Integrating the product of a real and a complex exponential

[seasponges]

Homework Statement

$\Psi(x,t) = \int^{\infty}_{-\infty} C(p)\Psi_{p}(x,t) dp$

is a solution to the Schroedinger equation for a free particle, where

$\Psi_{p}(x,t) = Ae^{i(px-Ept)/\hbar}$.

For the case $C(p) = e^{-(p-p_{0})^{2}/\sigma}$

where $\sigma$ is a real constant, compute the wavefunction at time t=0.

Homework Equations

$\int^{\infty}_{-\infty} e^{-αp^{2}+βp} = \sqrt{\frac{\pi}{α}}e^{\frac{β^{2}}{4\alpha}}$

where α is a positive real constant and β may be complex.

2. The attempt at a solution

This is the first part of one questions on a set of QM problems I've been given. I've made no progress with this part, because I don't know how to integrate the product of an exponential to a real number and an exponential to an imaginary number.

 

[vela]

What's keeping you from plugging everything in and using the relevant equation/hint?

 

[seasponges]

I can't isolate the $p^{2}$ term.

When I take the treat the integrand as an exponential to a complex power, I wind up with

$e^{-\frac{(p^{2}-2pp_{0}+p_{0}^{2})}{\sigma}+\frac{ix}{\hbar}p}$

 

[vela]

I guess I'm still kind of confused about where you're getting stuck. This is just basic algebra.
$$-\frac{p^2 - 2pp_0 + p_0^2}{\sigma}+\frac{ix}{\hbar}p = -\frac{1}{\sigma}p^2 + \left(\frac{2p_0}{\sigma}+\frac{ix}{\hbar}\right) p - p_0^2.$$ You may, however, find it easier to change variables in the integral first using the substitution $p' = p-p_0$, and then deal with the exponentials.

 

[seasponges]

Ahh sweet, that's cleared things up a little - many thanks!