I'm trying to find this integral
$$\int_{-\infty}^{\infty} \Psi^*(x,t)\left(-\hbar^2\frac{\partial^2}{\partial x^2}\right)\Psi(x,t)\ dx$$
where $$\Psi(x,t) = \frac{e^{-x^2/4(a^2+i\hbar t/2m)}}{\sqrt[4]{2\pi}\sqrt{a+i\hbar t/2ma}}$$
I'm reasonably certain that I calculated this wave function $\Psi$ correctly because the square of its absolute value is a Gaussian distribution which evolves in time which I know is what it's supposed to be.
However I'm having trouble simplifying the result of this integral. All of those constants (wrt $x$) are getting crazy.
For instance just the part of the expression in the integrand that I've gotten so far (which is maybe half of the part without $x$ in it) is
$$\frac{\hbar^2 a(a^4-\hbar^2t^2/(4m^2)-i\hbar t/m)}{2\sqrt{2\pi}(a^4+\hbar^2t^2/4m^2)^2}$$
This seems way too complicated and I haven't even gotten to the point where I can take the integral yet (which incidentally should be the easiest part). Mathematica isn't being especially useful at the moment in giving me shorter expressions. I suspect it's because it assumes all variables are complex. Is there some way to simplify all of this -- especially the complex algebra part -- that I'm missing? Thanks.
BTW, all variables/ constants are real. In fact all the constants ($a$, $\hbar$, and $m$) are positive real numbers and the variables $t$ and $x$ are real numbers.
Edit:
So setting
$$\Psi(x,t) = \frac{e^{-x^2/4(a^2+i\hbar t/2m)}}{\sqrt[4]{2\pi}\sqrt{a+i\hbar t/2ma}} = \frac{e^{-x^2/A}}{B},$$
evaluating the integral, and simplifying I'm getting
$$\frac{2\hbar^2}{|A|^2|B|^2}\frac{\sqrt{\pi}|A|\operatorname{Im}(A)\color{red}{iA^*}}{[\operatorname{Re}(A)]^{3/2}}$$
Before even plugging in my expressions for $A$ and $B$ I see something wrong: $-\hbar^2\frac{\partial^2}{\partial x^2}$ corresponds to an observable (momentum squared) and thus its expectation value should be real. But all of the numbers in this last expression are real except the part in red which will be complex. Thus I've made some mistake.
I can't see a mistake in my calculations. Where am I going wrong here?
Re-writhe $\Psi(x)=Ae^\frac {-x^2} s$, where $s=4(a^2+i\hbar t/2m)$ and $A=\frac{1}{\sqrt[4]{2\pi}\sqrt{a+i\hbar t/2ma}}$. Then $$\frac{d^2}{dx^2}(\Psi(x))=\frac{-2A(s-2x^2)}{s^2}e^{\frac{-x^2}{s}}$$ Then expand $s$ and calculate $\Psi(x)^*$
The integral is standard then.