Find the probability of measuring a particle's energy E

[Leechie]

Homework Statement

I need to use the overlap rule to find the probability that a measurement of a particle's energy at time t=0 will give the ground state E₀

The normalized wave functions I have are:

$$Ψ(x,0)=\left(\frac{2a}{\pi}\right)^{1/4}e^{-ikx-ax^2}$$$$ψ_0(x)=\left(\frac{2a}{\pi}\right)^{1/4}e^{-ax^2}$$

Homework Equations

The overlap integral equation I am trying to use is:
$$p_x=\left|\int_{-\infty}^{\infty}ψ^*_0\left(x\right)Ψ\left(x,0\right)dx\right|^2$$
I've also got the standard integrals which may help:
$$\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}$$$$\int_{-\infty}^{\infty}e^{-x^2}e^{-ikx}dx=\sqrt{\pi}e^{-\frac {k^2}{4}}$$

The Attempt at a Solution

So far, I've gone through the following steps:
$$p_x=\left|\int_{-\infty}^{\infty}\left(\frac{2a}{\pi}\right)^{1/4}e^{+ax^2}\left(\frac{2a}{\pi}\right)^{1/4}e^{-ikx-ax^2}dx\right|^2$$$$p_x=\left|\frac {\sqrt{2a}}{\sqrt{\pi}} \int_{-\infty}^{\infty}e^{+ax^2}e^{-ikx-ax^2}dx\right|^2$$$$p_x=\left|\frac {\sqrt{2a}}{\sqrt{\pi}} \int_{-\infty}^{\infty}e^{-ikx}dx\right|^2$$
But that's as far as I seem to be able to get. I've either done something wrong to get to here or I'm missing something from here on. My thoughts are I need to use one of the standard integrals but I can't seem to figure out how to. Can anyone offer any advice on how to proceed with this? Thanks

 

[DrClaude]

$\psi_0$ is real, so $\psi_0^* = \psi_0$.

 

[Leechie]

Thank you so much! I can't believe I didn't spot that.