Matrix for a Stern-Gerlach apparatus

[damosuz]

I am trying to find the matrix representing a modified Stern-Gerlach apparatus (as proposed in the Feynman lectures) with its magnetic field in the z direction and a filter that blocks spin 1/2 atoms that are in the |-z> state (thus I'll call the apparatus a SG+z apparatus). I want to use |+x> and |-x> as base states.

For an atom in the |+y> = 0,707|+x> + 0,707|-x> state entering the SG+z apparatus to exit in the |+z> = 0,707|+x> + 0,707 i |-x> state, the apparatus would have to correspond to the matrix $$\left( \begin{array}{cc} 1 & 0 \\ i & 0 \end{array} \right)$$.

The same matrix will transform the states |-y> = 0,707|+x> - 0,707|-x>, |+z>, |-z> = 0,707|+x> - 0,707 i |-x> and |+x> into the state |+z> as required, but it doesn't work for the state |-x> (it gives 0!). Am I trying to do something that is impossible or incorrect here?

I am aware my question might need some clarification, which I will be happy to provide.

 

[kith]

I am trying to find the matrix representing a modified Stern-Gerlach apparatus (as proposed in the Feynman lectures) with its magnetic field in the z direction and a filter that blocks spin 1/2 atoms that are in the |-z> state (thus I'll call the apparatus a SG+z apparatus).

If your apparatus transmits +z and blocks -z the matrix is determined by A|+z> = |+z> and A|-z> = 0. So in the z-basis it reads
$$\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right)$$
Note that an initially normalized superposition state won't be normalized anymore after applying the matrix. This is because the coefficients represent the probability to get the corresponding final state.

 

[atyy]

I don't think you can get a matrix which is independent of the initial state and gets you a normalized final state, since you are describing wave function collapse, which is non-unitary. The matrix should project onto |+z>, so it should represent the operator |+z><+z| = |+x><+x| - i|+x><-x| + i|+x><-x| + |-x><-x|, which will have a matrix representation in the x-basis $$\left( \begin{array}{cc} 1 & -i \\ i & 1 \end{array} \right)$$

 

[damosuz]

Thank you! It made me realize I had made a mistake in my post when I wrote that the SG+z apparatus could transform state |-z> into state |+z>. The matrix you propose atyy gives the null state when applied to state |-z>, as required.

 

[atyy]

I don't know whether my matrix is exactly right, but I'm sure kith's is (because the answer is obvious in the z-basis), so one way to check it is to check that his answer and my proposal are related by a coordinate transformation from the z-basis to the x-basis.