It is well known that in elementary QM the so-called destruction/creation operators
$$ a_i = \frac{Q_j + i P_j }{\sqrt{2}}, \quad a_i^* = \frac{Q_j - i P_j }{\sqrt{2}},$$
are introduced when studying the $n$-dimensional harmonic oscillator, where operators $Q_j$ and $P_j$ are defined by
$$(Q_j \psi) (x) = x_j \psi(x), \quad (P_j \psi) (x) = -i\hbar\frac{\partial\psi}{\partial x_j}(x),$$
on suitable domains in $L^2(\mathbb{R}^n)$, say the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ of smooth "rapidly decreasing" functions. So they are operators $L^2(\mathbb{R}^n) \supset \mathcal{S}(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$.
However, in Second Quantization formalism we have destruction/creation operators acting on the Fock space $\mathcal{F}(\mathfrak{h}) = \bigoplus_{k=0}^\infty \mathfrak{h}^{\otimes^k}$, where $\mathfrak{h}$ is some one-particle Hilbert space, and they are defined as operator-valued distributions $\psi \to a(\psi), \psi \to a(\psi)^*$, where $\psi \in \mathfrak{h}$.
Is there is any connection between the two notion, e.g. by taking as one-particle Hilbert space exatcly $\mathfrak{h} = L^2({\mathbb{R}^n})$, or the fact that they share the same name is just an "accident"? In the former case, I tried to exploit the very definition of destruction/creation operators in second quantization for the harmonic oscillator eigenfunctions (or Hermite functions) $\varphi_k$ by defining $a_k \equiv a(\varphi_k)$, but I think this approach fails miserably if I'm looking for any link between the two definitions...