- #1
phosgene
- 146
- 1
Homework Statement
Suppose a Gaussian wave packet ψ(x,0) is built out of plane waves according to the amplitude distribution function
[itex]A_{k} = \frac{Ca}{\sqrt{\pi}}e^{(-a^2(k-k_{0} )^2)}[/itex]
Calculate ψ(x,t) for this packet and describe its evolution.
Homework Equations
[itex]ψ(x,t) = ∫^{∞}_{-∞} A_{k} e^{i(kx - wt)}dk[/itex]
Gaussian integrals:
[itex]∫^{∞}_{-∞} e^{-Au^2}du = \sqrt{\frac{\pi}{A}}[/itex]
[itex]∫^{∞}_{-∞} u^2 e^{-Au^2}du = \frac{\sqrt{\pi}}{2A^{3/2}}[/itex]
The Attempt at a Solution
I'm fairly certain I'm supposed to put the equation for ψ(x,t) into the form of one of the above Gaussian integrals and use those identities to integrate it. But I'm really stuck as to how to do this. This is what I have to begin with:
[itex]ψ(x,t) = ∫^{∞}_{-∞} \frac{Ca}{\sqrt{\pi}}e^{(-a^2(k-k_{0} )^2)} e^{i(kx - wt)}dk[/itex]