Now define a i so that it applies a to | n i ⟩ ⟩ . Correspondingly, define a i † † as applying a † † to | n i ⟩ ⟩ . Thus, for example, the net effect of a i − − 1 a i † † is to move a particle from the ( i − − 1 ) -th to the i -th site while multiplying with the appropriate factor.