In this chapter, we had only expressed eigenstates of the harmonic oscillator Hamiltonian through repeated action of the raising operator, \(\hat{a}^{\dagger}\). This gives us a concrete algorithm for expressing the energy eigenstates in position space, which we will do here.

(a) In Eq. (6.75), we had found the ground state of the harmonic oscillator in position space, up to a normalization constant \(N\) :\[\begin{equation*}\psi_{0}(x)=N e^{-\frac{m \omega}{2 \hbar} x^{2}} \tag{6.120}\end{equation*}\]

What is \(N\) ?

(b) In Example 6.2, we constructed the first excited state of the harmonic oscillator in position space. Extending that procedure, determine the wavefunctions for the second and third excited states of the harmonic oscillator, \(\psi_{2}(x)\) and \(\psi_{3}(x)\).