Baryons and their quark constituents

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The discussion centers on the properties of baryons and their quark constituents, specifically addressing the spin of baryons like delta +1 and protons. It clarifies that spins are not simply additive in the traditional sense; rather, they are represented as spinors, leading to different addition rules. A system of three spin 1/2 particles can yield total spins of either 1/2 or 3/2, depending on their alignment. The confusion arises from the misconception that spins can be summed numerically, which is incorrect. Understanding these principles is crucial for demonstrating how baryons derive their properties from quarks.
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I have been given an assingment where I must show that the properties of the up, down and strange quarks follow from the properties of the baryons. I am just wondering why delta +1 (a baryon) has a spin 3/2 when it is, like a proton, composed of up, up, down. Aren't spins additive? and how do all of the 3/2 and 1/2 spin baryons follow from the combination of the up, down, and strange quarks alone? I can count only 10 combinations of those 3 types.
 
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If you think spins are additive in the sense of 1/2+1/2+1/2=3/2 you would be wrong. And if you do think this you should be wondering why the proton has spin 1/2 rather than 3/2. A system of three spin 1/2 particles can be in a total spin state of either 1/2 or 3/2.
 
spins are "additive" but they are not numbers, they are spinors. addition rules are different.. but still makes a lot sense, eg. spin +1/2 and spin -1/2 adds to spin 0 (this is just means the spins of the two constitutents are anti-align. so with 3 spins states, you can have \pm 1/2 \text{or} \pm 3/2 that means you get a spin 1/2 or 3/2 particle.
 

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