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!