- #1
Lambda96
- 233
- 77
- Homework Statement
- Calculate the Fourier transform of ##\frac{4 \pi d_0^2}{\hbar} \sin(\omega_0(t-t'))##
- Relevant Equations
- none
Hi,
I'm not sure if I have Fourier transformed the expression correctly
For the Fourier transformation, I used the following formula ##\int_{-\infty}^{\infty} f(t) e^{i \omega t}dt##
$$\frac{4 \pi d_0^2}{\hbar}\int_{-\infty}^{\infty} \sin(\omega_0(t-t')) e^{i \omega t}dt$$
$$\frac{4 \pi d_0^2}{\hbar}\int_{-\infty}^{\infty} \frac{1}{2i} \Bigl( e^{i \omega_0(t-t')} -e^{-i \omega_0(t-t')} \Bigr) e^{i \omega t}dt$$
$$\frac{4 \pi d_0^2}{\hbar} \frac{1}{2i} \Bigl(e^{-i \omega_0t'} \int_{-\infty}^{\infty} e^{it(\omega_0+\omega)} -e^{i \omega_0t'} \int_{-\infty}^{\infty} e^{it(\omega-\omega_0)} \Bigr) dt$$
To solve the integrals, I then applied the identity ##\int_{-\infty}^{\infty} e^{ik(x-x')} dk=2\pi \delta(x-x')## and obtained the following solution:
$$\frac{4 \pi d_0^2}{\hbar} \frac{1}{2i} \Bigl(e^{-i \omega_0t'} \delta(\omega_0 + \omega) -e^{i \omega_0t'} \delta(\omega - \omega_0) \Bigr)$$
Is that correct?
I'm not sure if I have Fourier transformed the expression correctly
For the Fourier transformation, I used the following formula ##\int_{-\infty}^{\infty} f(t) e^{i \omega t}dt##
$$\frac{4 \pi d_0^2}{\hbar}\int_{-\infty}^{\infty} \sin(\omega_0(t-t')) e^{i \omega t}dt$$
$$\frac{4 \pi d_0^2}{\hbar}\int_{-\infty}^{\infty} \frac{1}{2i} \Bigl( e^{i \omega_0(t-t')} -e^{-i \omega_0(t-t')} \Bigr) e^{i \omega t}dt$$
$$\frac{4 \pi d_0^2}{\hbar} \frac{1}{2i} \Bigl(e^{-i \omega_0t'} \int_{-\infty}^{\infty} e^{it(\omega_0+\omega)} -e^{i \omega_0t'} \int_{-\infty}^{\infty} e^{it(\omega-\omega_0)} \Bigr) dt$$
To solve the integrals, I then applied the identity ##\int_{-\infty}^{\infty} e^{ik(x-x')} dk=2\pi \delta(x-x')## and obtained the following solution:
$$\frac{4 \pi d_0^2}{\hbar} \frac{1}{2i} \Bigl(e^{-i \omega_0t'} \delta(\omega_0 + \omega) -e^{i \omega_0t'} \delta(\omega - \omega_0) \Bigr)$$
Is that correct?