The harmonic oscillator is defined by the mean value energy $\langle E\rangle=\frac{2}{3} \hbar\omega$. Can we have a wavefunction which describes such a state?
Any help is appreciated. Is it possible to have such state, as we know the energy is $E=\hbar\omega(n+\frac{1}{2})$?
In general $$ \psi(x)=\sum_n c_n \psi_n(x) \tag{1} $$ where $\psi_n(x)$ are eigenstates with eigenvalue $(n+1/2)\hbar\omega$. The average energy of the state in (1) would be $$ \langle E\rangle = \sum_n \vert c_n\vert^2 (n+1/2)\hbar\omega \tag{2} $$ so you need to find any set of numbers $c_n$ such that the right hand side of (2) gives $2\hbar\omega/3$, subject to the normalization constraint that $\sum_n\vert c_n\vert^2=1$. Indeed there is at least one solution, which you can find by yourself.
