How to solve the tranmission probability in an evolution of a quantum system

Question

I've just learned the evolution of some quantum system for about a week, and our homework sometimes something like this. I don't quite have any idea of solving this kind of problem.

Can you help giving me some fundamental enlightment on these problems, say one like this:

" Assume the spin number of a particle is 1, if the measure of the x component of spin is +1 at first, then the y component is measured -1. What is the probability that the x component is measured to be +1 again? "

Thank you very much!

Answer 1

You need to know how to rotate wavefunctions from one z axis to another z axis, that's it. I'll do it for spin 1/2, and you can work out the same for spin 1.

For spin 1/2, the operator which generates rotations around the x-axis is:

$$ S_x = {1\over 2} \sigma_x$$

Notice that $\sigma_x^2=1$, so that using the Taylor series of the exponential:

$$ e^{i\theta S_x} = \cos({\theta\over 2}) I + i \sin({\theta\over 2}) \sigma_x $$

If you rotate by a $\theta$ of 90 degrees, the resulting matrix is

$$ {1\over\sqrt{2}} ( I + i \sigma x ) $$

Applying this to the state (1,0), you get the state $(1, i)$, up to a normalizing factor of $\sqrt{2}$. This is the state of +1 spin in the y direction, and you can check that it is an eigenvector of $\sigma_y$.

So if you prepare the state $(1,i)$, the amplitude to be in the + or - z spin state is 1,i. The square of this is the probability you want.

To repeat this for spin 1, or any spin, you can use tensors of spin-1/2, or D matrices in elementary quantum mecanics books.

This is not a transmission of anything, it's just asking what the probability of state A to be found as state B in quantum mechanics without any time evolution, just using idealized measurements. This problem is also discussed very well from a physical point of view for spin 1/2 in Feynman's lectures III.