QM - Ladder Operator QHO - factorization

[JaneHall89]

Hi, quick question with A being the lowering operator and A^† the raising operator for a QHO

(A A^† - 1 + 1/2) ħω [Aψ] = A (A^† A - 1 + 1/2) ħω ψ

By taking out a factor of A. Why has the ordering of A A^† swapped around? I would have thought taking out a factor of A would leave it as

A (A^† - 1 + 1/2) ħω ψ

Jane

 

[blue_leaf77]

There is another A in [Aψ]. Pull this A into the bracket from the right to get (AA^†A - A + 1/2A). Now pull out A out to the left of the bracket.