Rotations in Bloch Sphere about an arbitrary axis

[polyChron]

Hey,
(I have already asked the question at http://physics.stackexchange.com/questions/244586/bloch-sphere-interpretation-of-rotations, I am not sure this forum's etiquette allows that!)

I am trying to understand the following statement. "Suppose a single qubit has a state represented by the Bloch vector $\vec{\lambda}$. Then the effect of the rotation $R_{\hat{n}}(\theta)$ on the state is to rotate it by an angle $\theta$ about the $\hat{n}$ axis of the Bloch sphere. This fact explains the rather mysterious looking factor of two in the definition of the rotation matrices."
I could work out that the rotation operators $R_x(\theta)$, $R_y(\theta)$ and $R_z(\theta)$ are infact rotations about the $X,Y$ and $Z$ axis. But how do I extend this for $R_{\hat{n}}(\theta)$ and prove the above statement. Please point me in the right direction.

Thanks.

 

[jfizzix]

I think what you want is what the pauli spin observable is along the "\hat{n}" axis.
Where \hat{n} is a unit vector with components (n_{x},n_{y},n_{z}),

\sigma_{\hat{n}} = \sigma_{x}n_{x} + \sigma_{y} n_{y} + \sigma_{z} n_{z}.

If you know what R_{x}(\theta) is in terms of \sigma_{x}, then I think you can work out what R_{\hat{n}}(\theta) is in terms of \sigma_{\hat{n}}.

 

[polyChron]

I think what you want is what the pauli spin observable is along the "\hat{n}" axis.
Where \hat{n} is a unit vector with components (n_{x},n_{y},n_{z}),

\sigma_{\hat{n}} = \sigma_{x}n_{x} + \sigma_{y} n_{y} + \sigma_{z} n_{z}.

If you know what R_{x}(\theta) is in terms of \sigma_{x}, then I think you can work out what R_{\hat{n}}(\theta) is in terms of \sigma_{\hat{n}}.

Thanks for your reply.
Yes I have figured that out as well. Let me explain.
I know the rotation matrices in terms of the Pauli matrices, i.e R_x(\theta) = e^{-i\sigma_x /2} and the rotation matrices for \sigma_y and \sigma_z follows in the same manner. I could also prove that R_{\hat{n}}(\theta) = cos(\frac{\theta}{2})I - i sin(\frac{\theta}{2})(n_x\sigma_x+n_y\sigma_y+n_z\sigma_z) using the Taylor expansion. But the difficulties for me start from here. How do I show that R_{\hat{n}}(\theta) is infact a rotation about \hat{n} axis by \theta. How can I construct a concrete proof?

 

[jfizzix]

Thanks for your reply.
Yes I have figured that out as well. Let me explain.
I know the rotation matrices in terms of the Pauli matrices, i.e R_x(\theta) = e^{-i\sigma_x /2} and the rotation matrices for \sigma_y and \sigma_z follows in the same manner. I could also prove that R_{\hat{n}}(\theta) = cos(\frac{\theta}{2})I - i sin(\frac{\theta}{2})(n_x\sigma_x+n_y\sigma_y+n_z\sigma_z) using the Taylor expansion. But the difficulties for me start from here. How do I show that R_{\hat{n}}(\theta) is infact a rotation about \hat{n} axis by \theta. How can I construct a concrete proof?

I think what you should do is a proof by demonstration. Compare a bloch vector before \hat{u} and after \hat{u}' a rotation about the n-axis.
Both \hat{u} and \hat{u}' dotted with \hat{n} should give the same value, and the vectors themselves should have the same magnitude. This proves that R_{n}(\theta) is at least some sort of rotation about the \hat{n}-axis.

To find the angle, you need to project \hat{u} and \hat{u}' onto the plane perpendicular to \hat{n}. The dot product of these projected vectors will be the magnitude of each times the cosine of the angle between them, and hopefully that angle will be none other than \theta.