Measuring sum of two components of spin angular momentum

[FarticleFysics]

Homework Statement

What are the possible results of the measurement of the sum of the x and z components of the spin angular momentum of a spin-1/2 particle?

Sx = Spin angular momentum operator x
Sz = Spin angular momentum operator x

Homework Equations

The Attempt at a Solution

I started by applying the spin-up eigenket to the sum of the spin angular momentum operators.

(Sx + Sz) | up > = Sx | up > + Sz | up > = -1/2h_bar + 1/2h_bar = 0

The text I'm using proves that Sx | up > = 1/2h_bar | down >
and like wise Sx | down > = 1/2h_bar | up >

Does this mean if you were to probe the particle for spin up in the x direction you would actually see spin down?

then I applied the spin down eigen-ket to pull the possible eigenvalues from the operators.

(Sx + Sz) | down > = Sx | down > + Sz | down > = +1/2h_bar + (-1/2h_bar) = 0

I feel that there is something wrong with how I've gone about calculating this.. I know that Sx and Sz don't commute therefore you cannot measure their eigenvalues simultaneously. Since I get zero did I show this correctly?

Also, the eigenkets for spin up and spin down are considered the eigenvectors, correct?

I am still trying to get the hang of the linear algebra and what everything means so any help would be amazing!

Thanks

 

[vela]

One problem with your work is that saying, for example, that $\hat{S}_x \lvert \text{up} \rangle = \frac{\hbar}{2}$ doesn't make sense. This is akin to writing
$$\frac{\hbar}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{\hbar}{2}.$$ The lefthand side is a vector, but the righthand side is a scalar.

Does this mean if you were to probe the particle for spin up in the x direction you would actually see spin down?

No. Rid yourself of the notion that taking a measurement means applying the associated operator to a state.

The result of a measurement is one of the eigenvalues of the associated observable. Given the state of a particle, you can calculate the probability amplitude for a particular result by finding the projection of this state in the direction of the corresponding eigenstate. For example, if you were measuring the spin in the z-direction, you would need to know the eigenvalues of $\hat{S}_z$, which are $\pm \hbar/2$, and the corresponding eigenstates, $\lvert + \rangle$ and $\lvert - \rangle$. If the particle is in state $\psi$, the probability amplitude for finding the particle in the spin-up state is $\langle + \vert \psi \rangle$, which is generally a complex number. The probability is given by square of the modulus of the amplitude.