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I'm totally confused on the relationship between kets written

[tex]| \uparrow \rangle, | \downarrow \rangle[/tex]

and

[tex]| \uparrow \uparrow \rangle, | \uparrow \downarrow \rangle | \downarrow \uparrow \rangle, | \downarrow \downarrow \rangle[/tex]

(

(

[tex] |s=0,m=0\rangle = \frac{1}{\sqrt{2}} (| \uparrow \downarrow \rangle - | \downarrow \uparrow \rangle) [/tex]

...but I'd like to illustrate it using bra-ket notation. The solution says to project [tex] | \uparrow \rangle \langle \uparrow | [/tex] onto the possible eigenstates of S_z for the first particle

[tex] (| \uparrow \rangle \langle \uparrow | + | \downarrow \rangle \langle \downarrow |)[/tex],

to get...

[tex]| \uparrow \uparrow \rangle \langle \uparrow \uparrow | + | \uparrow \downarrow \rangle \langle \uparrow \downarrow |[/tex]

First off, "projecting" here means matrix multiplication, right? (or direct product?) Second, how do we move from kets representing spin for one particle, to kets representing spin for two particles?

All help in understanding this is greatly appreciated!

[tex]| \uparrow \rangle, | \downarrow \rangle[/tex]

and

[tex]| \uparrow \uparrow \rangle, | \uparrow \downarrow \rangle | \downarrow \uparrow \rangle, | \downarrow \downarrow \rangle[/tex]

(

**Problem**) I have a system of two spin-1/2 particles, prepared in the singlet state, and I'd like to find the probability of measuring the first one and finding it to be spin-up. This being a homework problem from last week, I have the provided solution, but I don't understand it.(

**Attempt at Solution**) I*know*it is 50%, simply because no other measurements have been made of the system, and there's an equal chance of it being up or down, given that the singlet state can be described as:[tex] |s=0,m=0\rangle = \frac{1}{\sqrt{2}} (| \uparrow \downarrow \rangle - | \downarrow \uparrow \rangle) [/tex]

...but I'd like to illustrate it using bra-ket notation. The solution says to project [tex] | \uparrow \rangle \langle \uparrow | [/tex] onto the possible eigenstates of S_z for the first particle

[tex] (| \uparrow \rangle \langle \uparrow | + | \downarrow \rangle \langle \downarrow |)[/tex],

to get...

[tex]| \uparrow \uparrow \rangle \langle \uparrow \uparrow | + | \uparrow \downarrow \rangle \langle \uparrow \downarrow |[/tex]

First off, "projecting" here means matrix multiplication, right? (or direct product?) Second, how do we move from kets representing spin for one particle, to kets representing spin for two particles?

All help in understanding this is greatly appreciated!

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