Register to reply

Fourier Transform and Wave Function

by atomicpedals
Tags: fourier, function, transform, wave
Share this thread:
Apr28-12, 03:19 PM
P: 196
1. The problem statement, all variables and given/known data

a) Find the normalization constant N for the Gaussian wave packet [itex]\psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/itex]. b) Find the Fourier Transform and verify it is normalized.

2. The attempt at a solution

a) I think I've got
[tex]\psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/tex]
[tex]\int |N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}|^{2}dx = 1[/tex]
[tex]N = \frac{1}{\pi^{1/4}\sqrt K}[/tex]
b) This is where the trouble starts...
[tex]\psi (x) = \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/tex]
[tex]F(\omega)=\frac{1}{\sqrt{2\pi}} \int \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} e^{i \omega x}dx[/tex]
I think I can pull the normalization constant out of the integrand and get
[tex]F(\omega)=\frac{1}{\sqrt{2\pi}}\frac{1}{\pi^{1/4}\sqrt K} \int e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} e^{i \omega x}dx[/tex]
And the exponents should combine (here I'm not so sure)
[tex]F(\omega)=\frac{1}{\sqrt{2\pi}}\frac{1}{\pi^{1/4}\sqrt K} \int e^{\frac{-(x-x_{0})^{2}}{2K^{2}}+i \omega x}dx[/tex]
Assuming I've not gone horribly wrong earlier, evaluation of this integral stumps me. Any help and suggestions are much appreciated.
Phys.Org News Partner Science news on
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice
Apr28-12, 03:53 PM
P: 196
If I crank through it, I think the result is

[tex]F(\omega)=\frac{1}{\pi^{1/4}\sqrt{1/K^{}2}\sqrt K}e^{\frac{-K^{2}k{2}}{2}+i x_{0} \omega}[/tex]
Apr28-12, 05:07 PM
Sci Advisor
HW Helper
PF Gold
P: 11,688
There should be an ##\omega^2## somewhere in the exponent. Your answer seems to have other typos as well. It would help to see what you actually did, but I think you have it under control.

Register to reply

Related Discussions
Wave-Function, Fourier Transform, and Speed Quantum Physics 1
Does additivity apply to Fourier transform of the wave function Advanced Physics Homework 2
Fourier Transform of the Wave Eq. Calculus & Beyond Homework 2
New here: need help on Fourier transform of wave-function Quantum Physics 5
Fourier transform of a wave function Quantum Physics 1