Spin expectation value of singlet state from two axes

[bencmier]

Homework Statement

Homework Equations

The Attempt at a Solution

I am just trying to figure out how to start the problem. Any help would be greatly appreciated.

 

[Sourabh N]

Start by writing S₁ and S₂ in terms of the pauli matrices.

 

[bencmier]

Start by writing S₁ and S₂ in terms of the pauli matrices.

Would S₁=S_z and S₂ have cos(θ) instead of 1's in the matrix?

 

[Sourabh N]

Yes, but you need to be careful about the sign, whether it is cos(θ) or -cos(θ), since the question says -"makes an angle θ down with the z axis".

 

[bencmier]

Yes, but you need to be careful about the sign, whether it is cos(θ) or -cos(θ), since the question says -"makes an angle θ down with the z axis".

Ok, but what does the question mean by picking a single basis to work in? I just don't know what the question is saying.

 

[Sourabh N]

I believe they are trying to tell you to have both S_1 and S_2 in the same basis, i.e, either choose S_1 to lie along z-axis and S_2 to lie theta away from it, or choose S_2 to lie along z-axis and S_1 to lie -theta away from it.

Since they already chose the first of these options, the hint is redundant (as the author say so themselves!).

 

[andrien]

may be it will be clear if I show some solution,
Using the notation of question and I use + for up and - for down,

<00|S₁S₂|00>=-h^-/2.h^-/2 cosθ.(1/2)(<+-|+->+<-+|-+>),other two terms in brackets which you get are zero because of orthogonality condition.(The extra minus sign in front is just because cos(180⁰-θ)=-cosθ) and you will get this
=-h^-2/4.cosθ(because <+-|+->=1 and similarly for other)
edit:I hope this post will not be deleted like some of my previous ones.

 

[bencmier]

may be it will be clear if I show some solution,
Using the notation of question and I use + for up and - for down,

<00|S₁S₂|00>=-h^-/2.h^-/2 cosθ.(1/2)(<+-|+->+<-+|-+>),other two terms in brackets which you get are zero because of orthogonality condition.(The extra minus sign in front is just because cos(180⁰-θ)=-cosθ) and you will get this
=-h^-2/4.cosθ(because <+-|+->=1 and similarly for other)
edit:I hope this post will not be deleted like some of my previous ones.

Thank you andrien, your answer was clear and to the point. I don't know why your other posts would have been deleted but this one won't be.

 

[andrien]

Thank you andrien, your answer was clear and to the point. I don't know why your other posts would have been deleted but this one won't be.

it is just because I can not do anyone's homework in this section.I have already gotten infraction for this.But you saw it,so cheers!