Download 6.1.2. Number Representation: States

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)