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FarticleFysics
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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