Quantum mechanical operator

●OPERATOR :-

         

          Operator is a function over the state of system OR operator is mathematical rule in physics , when it apply on a state |Ψ >  then it can be change and provide same value of state  , An observable ca be represented by  ' ^ ' to  convert into operator ,

                               

                                  x = x^

●Types of operator :- 

           

Generally operator are many types , we discussed about few important types ,

1. Linear operator :-

   

If two operator A^ and B^ are                            following such relation

          ( A^ +B^ )|Ψ>  = A^| Ψ > + B^| Ψ >

 Then A^ and B^  are linear operator .

2. Null operator :-

 If a operator O^ apply on a state |Ψ >  then we get zero of its value . 

                   O^ | Ψ >  =  0

3. Unit operator :-

  If an operator I^ apply on a state | Ψ > After operating operator it state remains same  and it gives unit value then this operator is known as unit operator . It is represented by -

                           I^ | Ψ >  =  | Ψ > 

4. Hermitian operator :- 

 Ifan operator following such relation then it's operator known as hermitian operator .

                                 (A^)+  =  A^

  A+ is represented degar of operator A^

Two operator A^ and B^ are Hermitian operator then there product A^B^ is also Hermitian operator if and only if A^ and B^ commute. 

                     

            Given  , A^  is Hermitian operator 

            

            ( A^)+  =  A^       -----( 1 ) 

           

            B^    is Hermitian operator 

          

            (A^)+  =  A^       -----( 2 ) 

          ( A^B^ )+  =  A^B^       ---------( 3 ) 

        Now take L. H. S.  From equation ( 3 ) 

      

         ( A^B^ )+  =  (B^+) (A^+) 

         ( A^B^ )+  =   B^A^   ---------( 4 ) 

         Condition  , ( A^ and B^ commute) ! 

          Then  , [ A^ B^ ]  = 0

           

               [ A^B^ - B^A^ ]    =  0

              A^B^    =   B^A^   ---------( 5 ) 

       Put value in equation ( 4 ) from ( 5 ). 

              (A^B^)+  = A^B^   =  R.H.S.

5. Anti Hermitian operator :- 

 If operator A^ is following such relation .

                                A+  =   -A 

      A+ is represented degar of operator A^

6. projection operator :- 

   

If an operator P^ following such relation .

                                 (P^)+  =  p^

                                 

                                 (P^)² =  P^

           Then P^ is known as projection                           operator .

               (P^)+ = degar of operator P^

7. Unitary operator :- 

           

An operator U^ is such that  , 

         

         (U^)+   =  ( U^) inverse  ( power -1 ) 

Then U is known as unitary operator.

U+ is represented degar of operator U^

8. Inverse operator :- 

If operator B^ is following such relation .

                          BA  =  AB    =   I^  

then B is known as inverse operator of A^ and it can written as

                         A   =   B inverse 

      

 If operator A is define in matrix form.              Then,

          

             A inverse = adj ( A ) / | A |  , A not = 0


Other Some important operator:- 

● Del operator -

Basically del operator have three types