Equation problem. How to elimintate t?

[intijk]

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

A and a are positive real constant.

Use Normalization to get A, the answer says that:

[itex]1=2|A|^2 \int_0^\inf e^{-2amx^2/h}dx[/itex] (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
[tex]\Psi(x,t)= Ae^{-amx^2/h}e^{ait}[/tex]

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

 

[intijk]

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

Thank you so much!