Expectation value in quantom mechanics (a general question)

[itaibh1]

Homework Statement

Hello, I'm a bit confused about the calculation of the expectation values. Normally, when I have a wave function of sort and I want to calculate the expectation value of some operator, I just insert it into the braket <ψ|A|ψ>, where ψ for example is a wave function composed out of eigenstates ψ=1/√3⋅(φ₁+φ₂+φ₃)
and I will just multiply those accordingly.
My problem started when I was asked to calculate uncertainty for L_x and L_y, I keep seeing the expectation value it's being calculated like this:
L_x=1/2⋅<lm|L₊+L_-|lm>=0, with both bra-ket are for the same state. I don't understand, why can't the wavefunction be one that is composed out of several different eigenstates?

Homework Equations

The Attempt at a Solution

When I calculated it at first, I did:
<L_x>=1/2⋅<lm|L₊+L_-|l'm'>, which confused me even more, since this is actually a calculation for the L_x matrix elements. What am I doing wrong here?

 

[BvU]

So far I see nothing wrong.

I keep seeing the expectation value it's being calculated like this:
Lx=1/2⋅<lm|L++L-|lm>=0, with both bra-ket are for the same state

So what you see there is that they calculate L_x for a state |lm> that is an eigenstate of both L² and L_z.
Apparently you have a different state Ψ so you get something different. Where 's the contradiction you see ?