**Quantum Uncertainty:**

In quantum mechanics, the Heisenberg uncertainty principle, dictates the uncertainty product of position and momentum of a quantum mechanical system. The inequality arises from definitions of variance and the Cauchy-Schwarz inequality. The proof can be found on wikipedia.

Now, a quantum mechanical state has a specific position and momentum with uncertainty (model 1). A similar analogy exists in the quadratures of light (model 2).

Position and momentum operators (or in-phase and out-of-phase quadratures of light) can be represented as Hermitian operators and respectively.

These operators are canonically conjugate variables, which means they satisfy the following commutation relation

Then the uncertainty relation for a given wavefunction can be expressed as follows.

where

and

Naturally occurring states are the number states

The uncertainty dependence on photon number/principle quantum number is

The uncertainty increases with increase in photon/quantum number. Hence the ground/vacuum state is a minimum uncertainty state.

All minimum uncertainty states are called *coherent states*. A coherent state is hence, a ground state/vacuum state *displaced* by . The solution for the minimum uncertainty condition turns out to be a Gaussian.

The transformation from a ground/vacuum state to a coherent state is dictated by the Weyl operator.

Conceptually, the uncertainty in position(or in-phase quadrature) and momentum(or out-of-phase quadrature) are related in a way that, if one increases, the other must decrease, to conserve the product. So for a particle whose position is accurately known, the uncertainty in momentum of the particle is infinitely high and vice versa.

**Fourier Uncertainty:**

In signal processing, a signal can be represented in time domain or frequency domain. A signal that is ‘spread out’ and unbounded in the time domain, is a ‘squeezed’ and bounded in the frequency domain and vice versa. A function in time domain, and its Fourier representation are also conjugate variables.

Fractional Fourier Domain by Ozaktus and Aytur offers a good fundamental understanding of the time-frequency representation, and the transform between the various bases that lie in the time-frequency plane.

For a signal and a coordinate multiplication operator defined in domain parametrized by ,

It can be proved that and are related by a similar commutation and uncertainty equation

The transformation from one domain to the other is performed using Fractional Fourier Transform.

Chirp and Wavelet transforms are special forms of Fractional Fourier Transforms (ref).