Fourier Transforms in quantum mechanics

A Fourier series represents any periodic function as a sum of sine and cosine functions with appropriate coefficients. Since the sinusoids each have a representative frequency a periodic function in time can be analyzed in terms of its frequency. For non-periodic functions we use a Fourier transform to decompose a function of time (arbitrary) into its frequency components. We call time and frequency conjugate variables. Likewise, position and momentum are conjugate variables. For any conjugate variables x and k we can write a Fourier transform as

 

If we consider the time/frequency pair we have

The caveat here is that we might integrate from 0 to instead of from to . For a gaussian function we obtain the same answer and should be generally true. In quantum mechanics we have the following relations

Using these relations we can express a Fourier transform in terms of the quantum mechanical conjugate pairs. For example the position/momentum Fourier transform conjugate pair is:

In reality there are only a few functional forms that we routinely use in spectroscopy. These are

Exponential à FT is Lorentzian

Derivation of the exponential FT

Gaussian à FT is Gaussian

Derivation of the Gaussian FT

Square à FT is sin(x)/x

Derivation of the square FT

These forms are all that is required to describe a wealth of spectroscopy.