They only need to be different.

That means a tiny and experimentally trivial difference could qualify as "different"? So that you might need a really long run of identical experiments to bring out that tiny difference in state and prove that they are not violating Pauli's principle?

 

[PeterDonis]

That means a tiny and experimentally trivial difference could qualify as "different"?

If there is a continuous degree of freedom involved, yes. For example, two identical fermions could differ in position by a very small amount, while being in identical spin states, without violating the Pauli exclusion principle. However, if the two identical fermions had exactly the same position, then they could not have spin states that differed by a very small amount, because the spin degree of freedom is not continuous.

 

[Isaac0427]

If you actually try to write down a valid state for a quantum system consisting of two identical fermions in a single energy level of a bound system, you will find that you cannot write down a state that could be described in words as "one particle is in the state (01)(01)\begin{pmatrix} 0\\ 1\\ \end{pmatrix} and one particle is in the state (.1√.99)(.1.99)\begin{pmatrix} .1\\ \sqrt{.99}\\ \end{pmatrix}". There is no such state that satisifies the constraints. By contrast, if you try to write down a valid state for a quantum system consisting of two identical fermions in free space that both have the same energy, you will find that you can write down such a state. But you actually have to go through the exercise of trying to write down actual states of the actual quantum system under discussion; just naming one-particle spin states won't do it.

Can you help me with this? I am not exactly sure what constraints you are talking about.

 

[PeterDonis]

I am not exactly sure what constraints you are talking about.

Have you tried to answer the questions I have been asking? Start with this one: what is a valid basis of the Hilbert space of the quantum system consisting of two identical fermions?

 

[Isaac0427]

Have you tried to answer the questions I have been asking? Start with this one: what is a valid basis of the Hilbert space of the quantum system consisting of two identical fermions?

I really don’t know what you are asking me right now. Can I have some help arriving at the answer? I understand what a Hilbert space is, but I always thought that a valid basis would be a spin up state and a spin down state, or if we are talking about energy, the ground state, the first excited state, the second excited state, etc.