My book has started using the wave packet definition as follows (time independent form):
$$\Psi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} A(k) \ e^{ikx}dx$$
I do not understand where the $1/\sqrt{2\pi}$ comes from in this definition. First, I thought it has something to do with normalization however, I can't seem to prove this to myself.
$$\Psi'(x) = N \int_{-\infty}^{\infty} A(k) \ e^{ikx}dx$$ $$\Psi'(x)^{\ast} = N \int_{-\infty}^{\infty} A^{\ast}(k) \ e^{-ikx}dx$$
$$\Psi'(x) \Psi'(x)^{\ast}= N^2 \int_{-\infty}^{\infty} A(k) \ e^{ikx}dx \int_{-\infty}^{\infty} A^{\ast}(k) \ e^{-ikx}dx = N^2 \int_{-\infty}^{\infty} A(k) A^{\ast}(k)dx = N^2 A(k) A^{\ast}(k)$$
The last step I justify by the conditions that the wave functions must approach zero as you go from $\pm \infty$.
$$P = 1 = N^2 \int_{-\infty}^{\infty} A(k) A^{\ast}(k)dk$$
I am not sure where to go from here. Does this term actually come from the normalization? If so, how can I show this.