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headbomb

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First, this is my 1st post on physicsforums.com, so I'm not too sure I'm posting this in the right place, but guts tell me this is a good place to start.

Second I'm sure you've all heard this question a million times. I've seen it answered here a couple of times, but what I've seen is of no help. I'm trying to derive various hadronic wavefunctions, but I'm stuck on this one for now (and if I can't do this one, this means I don't really understand what I'm doing for the others even though they seem to work). It is my understanding that baryons are made of quarks, and have 4 degrees of freedom: Space, flavour, spin, and color. The spatial part of the wavefunction is assumed symmetric under particle exchange, and in the case of a uuu baryon, the flavour part is symmetric as well (which is trivial to prove). If this state of the uuu is to be seen, this leaves the spin+colour part to be globally antisymmetric, so the overall function is antisymmetric (per Pauli).

Now everywhere, people invoke Pauli to exclude a spin 1/2 uuu baryon (or any spin 1/2 single-flavoured baryon). But it seems to me that if colour is chosen antisymmetric, then a spin 1/2 config is permitted.

Let r, g, and b denote the three colours.

You have 6 permutations of colours (rgb, rbg, grb, gbr, brg, bgr), so the wavefunction is

|colour> = A |rgb> + B|rbg> + C|grb> + D|gbr> + E|brg> + F|bgr>

Now under the constraints |psi 123> = -|psi 213> (A=-C, B=-E, D=-F) and |psi 123> = -|psi 321> (A=-F, B=-D, C=-E), a solution is possible, namely A=-B=-C=D=E=-F, which can be normalized to A=exp(i\phi)/sqrt{6}. And thus you get a colour wavefunction of

|colour> = exp (i\phi)/sqrt{6} (|rgb> - |rbg> - |grb> + |gbr> + |brg> - |bgr>)

I see this in books everywhere, so this tells me I'm doing something right here (although I'm perhaps seeing it in a different context than here). Now if I do the same for spins in a symmetric spin-1/2 config, I also get something.

Let u and d denoted the spin up and spin down quark configs (

You have three permutations of spin 1/2 (uud, udu, duu), so the wavefunction is

|spin> = A|uud> + B|udu> + C|duu>

Now under the constraints |psi 123> = |psi 213> (A=A, B=C) and |psi 123> = |psi 321> (A=C, B=B), a solution is possible, namely A=B=C, which can be normalized to exp(i/phi)/sqrt{3}.

|spin> = exp(i\phi)/sqrt{3} (|uud> + |udu> + |duu>)

Now this puzzles me a great deal. I don't see anything wrong with what I did above, but yet this contradicts the explanation for the lack of uuu baryons in spin 1/2 config.

I thought that I did something wrong above, because if colour is antisymmetric, and spin symmetric, then this

Second I'm sure you've all heard this question a million times. I've seen it answered here a couple of times, but what I've seen is of no help. I'm trying to derive various hadronic wavefunctions, but I'm stuck on this one for now (and if I can't do this one, this means I don't really understand what I'm doing for the others even though they seem to work). It is my understanding that baryons are made of quarks, and have 4 degrees of freedom: Space, flavour, spin, and color. The spatial part of the wavefunction is assumed symmetric under particle exchange, and in the case of a uuu baryon, the flavour part is symmetric as well (which is trivial to prove). If this state of the uuu is to be seen, this leaves the spin+colour part to be globally antisymmetric, so the overall function is antisymmetric (per Pauli).

Now everywhere, people invoke Pauli to exclude a spin 1/2 uuu baryon (or any spin 1/2 single-flavoured baryon). But it seems to me that if colour is chosen antisymmetric, then a spin 1/2 config is permitted.

Let r, g, and b denote the three colours.

You have 6 permutations of colours (rgb, rbg, grb, gbr, brg, bgr), so the wavefunction is

|colour> = A |rgb> + B|rbg> + C|grb> + D|gbr> + E|brg> + F|bgr>

Now under the constraints |psi 123> = -|psi 213> (A=-C, B=-E, D=-F) and |psi 123> = -|psi 321> (A=-F, B=-D, C=-E), a solution is possible, namely A=-B=-C=D=E=-F, which can be normalized to A=exp(i\phi)/sqrt{6}. And thus you get a colour wavefunction of

|colour> = exp (i\phi)/sqrt{6} (|rgb> - |rbg> - |grb> + |gbr> + |brg> - |bgr>)

I see this in books everywhere, so this tells me I'm doing something right here (although I'm perhaps seeing it in a different context than here). Now if I do the same for spins in a symmetric spin-1/2 config, I also get something.

