I've been trying to form a proof using , using majorly dirac notation.There has been claims that its much better to use in QM. The question i wanted to generally show that the expected value is Zero for all odd energy levels.I believe i have solved the question but im a bit Iffy about a step i took: <x>n = <Ψn|x|Ψn> = L for a given Ψn = (A+)n(n!)-2 Energy eigen functions have definite parity, assume for all odd n's if one is zero the rest should follow. Take n = 1 => L = <(A+)(n!)-2|x|(A+)(n!)-2> = (n!)-1 <(A+)|x|(A+)> B = <(A+)|x|(A+)> Define A+ = Lx + iC : B,C are Real => <Lx+iC|x|Lx+iC> (Bit iffy after these steps) = <Lx|x|Lx|> + <iC|x|iC> = <L|x3|L>+<C|x|C> as ∫x2n+1dx for limits [-∞,∞] and n =0,1,2,3.... => 0 we find B=0 therefore <x>n = 0 For odd ns.