Question

HELP ME! Wave Packet equation

Answer 1

The superposition is called a wave packet, and it takes the form:
\[\psi (x,t) = \int\limits_{-\infty}^{\infty} dk. A(k)e^{i(kx- \omega t)}\]
we begin by considering the wave packet at time t=0
\[\psi (x,0) = \int\limits\limits_{-\infty}^{\infty} dk. A(k)e^{ikx} \]
and ilustrate it by considering a special form, called the gaussian form
\[A(k) = e^{-\alpha (k-k_{o})^{2}/2}\]
proof that:
\[\psi (x,0) = \sqrt{\frac{ 2 \pi }{ \alpha }} e^{ik_{o}x} e^{-x^{2}/2\alpha} \]

Answer 2

Well, this would suit better in a mathematical question. Anyways, as physics is mixed with a lot of maths, this will help you,
http://en.wikipedia.org/wiki/Gaussian_integral

Answer 3

May be you have found the answer. But in case you have not, this is my proposition.
\[\psi(x,0)=\int dk\ e^{-\alpha(k-k_0)^2/2+ikx}=\int dk\ e^{-\alpha(k^2+k_0^2-2kk_0)/2+ikx}=\\
=\int dk\ e^{\alpha k^2/2+\alpha k_0k+ikx-\alpha k_0^2/2}=\int dk\ e^{\alpha k^2/2+k(\alpha k_0+ix)-\alpha k_0^2/2}\]
Apply,
\[\int dx\ e^{-ax^2+bx+c}=\sqrt{\frac{\pi}{a}}e^{b^2/(4a)+c}\]And you have,
\[\sqrt{\frac{2\alpha}{\pi}}e^{(\alpha k_0+ix)^2/(2\alpha)-\alpha k_0^2/2}=
\sqrt{\frac{2\alpha}{\pi}}e^{ixk_0}e^{-\alpha x^2/(2\alpha)}\]