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6.1.2. Number Representation: States Consider a set of complete, orthonormal 1-particle (1-P) basis. For the sake of clarity, we shall assume the quantum numbers to be discrete. (Results for the continuous case can be obtained by some limiting procedure). To begin, we arrange the 1-P states by some rule into a unique sequence 0,1,2, of monotonically increasing energy so that 0 is always the 1-P ground state. For example, the one electron spinless states nlm in a hydrogenic atom can be arranged as 0 100 , 1 11 1 , 2 110 , 3 111 , … . The basis in the number (n-) representation consists of all the eigenstates of the number operator n̂ : nˆ n0 , n1, n n0 , n1, , n , , n , where n is the number of particles in the 1-particle state . We shall assume orthonormality: n0 , n1, , n , n0 , n1, n0n0 n1n1 , n , n n (6.2) Completeness of the basis means the state vector of the system can be written as t n0 , n1 , , n , n0 , n1 , , n , t n0 ,n1 , ,n , n0 ,n1 , ,n , n0 , n1, , n , n0 ,n1 , ,n , t (6.3)