A real scalar free field of mass $m$ can be represented as:
$$\hat\phi(\mathbf{x}) = \int \frac{d^3\mathbf{k}}{(2\pi)^3\sqrt{2\omega_{\mathbf{k}}}}\hat a_{\mathbf{k}}e^{i(\omega_{\mathbf{k}}t - \mathbf{k} \cdot \mathbf{x})} + \hat a^\dagger_{\mathbf{k}} e^{-i(\omega_{\mathbf{k}} t + \mathbf{k} \cdot \mathbf{x})},$$
where $\omega_{\mathbf{k}} = \sqrt{|\mathbf{k}|^2 + m^2}$.
What wavefunction does the creation operator $\hat a^\dagger_{\mathbf{k}}$ create from the vacuum state? Is it the momentum eigenstate $e^{i\mathbf{k}\cdot\mathbf{x}}$? If so, how can I show this?
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