In this chapter, we noted that the Hilbert space of the harmonic oscillator corresponds to all those states that can be accessed from the ground state through action by an anatic function of \(\hat{a}^{\dagger}\) about \(\hat{a}^{\dagger}=0\). We'll study a bit more why this is the case in this problem.

(a) In the basis formed from eigenstates of the harmonic oscillator Hamiltonian, write down the raising operator \(\hat{a}^{\dagger}\) as a matrix. Do this in two ways: first, as an explicit row-column matrix and second, as a sum over outer products of eigenstates of the Hamiltonian. For the explicit row-column matrix, just write down the first three rows and three columns from the upper-leftmost element.

(b) With this representation of the raising operator as a matrix in the basis of energy eigenstates of the harmonic oscillator, express the position and momentum operators \(\hat{x}\) and \(\hat{p}\) as matrices in this same basis.
(c) What is the determinant of the raising operator \(\hat{a}^{\dagger}\) on the space of energy eigenstates?
(d) At a point where a function is non-analytic, we can express the function through an extension of the Taylor series to include negative powers of the argument. Such an expansion is called a Laurent series, and, if we assume that \(g\left(\hat{a}^{\dagger}\right)\) is non-analytic at \(\hat{a}^{\dagger}=0\), its Laurent expansion can be expressed as

\begin{equation*}
g\left(\hat{a}^{\dagger}\right)=\sum_{n=-\infty}^{\infty}c_{n}\left(\hat{a}^{\dagger}\right)^{n} = \cdots + c_{-2}\left(\hat{a}^{\dagger}\right)^{-2} + c_{-1}\left(\hat{a}^{\dagger}\right)^{-1} + c_{0} + c_{1} \hat{a}^{\dagger} + c_{2}\left(\hat{a}^{\dagger}\right)^{2} + \cdots \tag{6.121}
\end{equation*}

where the \(c_{n}\) are some complex-valued numbers. For the raising operator on the space spanned by the energy eigenstates of the harmonic oscillator, can this general Laurent expansion exist? Are there constraints we must impose on the coefficients \(c_{n}\) to ensure that \(g\left(\hat{a}^{\dagger}\right)\) exists?