1. The problem statement, all variables and given/known data Calculate the expectation value for a harmonic oscillator in the ground state when operated on by the operator: $$AAAA\dagger A\dagger - AA\dagger A A\dagger + A\dagger A A A\dagger)$$ 2. Relevant equations $$AA\dagger - A\dagger A = 1$$ I also know that an unequal number of raising and lowering operators gives a zero expectation value due to orthogonality requirements. 3. The attempt at a solution I guess that the first term in brackets gives a zero expectation value as it leads to a function which is orthogonal to $$\psi_0$$ If I say n=1 then: $$A\dagger\psi_0 = \psi_1$$ And $$A \psi_1 = \psi_0$$ I've tried taking the third term and saying: $$A\dagger A(1+ A\dagger A) = A\dagger A + A\dagger A A\dagger A$$ Then doing the same thing with the second term to get $$ 1+ 2A\dagger A +A\dagger A A\dagger A$$ Then I subtract this term from the third and I get $$-A A\dagger $$ But this, operating on $$\psi_0$$ seems to give me an expression which results in an infinity when integrated over all space. Can someone tell me where I went wrong?