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Quantum Spin problem

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  1. Apr 21, 2016 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations

    Su22=cosθ*Sz2+sinθ*Sx2

    Also use |00>=1/√(2) * (|↓↑>-|↑↓>)

    3. 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.
     
  2. jcsd
  3. Apr 21, 2016 #2

    blue_leaf77

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    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.
     
  4. Apr 21, 2016 #3

    TSny

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    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]
     
  5. Apr 21, 2016 #4
    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
     
  6. Apr 21, 2016 #5

    TSny

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    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.
     
  7. Apr 21, 2016 #6

    TSny

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    OK

    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) = - <↓|⊗<↑| = - <↓↑|.
     
  8. Apr 21, 2016 #7

    blue_leaf77

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    The OP seems to be not quite familiar with the direct product, which is necessary if he/she wants to work in the matrix representation. In fact, the OP does not need to resort to the matrix representation if he knows how to proceed with the braket notation (this is what he has been doing actually only that the bras and kets are written in matrix form which can turn out to be misleading).
     
  9. Apr 21, 2016 #8

    TSny

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    I agree. It's easier to stay with the arrow notation as long as the student knows the effect of σx on |↑> or |↓>.
     
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