Homework Help: String Theory, Number Operator , Mass of States

1. Nov 12, 2017

binbagsss

1. The problem statement, all variables and given/known data

I have the following definition of the space-time coordinates

2. Relevant equations

Working in a certain gauge we can also do:
From which we can find:

Where $N_{lc}$ sums over the transverse oscillation modes only.
3. The attempt at a solution

MY QUESTION:

I don't understand the RHS of the first excited state, how does $N$ take integer values? what exactly are the $\alpha$?

So (here the summation is over all modes, whereas above it is over the transvere oscillation modes only but ignoring this)

so for the first excited state do we have all alpha modes are zero except $\alpha_{-1}$ ?

So Then $N= \alpha^{j}_{-1}$ ? how is this $N=1$?

So for the next excited state I expect:

$m^2 \alpha_{-1}^i \alpha_{-1}^j |p^i>=m^2\alpha^{j}_{-2}|p^i>=(2-a) \alpha_{-1}^i \alpha_{-1}^j |p^i>=(2-a) \alpha_{-2}^j |p^i>$ right?

And I am told in my notes that this can be achieved by either $\alpha_{-1}^i \alpha_{-1}^j |p^i>$ or $\alpha_{-2}^i$

How does this get $N=2$? since, well looking at $\alpha_{-1}^i \alpha_{-1}^j$, the sum in $N$ is over $n$ not $i$ or $j$ so am I looking at two different number operators here?

For the other expression So Then $N= \alpha^{j}_{-2}$ ? how is this $N=2$?

I'm just really confused as you can tel..