The fundamental idea of these operators is that we can use them to add particles to our system to a specific eigenstate. Now my book has examples of these operators of which the harmonic oscillator ladder operators are used. But thinking about it, this example does not make sense for me. The idea of the ladder operators for the SHM was that when used on eigenstate number n it took us to eigenstate n+1. How is this the same as adding a new particle to the system? E.g. if we let a_+ denote the creation operator in second quantization and l1> the first excited state my books idea is that: a_+l1> = l2> But this is not true? a_+ acts on a vector specifying the number of particles in each eigenstate and by using it on l0,1,0,0,0,0...> we get the vector l0,1,1,0,0,0,0,0,0...>, that is a 2-particle state - one in first singleparticle eigenstate and one in the second. What am I missing?