Invariance of a spin singlet under rotation

[Silicon-Based]

Homework Statement

Show that a spin singlet state remains the same under a rotation about the $y$-axis by an angle $\alpha$

Relevant Equations

$|\text{singlet}\rangle = \frac{1}{\sqrt{2}}(|S_{1z}+\rangle |S_{2z}-\rangle - |S_{1z}-\rangle |S_{2z}+\rangle)$

$\mathcal{D_y(\alpha)} = e^{-i\sigma_y\alpha/2} =\mathsf{1}\cos(\alpha/2) - \sigma_y\sin(\alpha/2)$

$\mathcal{D_{1y}(\alpha)} \otimes \mathcal{D_{2y}(\alpha)} = e^{-i\sigma_{1y}\alpha/2} \otimes e^{-i\sigma_{2y}\alpha/2}$

I have tried doing the obvious thing and multiplied the vectors and matrices, but I don't see a way to rearrange my result to resemble the initial state again:

$(\mathcal{D_{1y}(\alpha)} \otimes \mathcal{D_{2y}(\alpha)} )|\text{singlet}\rangle = \frac{1}{\sqrt{2}}\left[ \begin{pmatrix} \cos(\alpha/2)\\ \sin(\alpha/2) \end{pmatrix} \begin{pmatrix} -\sin(\alpha/2)\\ \cos(\alpha/2) \end{pmatrix} - \begin{pmatrix} -\sin(\alpha/2)\\ \cos(\alpha/2) \end{pmatrix} \begin{pmatrix} \cos(\alpha/2)\\ \sin(\alpha/2) \end{pmatrix}\right]$

 

[vanhees71]

Write out the result in terms of the basis vectors and multiply out the tensor products! The equation makes not too much sense to me, because on the left-hand side you have a ket, and on the right-hand side some strange notation of components.

 

[Abhishek11235]

As @vanhees71 tells,you should use tensor product defined as:
$$S \otimes T= ST^T$$
Where S and T are vectors. Once you do this,do you recognise the matrix you got and how it affects singlet state?

 

[Silicon-Based]

Write out the result in terms of the basis vectors and multiply out the tensor products! The equation makes not too much sense to me, because on the left-hand side you have a ket, and on the right-hand side some strange notation of components.

I'm not sure I understand why I need to do this. Don't the rotation operators act only the corresponding spin states in their Hilbert space, in which case I wouldn't need to find the tensor product? This worked for me when trying to show invariance under rotation about z, unless that was purely coincidental.

I've not done tensor products before having been assigned this problem so I'm pretty sure there should be a different way to do this.

I don't see the issue with the equation, I've written the kets in terms of their components because they are no longer simply spin up or down but a superposition of both.

 

[Silicon-Based]

So I've tried factoring out the eigenkets from the superposed kets in my equation, e.g. $\cos(\alpha/2)|+\rangle + \sin(\alpha/2)|-\rangle$, and found out that most of the terms cancel. I ended up with the expression $\cos(\alpha)|\text{singlet}\rangle$. Now I'm only unsure how to rationalize the factor of $\cos(\alpha)$.

 

[vanhees71]

Then there must be some mistake since the rotation is of course a unitary operator, i.e., the norm of your singlet state cannot change.

 

[Silicon-Based]

Then there must be some mistake since the rotation is of course a unitary operator, i.e., the norm of your singlet state cannot change.

That's right; I missed a minus sign which normalized everything.