Given the Fock space: $$\mathcal{F}(\mathcal{h}):=\bigoplus_0^\infty\mathcal{h}^{n}\text{ with } \mathcal{h}^{n}:=\bigotimes_1^n \mathcal{h},\mathcal{h}^0:=\mathbb{C}$$
Define the creation and annihilation operators by: $$a(f)\alpha_1\otimes\ldots\otimes\alpha_n=\sqrt{n}\langle f,\alpha_n\rangle\alpha_1\otimes\ldots\otimes\alpha_{n-1}\qquad\alpha_i\in\mathcal{h}$$ $$a(f)^*\alpha_1\otimes\ldots\otimes\alpha_n=\sqrt{n}\langle f,\alpha_n\rangle\alpha_1\otimes\ldots\otimes\alpha_{n}\otimes f\qquad\alpha_i\in\mathcal{h}$$
Given this definition of the Fock Space and $\alpha \in \mathcal{h}$ how do I apply this to the following situation: A spin is coupled to a single harmonic oscillator. My prof then told me that for this case we have $\mathcal{h}=\mathbb{C}$. Is my Fock Space then only given by the following? (As I only have one harmonic oscillator) $$\mathcal{F}(\mathcal{h}):=\bigoplus_0^1\mathcal{h}^{n}$$ He also said that the creation and annhilation are then given by simply $a$ and not $a(1)$ for $1\in \mathbb(C)$. So how do these operators then act on an element of the Fock Space? I assume the following but not sure if this is correct: $$ a[\alpha]=\sqrt{1}<1,\alpha>=\alpha$$ and $$ a^*[\alpha]=\sqrt{1+1} 1\otimes \alpha=\sqrt{2}\alpha$$ as $1,\alpha \in \mathbb{C}$. I can see that this cannot be correct as $a$ not equal to the identity operator but then how does it act? It seems I am a bit confused on the Fock Space and how the operators act. Any help for clarification would be appreciated.