Direct Product of Two Spin-Up Vectors: What Is It?

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SUMMARY

The direct product of two spin-up vectors, denoted as |up1> x |up2>, results in the state |up1, up2>, which represents an ordered pair in a larger vector space. The notation |1> is used to label the total z angular momentum of this combined state, which equals 1ħ. The discussion clarifies that the direct product does not necessarily imply a larger vector space but rather creates a composite state from the original vectors.

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Hi
I have just started looking at direct products and came across the following which i don't understand :

the direct product of two spin -up vectors = | 1 > which is in a bigger vector space

I don't understand how the direct product is | 1 > ? and in this case is it always a bigger vector space or the same size as the original vector spaces ?
 
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That's just sloppy notation. If you are trying to understand direct product, it's basically just making an ordered pair of the two vectors.
|up1> x |up2> = |up1, up2>
Now clearly, the total z angular momentum of |up1, up2> is 1hbar so we can also label this state |1>. But that's just notation, not math.
 

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