Past exam of the quantum theory course of the University of Oxford / 2015 / 2 / b / ii / Solution

The squared norm is obtained by moving the adjoint operator to the other side of the inner product:For an energy eigenstate, this isSince a squared norm is nonnegative, every nonzero energy eigenstate has $E\ge \hslash \omega /2$. Moreover $a_{-}\psi =0$ exactly when $E=\hslash \omega /2$.