- #1

Mutatis

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## Homework Statement

Find the noralization constant ##A## of the function bellow: $$ \psi(x) = A e^\left(i k x -x^2 \right) \left[ 1 + e^\left(-i \alpha \right) \right], $$ ##\alpha## is also a constant.

## Homework Equations

##\int_{-\infty}^{\infty} e^\left(-\lambda x^2 \right) \, dx = \sqrt {\frac {\pi} {\lambda} }##

## The Attempt at a Solution

Well, first I've tried to find the density of probability: $$ \left| \psi(x) \right|^2 = 2 \left| A \right|^2 e^\left( -2 x^2 \right) + e^\left( -2 i \alpha x^2 \right) +e^\left(2 i \alpha x^2 \right) .$$ Then I have no idea how to solve the integral to normalize ##\psi(x)##... The fisrt two terms I've solved using the gaussian integral above, but I can't do it to the third term: $$ 2 \left| A \right|^2 \left( \int_{-\infty}^{\infty} e^\left( -2 x^2 \right) dx + \int_{-\infty}^{\infty} e^\left( -2 i \alpha x^2 \right) dx + \int_{-\infty}^{\infty} e^\left(2 i \alpha x^2 \right) dx \right)= 1 \\ 2 \left| A \right|^2 \left( \frac {\sqrt {2 \pi}} {2} + \frac {\sqrt {2 i \alpha \pi}} {2 i \alpha} + \int_{-\infty}^{\infty} e^\left(2 i \alpha x^2 \right) dx \right)= 1$$.

I don't know if I'm doing it right, but I have no idea to get this solved... Help!