Equation problem. How to elimintate t?

[intijk]

$\Psi(x,t)=Ae^{-a[(mx^2/h)+i t]}$ (1)

A and a are positive real constant.

Use Normalization to get A, the answer says that:

$1=2|A|^2 \int_0^\inf e^{-2amx^2/h}dx$ (2)

Can you show me how to do the transform to get the righside of the equation (2)?

 

[HallsofIvy]

First, you can write the equation as
$$\Psi(x,t)= Ae^{-amx^2/h}e^{ait}$$

To "normalize" that function means to find A such that the integral of $|\Psi|^2= (\Psi)(\Psi^*)$, the product of $\Psi$ and its complex conjugate, over all "space", is 1. The only "i" is in $e^{ait}$ and, of course, $(e^{ait})(e^{-ait})= 1$. Since this has only one space variable, x, that should be for x from $-\infty$ to $\infty$. Of course, the function is even in x so you can just integrate from 0 to $\infty$ and then multiply by 2.

 

[intijk]

Thank you, HallsofIvy.
I know it now. In $\Psi^*$ there is a $e^{-ait}$.
So, $\Psi\Psi^*$ will cause $e^{ait}e^{-ait}=1$, then t is eliminated.

Thank you so much!