= Solution
Start with a nonzero <eigenfunction> of energy $E$ and apply the lowering <ladder operator> repeatedly. Whenever the resulting wavefunction is nonzero, part (b) makes it an <eigenfunction> of energy $E-k\hbar\omega$. This cannot remain nonzero for every $k$, since part (e) rules out negative energies. Let $\psi=\hat A^n\phi$ be the last nonzero member. Then $\hat A\psi=0$, and part (d) forces its energy to be $\hbar\omega/2$. Therefore \b[every energy eigenvalue has the form]
$$
\boxed{E=(n+\tfrac12)\hbar\omega,\qquad n=0,1,2,\ldots.}
$$
These levels actually occur: solving $\hat A\psi_0=0$ with $\hat p=-i\hbar\,d/dx$ gives the normalizable Gaussian $\psi_0\propto\exp(-m\omega x^2/(2\hbar))$. The raising operator $\hat B$ increases the energy by $\hbar\omega$, and $\|\hat B\psi\|^2=2m(E+\hbar\omega/2)\|\psi\|^2>0$. Repeated raising therefore produces a nonzero normalizable state at every listed level.
Back to article page
