# Quantum Spin problem

## Homework Statement

It was noted that <00|Su11Su22|00> = -ħ2/4 * u1⋅u2.

where u1 and us 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

Su11=Sz1

and

Su22=cosθ*Sz2+sinθ*Sx2

Also use |00>=1/√(2) * (|↓↑>-|↑↓>)

and that Su11 and Su22 act on particle 1 and 2, respectively.

## Homework Equations

Su22=cosθ*Sz2+sinθ*Sx2

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|Su11Su22|

I get -ħ2/4√(2)* {(0,1)*σz*[(1,0)*σzcosθ+sinθ*(1,0)*σx]-(1,0)*σz*[(0,1)*σz*cosθ+sinθ*(0,1)*σx]}

where the σ are the pauli spin matricies.

my answer for <00|Su11Su22| is then just zero.

blue_leaf77
Homework Helper
writing out

|↓↑>=|↓>*|↑> = (0,1)*(1,0)
That's not the way you translate a composite system into the matrix representation. Two or more systems paired together to form a new vector space becomes a single vector space and all vectors in this space must admit the usual column matrix notation. Here you are pairing two two-dimensional vector spaces, the result is a 4 dimensional vector space and the state vector in there will be a 4 elements column matrix.

TSny
Homework Helper
Gold Member
I get -ħ2/4√(2)* {(0,1)*σz*[(1,0)*σzcosθ+sinθ*(1,0)*σx]-(1,0)*σz*[(0,1)*σz*cosθ+sinθ*(0,1)*σx]}

I think this is OK, but the notation is a bit intimidating. I don't think you should get zero.

For example, what do you get when you simplify this part: (0,1)*σz*[(1,0)*σzcosθ]

This will not cancel out with the other cosθ term: -(1,0)*σz*[(0,1)*σz*cosθ]

I think this is OK, but the notation is a bit intimidating. I don't think you should get zero.

For example, what do you get when you simplify this part: (0,1)*σz*[(1,0)*σzcosθ]

This will not cancel out with the other cosθ term: -(1,0)*σz*[(0,1)*σz*cosθ]
I get zero for both terms, with σz=[(1,0),(0,-1)] (the 2x2 form)

(0,1)*[(1,0),(0,-1)] =(0,-1)

then (0,-1)*(1,0)=0

TSny
Homework Helper
Gold Member
That's not the way you translate a composite system into the matrix representation.

I believe Cracker Jack is just using direct product notation: |↓↑>=|↓>*|↑> means |↓↑>=|↓>⊗|↑> = (0,1)T⊗(1,0)T where Cracker Jack did not bother to write the transpose symbols to indicate column vectors.

TSny
Homework Helper
Gold Member
with σz=[(1,0),(0,-1)] (the 2x2 form)

(0,1)*[(1,0),(0,-1)] =(0,-1)
OK

then (0,-1)*(1,0)=0
No. You cannot multiply these two vectors like this. The two "factors" correspond to different particles.
The first factor is (0, -1) = -(0, 1) = - <↓| and refers to particle 1. The second factor is (1, 0) = <↑| and refers to particle 2.
So, (0,-1)*(1,0) = - <↓|⊗<↑| = - <↓↑|.

blue_leaf77