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In this problem I am given a function [tex] a(k) = \frac{C \alpha}{\sqrt{ \pi}} e^{-\alpha ^2 k^2} [/tex]

where alpha and C are both constants

Now I am supposed to construct [tex] \psi (x,t) [/tex]

[tex]\psi (x,t) = \int_{-\inf}^{\inf} a(k) e^{i[kx - \omega (k) t]} dk [/tex]

pull out the constants from our given function and join exponentials to get

[tex]\psi (x,t) = \frac{C \alpha}{\sqrt{\pi}} \int_{-inf}^{inf} e^{i[kx - \omega (k) t] - \alpha ^2 k^2} dk [/tex]

[tex]\psi (x,t) = \frac{C \alpha}{\sqrt{\pi}} \int_{-inf}^{inf} e^{-(\alpha k - kx/2)^2 - (kx/2)^2 - \omega (k)t} dk [/tex]

Where do I go from here? What is the best way to evaluate this integral?

where alpha and C are both constants

Now I am supposed to construct [tex] \psi (x,t) [/tex]

**My work:**[tex]\psi (x,t) = \int_{-\inf}^{\inf} a(k) e^{i[kx - \omega (k) t]} dk [/tex]

pull out the constants from our given function and join exponentials to get

[tex]\psi (x,t) = \frac{C \alpha}{\sqrt{\pi}} \int_{-inf}^{inf} e^{i[kx - \omega (k) t] - \alpha ^2 k^2} dk [/tex]

**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[tex]\psi (x,t) = \frac{C \alpha}{\sqrt{\pi}} \int_{-inf}^{inf} e^{-(\alpha k - kx/2)^2 - (kx/2)^2 - \omega (k)t} dk [/tex]

Where do I go from here? What is the best way to evaluate this integral?

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