B.3.b. Number Representation For Many States

Let the complete set of 1-particle eigenstates be arranged in a sequence according to some rules so that each of them can be referred to by a single integer α=0,1,, .  For example, one can arrange them in the order of increasing energies so that α=0  is the ground state.  The case of continuous quantum numbers can be treated either by replacing a with real numbers or replacing the quantum numbers with their discrete approximations.  Thus, for the case of the momentum and spin eigenstates, a given value of a will represent a state with quantum numbers k α =( k α , σ α ) .

The basis states in the number representation are written as | n 0 , n 1 ,, n α ,, n  where n α  is the number of particles in the 1-particle state a.   The canonical (generalized coordinate and momentum) operators in the number representation are the creation a α +  and annihilation a α  operators with ( a α ) = a α + , where  is the adjoint (hermitian conjugate) operator.  Note that we have suppressed the ^ on both a α +  and a α  since they are always used as operators.  By definition, all dynamical operators of the system can be written as functions of a α +  and a α .  For example, n ˆ α = a α + a α .