# Fourier transform

## Homework Statement

An atom raised at t=0 to an excited state with energy [itex] E_0= \hbar \omega_0 [itex] has the time dependence [itex] T(t)=\frac{1}{\sqrt \tau}e^{-t/ 2 \tau} [itex] for t>0 and T(t)=0 for t<0. Thus the probability of being in an excited state decays exponentially with time.

[itex] T(t)^2= \ frac {1} {\tau} e^ {-t/ \tau} [itex]

a.) Find the transform b(w) of T(t).

('w' is omega, frequency)

b.) Plot |b(w)|^2 as a function of w.

c.) show that b(w) traces a circle on the complex plane as w runs from well below w_0 to well above it.

## Homework Equations

$$b(\omega)=\frac{1}{\sqrt{2 \pi}}\int{T(t)e^{i \omega t}}$$

The integral runs from -infinity to infinity.

## The Attempt at a Solution

$$b(\omega)=-\frac{1}{\sqrt{2 \pi \tau}}\frac{1}{i(\omega - \omega _0)-\frac{1}{2 \tau}}$$

When I do plot for part b, I get exponential incerase. Does that sound right?

But when I do part (c), I don't get a circle nor elipse.