Let u and d denoted the spin up and spin down quark configs (

*not*the up and down quarks).You have three permutations of spin 1/2 (uud, udu, duu), so the wavefunction is

|spin> = A|uud> + B|udu> + C|duu>

Now under the constraints |psi 123> = |psi 213> (A=A, B=C) and |psi 123> = |psi 321> (A=C, B=B), a solution is possible, namely A=B=C, which can be normalized to exp(i/phi)/sqrt{3}.

|spin> = exp(i\phi)/sqrt{3} (|uud> + |udu> + |duu>)

Now this puzzles me a great deal. I don't see anything wrong with what I did above, but yet this contradicts the explanation for the lack of uuu baryons in spin 1/2 config.

I thought that I did something wrong above, because if colour is antisymmetric, and spin symmetric, then this

*is**overall antisymmetric. Which books and reality tells me is impossible. So I thought that perhaps you have to deal with the "colourspin" wavefunction in order to get the proper result. However I still get something.*

You have three permutations of spin (uud, udu, duu) and six permutations of colour (rgb, rbg, grb, gbr, brg, bgr) so you have a wavefunction with 3 x 6 = 18 parts before symmetrisation:

|psi> =

+ A|uud/rgb> + B|uud/rbg> +C|uud/grb> +D|uud/gbr> +E|uud/brg> +F|uud/bgr>

+ G|udu/rgb> + H|udu/rbg> +I|udu/grb> +J|udu/gbr> +K|udu/brg> +L|udu/bgr>

+ M|duu/rgb> + N|duu/rbg> +O|duu/grb> +P|duu/gbr> +Q|duu/brg> +R|duu/bgr>

Under the |psi 123> = -|psi 213> constraint, you have

A=-C, B=-E, D=-F, G=-O, H=-Q, I=-M, J=-R, K=-N, L=-P

Under the |psi 123> = -|psi 321> constraint, you have the following conditions on the A-R constants:

A=-R, B=-P, C=-Q, D=-N, E=-O, F=-M, G=-L, H=-J, I=-K

Which is basically French for

A=-C=Q=-H=J=-R;

B=-E=O=-G=L=-P;

D=-F=M=-I=K=-N

So you get

|psi> =

+A ( |uud/rgb> - |uud/grb> + |duu/brg> - |udu/rbg> + |udu/gbr> - |duu/bgr> )

+B ( |uud/rbg> - |uud/brg> + |duu/grb> - |udu/rgb> + |udu/bgr> - |duu/gbr> )

+D ( |uud/gbr> - |uud/bgr> + |duu/rgb> - |udu/grb> + |udu/brg> - |duu/rbg> )

with A, B, and D chosen so A*A + B*B +D*D = 18. I'm also kinda worried that I can't get a unique A:B:D ratio.

Now it strikes me that between a scenario where I'm wrong because of a stupid mistake or me not seeing things through the correct glasses, and the scenario where all textbooks and web resources all being wrong on this, reality would choose the former over the latter.

So what is it I'm not seeing or what is it I'm doing wrong? I'm greatly puzzled by this.You have three permutations of spin (uud, udu, duu) and six permutations of colour (rgb, rbg, grb, gbr, brg, bgr) so you have a wavefunction with 3 x 6 = 18 parts before symmetrisation:

|psi> =

+ A|uud/rgb> + B|uud/rbg> +C|uud/grb> +D|uud/gbr> +E|uud/brg> +F|uud/bgr>

+ G|udu/rgb> + H|udu/rbg> +I|udu/grb> +J|udu/gbr> +K|udu/brg> +L|udu/bgr>

+ M|duu/rgb> + N|duu/rbg> +O|duu/grb> +P|duu/gbr> +Q|duu/brg> +R|duu/bgr>

Under the |psi 123> = -|psi 213> constraint, you have

A=-C, B=-E, D=-F, G=-O, H=-Q, I=-M, J=-R, K=-N, L=-P

Under the |psi 123> = -|psi 321> constraint, you have the following conditions on the A-R constants:

A=-R, B=-P, C=-Q, D=-N, E=-O, F=-M, G=-L, H=-J, I=-K

Which is basically French for

A=-C=Q=-H=J=-R;

B=-E=O=-G=L=-P;

D=-F=M=-I=K=-N

So you get

|psi> =

+A ( |uud/rgb> - |uud/grb> + |duu/brg> - |udu/rbg> + |udu/gbr> - |duu/bgr> )

+B ( |uud/rbg> - |uud/brg> + |duu/grb> - |udu/rgb> + |udu/bgr> - |duu/gbr> )

+D ( |uud/gbr> - |uud/bgr> + |duu/rgb> - |udu/grb> + |udu/brg> - |duu/rbg> )

with A, B, and D chosen so A*A + B*B +D*D = 18. I'm also kinda worried that I can't get a unique A:B:D ratio.

Now it strikes me that between a scenario where I'm wrong because of a stupid mistake or me not seeing things through the correct glasses, and the scenario where all textbooks and web resources all being wrong on this, reality would choose the former over the latter.

So what is it I'm not seeing or what is it I'm doing wrong? I'm greatly puzzled by this.

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