I have several problems with General Definitions of an Operator and Commutator :
- the product of operators is generally not commutative: $$\hat A \hat B \not= \hat B\hat A .$$ what is this means that generally is not commutative? and , is the product of two operators $\hat A$ Hermitian Adjoint $\dagger$ then $\hat B$ (because in my book it is just $\hat A \hat B$ )? $$\hat A^{\dagger} \hat B \not= \hat B^{\dagger}\hat A .$$
Remark 2. Note that the Hermitian adjoint of an operator is not, in general, equal to its complex conjugate: $$\hat A^{\dagger} \not= \hat A^* .$$
What happens if the Hermitian adjoint of an operator is, in special, equal to its complex conjugate!? $$\hat A^{\dagger} = \hat A^* .$$
Commutator Algebra
- the commutator of two operators $\hat A$ and $\hat B$, denoted by: $$[\hat A , \hat B]=\hat A \hat B - \hat B \hat A .$$
again i do not know that, is the product of two operators $\hat A$ Hermitian Adjoint $\dagger$ then $\hat B$? $$[\hat A , \hat B]=\hat A^{\dagger} \hat B - \hat B^{\dagger} \hat A$$