Angular Momentum Problem in Dirac Notation

[xago]

Homework Statement

http://img857.imageshack.us/img857/2079/dirac.png

Homework Equations

H|ψ> = E|ψ>
$L^{2}$|ψ> = l(l+1)$\hbar^{2}$|ψ>
$L_{z}$|ψ> = $m_{l}$$\hbar$|ψ>

The Attempt at a Solution

I know this problem is very simple since I've seen a very similar problem a while ago but I've completed forgot how to do it over the winter break.
As far as normalization goes its <ψ|ψ> = 1, so I simply multiply the given ket vector by the bra vector of the same state. However I can't for the life of me remember how the bra and ket vectors multiply to an equation is which you just solve for A. Say for the 3rd term, 2|$ψ_{2,1,-1}$> represents n=1, l=1, $m_{l}$ = -1
So then the Hamiltionian eigenvalue is 1, the $L^{2}$ eigenvalue is 1(1+1)$\hbar^{2}$ = 2$\hbar^{2}$ and the $L_{z}$ eigenvalue is -$\hbar$ but how do I put all this together?

FYI this is for the hydrogen atom |$ψ_{n,l,m_{l}}$>

 

[Jorriss]

This doesn't appear to be an angular momentum problem.

If you have a state c|n>, its corresponding bra is c*<n|, thus the normalization requirement is c*c = <n|n>. And remember orthogonality of eigenstates, <n|n'>= 1 if n=n' and 0 if n=/=n'.

 

[xago]

Thats what I originally thought but I kept thinking i was missing something lol. So is it just
A^2(6+1+4+9+16) = 1
A= 1/6