How Do You Construct the Wavefunction \(\psi(x,t)\) from a Given \(a(k)\)?

[Mindscrape]

In this problem I am given a function $$a(k) = \frac{C \alpha}{\sqrt{ \pi}} e^{-\alpha ^2 k^2}$$
where alpha and C are both constants

Now I am supposed to construct $$\psi (x,t)$$

My work:
$$\psi (x,t) = \int_{-\inf}^{\inf} a(k) e^{i[kx - \omega (k) t]} dk$$
pull out the constants from our given function and join exponentials to get
$$\psi (x,t) = \frac{C \alpha}{\sqrt{\pi}} \int_{-inf}^{inf} e^{i[kx - \omega (k) t] - \alpha ^2 k^2} dk$$
Here's where I am unsure. This is strange integral to evaluate, but my tactic was to complete the square, and hope for the best. The rest of my work is shown below, but I don't know if it is right or not
$$\psi (x,t) = \frac{C \alpha}{\sqrt{\pi}} \int_{-inf}^{inf} e^{-(\alpha k - kx/2)^2 - (kx/2)^2 - \omega (k)t} dk$$
Where do I go from here? What is the best way to evaluate this integral?

 

[Mindscrape]

Maybe quantum mechanics isn't intro level physics, should this be moved to advanced physics?

Also, what is the LaTeX command for infinity?

 

[OlderDan]

Maybe quantum mechanics isn't intro level physics, should this be moved to advanced physics?

Also, what is the LaTeX command for infinity?

It probably should be in the advanced section, but it is here now.

Have you tried expresseing the complex exponential in terms of sine and cosine? Have you tried using an online calculus tool to look up the integral?

$$\infty$$

Do you have a finctional form for $$\omega (k)$$?