# Uncertainty Relation

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 $\hat{q}$ and $\hat{p}$ respectively.

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

$[\hat{\textbf{q}},\hat{\textbf{p}}] = \hat{\textbf{q}}\hat{\textbf{p}} - \hat{\textbf{p}}\hat{\textbf{q}}$ $= i\hbar$

Then the uncertainty relation for a given wavefunction $\left|\psi\right\rangle$ can be expressed as follows.

$D(\hat{p})D(\hat{q}) \ge \frac{\hbar^2}{4}$

where

$D_\psi(\hat{q}) = ||(\hat{\textbf{q}}-{x})\psi||^2 = \left \langle \psi|(\hat{\textbf{q}}-x)(\hat{\textbf{q}}-x)|\psi\right\rangle\\ \label{D2} D_\psi(\hat{p}) = ||(\hat{\textbf{p}}-{y})\psi||^2 = \left \langle \psi|(\hat{\textbf{p}}-{y})(\hat{\textbf{p}}-{y})|\psi\right\rangle$

and

${x} = \left \langle \psi |\hat{\textbf{q}}|\psi\right\rangle\\ {y} = \left \langle \psi |\hat{\textbf{p}}|\psi\right\rangle$

q represents position and p represents momentum. D(q) and D(p) are variances. The radius of the circle is related to the uncertainty product. Minimum uncertainty condition is met when the circle has the smallest radius.

Naturally occurring states are the number states

The ground state (for quantum harmonic oscillator) or the vacuum state (for light quantization) are the minimum uncertainty states, centered at the origin. As the number of energy level/number of photons in the system (n) increases, the associated uncertainty also increases.

The uncertainty dependence on photon number/principle quantum number is

$D(\hat{q})D(\hat{p}) = \frac{\hbar}{2\omega}(1+2n)\times\frac{\hbar\omega}{2}(1+2n)$

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 $(x,y)$. The solution $\left\langle q|\psi\right \rangle$ 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 $\psi$ and a coordinate multiplication operator $X_a$ defined in domain $x_a$ parametrized by $a$,

$(X_a\psi)(x_a) = x_a\psi(x_a)$

$x_0$ is the time domain and $x_1$ is the frequency domain. $x_a$ and $x_a'$ are intermediate domains.

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

$[X_a,X_a'] = \frac{i}{2\pi}sin(\phi'-\phi)$

$D(x_a)D(x_a') \ge \frac{1}{16\pi^2}\sin^2(\phi'-\phi)$

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).