Construct an Operator (2-D QM)

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Matterwave
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Homework Statement


Construct the [tex]S_r[/tex] operator, which is the spin operator in the r direction.


Homework Equations


The pauli-matrices are useful, I would assume.

The Attempt at a Solution


So, the way we constructed the other 3 operators by noting their effects on a state |s m>

But I can't do that here because I have no idea what this operator [tex]S_r[/tex] will do to a state |s m>

I thought about looking at the fact that:

[tex] x = rsin(\theta)cos(\phi), y = rsin(\theta)cos(\phi), and z=rcos(\theta)[/tex]

and just replace all the x's with [tex]S_x[/tex]'s and r's with [tex]S_r[/tex]'s etc. But I don't think this will work as it will give me 3 different answers.

I need help on how to start with this.
 

Answers and Replies

  • #2
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If you have a vector A then how do you calculate its component in a certain direction n?

You take the inner product of the unitvector of the given direction n and the vector A.

So this is the same case. How would you do it here? ;)
 

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