Proving "If f is Simultaneously an Eigenfunction of L^2 & L_z

[Funzies]

Hello there,
I've got two short questions I was hoping you could help me with:

-I have to prove:
"if f is simulateneously an eigenfunction of L^2 and L_z, the square of the eigenvalue of L_z cannot exceed the eigenvalue of L^2"
He gives a hint that I should evaluate
$$<f|L^2|f>$$
But I don't have a clue what he means by this notation?!

-Why does this hold:
$$<S_x> = \chi^+S_x\chi$$
I am familiar with calculating <x> by doing:
$$<x>=\int_{-\infty}^{\infty}\psi(x)^*x\psi(x)dx$$
But I do not understand this different situation. Could you tell me the underlying differences/similarities?

Thanks!

 

[dextercioby]

The "S_x" is one of the spin-operators. He has a trivial action in the Hilbert space of angular momentul states. So just forget about the spin, remember what you have to prove.

Another hint would be to connect the square of L_z and the L^2.

 

[norousan]

$$<f|L^2|f>$$ is dirac notation and is more or less the same as $$\int_{-\inf}^{\inf}f(x)^*L^2f(x)dx$$ so its essentially the expectation value $$<L^2>$$ in this case. The difference is, dirac notation does not imply a particular representation for your state function f. When you write f(x), it is referred to as the coordinate representation, ie your state function is a function of position. You could also write your state function as a function of momentum, or whatever else.

In short, the $$<f|L^2|f>$$ is a more general way to represent the integral you are used to.