A very useful operator is the so-called number-operator. Most physics cases we will study in this text conserve the total number of particles. The number operator is therefore a useful quantity which allows us to test that our many-body formalism conserves the number of particles. In for example \( (d,p) \) or \( (p,d) \) reactions it is important to be able to describe quantum mechanical states where particles get added or removed. A creation operator \( a_\alpha^{\dagger} \) adds one particle to the single-particle state \( \alpha \) of a give many-body state vector, while an annihilation operator \( a_\alpha \) removes a particle from a single-particle state \( \alpha \).