My book states that the wavefunctions for the quantum harmonic oscillator are $$\psi_n(x)=(1/2)^{n/2}H_n \left(\sqrt{\frac {m\omega}\hbar}x \right) \exp \left( -\frac{m\omega}{2\hbar}x^2\right)$$ where $H_n$ are the Hermite Polynomials. It also says that $\Psi_n(x)$ are the energy eigenstates, would these be vectors in function space? My question is, given these wavefunctions, how would you define the state vector $|\Psi\rangle$? I had assumed that I could write the eigenstates as $$|\Psi_n\rangle=\int_{-\infty}^\infty dx|x\rangle \psi_n(x)$$ and then expand the state vector as $$|\Psi\rangle=\sum_nc_n|\Psi_n\rangle$$ Is this correct? If not, how would I go about defining the state vector?
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Yes, everything you have written is correct, although, maybe, it is better to clarify the meaning of some definitions.
The "wavefunctions" of the Quantum Harmonic Oscillator are nothing but the representations in the position basis of the eigenstates of the Hamiltonian associated to the harmonic oscillator. Let us term the latter as $H_{HO}$. Then, its eigenstates are $|\Psi_n\rangle$, with $H_{HO}|\Psi_n\rangle=E_n |\Psi_n\rangle$, where $E_n$ is the energy of the $n$th level. Next, we insert a resolution of the identity in order to find the position representation of $|\Psi_n\rangle$: $$ |\Psi_n\rangle=\int_{-\infty}^\infty |x\rangle\langle x|\Psi_n\rangle=\int_{-\infty}^\infty dx |x\rangle\psi_n(x), $$ where $\psi_n(x)=\langle x|\Psi_n\rangle$ are the wavefunctions and have the form given by your textbook. Note that we have recovered the integral you have written in the second formula.
Finally, the state of the system at a given time does not need to be an eigenstate of $H_{HO}$, but can be any state of our Hilbert space. This is what you refer to as "state vector" $|\Psi\rangle$. How can we express it? Well, we can choose the basis decomposition we prefer, for instance: $$ |\Psi\rangle=\sum_n c_n |\Psi_n\rangle=\int_{-\infty}^\infty dx |x\rangle\psi(x), $$ where $c_n=\langle \Psi_n|\Psi\rangle$ and $\psi(x)=\langle x|\Psi\rangle $. Both are perfectly equivalent representations of the same "physical reality" described by $|\Psi\rangle$, and you may choose one of them according to the problem you want to address.
