Spin-1/2 Particle: S_y Eigenvalue, Eigenvector, and Average Value Calculation

[lleee]

Homework Statement

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:
[0 -i]
[i 0]
So S_y=-(hbar/2)?

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)]
[sin(theta/2)]
and
<+| = 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...

 

[lleee]

any insights?

 

[Galileo]

Do you know the method for finding the eigenvalues and eigenvectors of an arbitrary matrix?

 

[lleee]

I guess not... :(

 

[malawi_glenn]

I guess not... :(

that method is introduced in introductory Linear Algebra courses, so it is very strange that you have Quantum mechanics before having enough math knowledge. But you learn it easy, just get a book from your library and study.

 

[nrqed]

I guess not... :(

Are you doing this problem because it was assigned in a course or you do it as part of self-study? If you haven't learned how to find the eigenvalues and eigenvectors of a matrix, the question is not at the right level for you, yet. If it was assigned in a class, you should talk to the instructor! If you do it as slef-study, you have to pick a book on lenear algebra and learn that first.