Eigenvalues/Eigenstates of Spin Operator S in xz Plane

[rsaad]

Homework Statement

Find the eigenvalues and eigenstates of the spin operator S of an electron in the direction
of a unit vector n; assume that n lies in the xz plane.

Homework Equations

S|m>= h m|m>

The Attempt at a Solution

This question is from Zettili QM and they have written:

n.S|m>= (h/2) m|m>

I do not understand why are they taking a half.
If I take m=1/2 in S|m>= (h/2) m|m>, I get h/4 but the answer should be h/2, by using S|m>= h m|m>.
So where am i going wrong?

 

[BruceW]

m is the quantum number. you need to check its definition

 

[rsaad]

Of course it is a quantum number but why is there a half?

 

[vela]

It's probably just a typo. It doesn't really matter, though. It just means the quantum numbers are $\pm1$ instead of $\pm 1/2$.

 

[BruceW]

I see now. The notation in the two equations is not consistent.
S|m>= h m|m>
n.S|m>= (h/2) m|m>
In the second equation, m is a 'quantum number' (i.e. integer), while in the first equation I guess you could interpret m as the value of the projection of angular momentum, in natural units.