1. find eigenvalue of operator S_y for spin 1/2 particle
2. find eigenvector for the same S_y
3. Find average value that'll be obtained after numerous independent measurement of S_y if electron is in spin state |+>
The Attempt at a Solution
1. the eigenvalue is just: S_y=(hbar/2)[matrix] where the matrix is:
How does the possibility of +(hbar/2) come in, mathematically..?
2. My book says that the eigenvector of S_y is:
|plus or minus> = (1/sqrt(2)) [ |+> plus or minus i|-> ]
I'm not really sure what to make of the notations.... :(
3. Since S_y=-(hbar/2) and
|+> = vertical maxtrix[cos(theta/2)]
<+| = horizontal matrix [ cos(theta/2) sin(theta/2)]
<+|S_y|+> = -(hbar/2)sin(theta)
Is that correct...? Because it's the same value as the expectation value for S_x, right? I'm not really sure what I should be expecting for these values...