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## Homework Statement

It was noted that <00|S

_{u1}

^{1}S

_{u2}

^{2}|00> = -ħ

^{2}/4 * u

_{1}⋅u

_{2}.

where u

_{1}and u

_{s}are unit vectors along which the two spin operators are measured and θ is the angle between. |00> is the singlet state that the two electrons are entangled in (corresponding to total spin values ). Prove this relationship using

S

_{u1}

^{1}=S

_{z}

^{1}

and

S

_{u2}

^{2}=cosθ*S

_{z}

^{2}+sinθ*S

_{x}

^{2}

Also use |00>=1/√(2) * (|↓↑>-|↑↓>)

and that S

_{u1}

^{1}and S

_{u2}

^{2}act on particle 1 and 2, respectively.

## Homework Equations

S

_{u2}

^{2}=cosθ*S

_{z}

^{2}+sinθ*S

_{x}

^{2}

Also use |00>=1/√(2) * (|↓↑>-|↑↓>)

## The Attempt at a Solution

I tried to use the matrix representations of these states (i.e. |↑>=(1,0) and |↓>=(0,1)) then writing out

|↓↑>=|↓>*|↑> = (0,1)*(1,0)

and trying to use the respective spin matrices on each particle but I keep getting zero each time I try.

i.e. when I try the first multiplication <00|S

_{u1}

^{1}S

_{u2}

^{2}|

I get -ħ

^{2}/4√(2)* {(0,1)*σ

_{z}*[(1,0)*σ

_{z}cosθ+sinθ*(1,0)*σ

_{x}]-(1,0)*σ

_{z}*[(0,1)*σ

_{z}*cosθ+sinθ*(0,1)*σ

_{x}]}

where the σ are the pauli spin matricies.

my answer for <00|S

_{u1}

^{1}S

_{u2}

^{2}| is then just zero.