Fourier Transforms and Quantum Uncertainty

June 2026

Fourier Transform

One of the most remarkable ideas in mathematics is that a complicated wave can be decomposed into simpler waves.

The Fourier Transform is the mathematical tool that performs this decomposition. It takes a complex waveform and expresses it as a combination of simpler oscillatory components, each with its own frequency.

Consider a simple sine wave:

f(x)=sin⁡(x)f(x) = \sin(x)

This wave contains a single frequency. If we perform a Fourier Transform, the result is a single peak in frequency space corresponding to that frequency.

Now consider:

f(x)=sin⁡(x)+sin⁡(3x)f(x) = \sin(x) + \sin(3x)

The signal now contains two frequencies. The Fourier Transform reveals this by producing two peaks in the frequency spectrum.

This idea appears throughoutin many engineering problems like Audio compression, Medical Imaging, Image compression etc.

Quantum Mechanics

One of the greatest conceptual shifts in physics came when we realized that particles do not behave like tiny billiard balls following exact trajectories.

In classical mechanics, we imagine a particle having a precise position and velocity at every instant.

At microscopic quantum scales, particles exhibit wave-like behavior.

Instead of describing a particle using an exact trajectory, we describe it using a wavefunction:

ψ(x)\psi(x)

The wavefunction contains information about all possible measurement outcomes.

More specifically:

∣ψ(x)∣2|\psi(x)|^2

represents the probability density of finding the particle near position (x).

This uncertainty is not merely a limitation of our instruments. It reflects something deeper about the nature of reality itself.

Oscillations and Momentum

A particle is described by its wave function

Position and Momentum of the particle - the principle states that if we try to localize the particle (single peak for x), we ought to be increasing the uncertainty in momentum of the particle.

To understand this we have to understand the idea of momentum in quantum mechanics

In classical mechanics:

p=mvp = mv

In Quantum mechanics, the pure momentum state is represented by a wave:

ψ(x)=eikx\psi(x) = e^{ikx}

The quantity that measures this oscillation density is k

Louis de Broglie proposed that every particle possesses an associated wavelength:

λ=hp\lambda = \frac{h}{p}

Rearranging:

p=hλp = \frac{h}{\lambda}

Substituting:

k=2πλk = \frac{2\pi}{\lambda}

and using

ℏ=h2π\hbar = \frac{h}{2\pi}

gives:

p=ℏkp = \hbar k

The tighter the oscillations, the larger the momentum. It becomes a measure of the spatial oscillation content of the wave.

Heisenberg’s Uncertainty Principle

Heisenberg Uncertainty Principle

In order to create a localized point for a particle - to get the closest position of the particle, we need multiple values of k, a single wave extends throughout all space and cannot represent a localized particle.

ψ(x)=======localized wave packet\psi(x) ======= \text{localized wave packet}

A single wave:

eikxe^{ikx}

extends throughout all space and cannot represent a localized particle.

To create localization, many waves with different wave numbers must interfere:

eik1x,eik2x,eik3x,...e^{ik_1x}, \quad e^{ik_2x}, \quad e^{ik_3x}, \quad ...

The spread of wave numbers is denoted by:

Δk\Delta k

The more localized we want the particle to be, the more wave numbers we must combine.

Consider a Gaussian wave packet:

ψ(x)=======e−x2/(2σx2)\psi(x) ======= e^{-x^2/(2\sigma_x^2)}

Its spatial width is:

Δx=σx\Delta x = \sigma_x

Performing a Fourier Transform yields another Gaussian:

ϕ(k)=======e−k2/(2σk2)\phi(k) ======= e^{-k^2/(2\sigma_k^2)}

whose width is:

Δk=σk\Delta k = \sigma_k

The widths satisfy:

σxσk=12\sigma_x \sigma_k = \frac{1}{2}

Therefore:

ΔxΔk=12\Delta x \Delta k = \frac{1}{2}

More generally:

ΔxΔk≥12\Delta x \Delta k \ge \frac{1}{2}

A wave cannot be simultaneously localized in space and concentrated into a single wave number.

Using:

p=ℏkp = \hbar k

we obtain:

Δp=ℏΔk\Delta p = \hbar \Delta k Δx(Δpℏ)≥12\Delta x \left( \frac{\Delta p}{\hbar} \right) \ge \frac{1}{2} ΔxΔp≥ℏ2\Delta x \Delta p \ge \frac{\hbar}{2}

This is the Uncertainity Principle.

Precise position requires uncertain momentum, Precise momentum requires uncertain position - this is Heisenberg’s uncertainty principle.

Fourier Transform - encoding k

Because a quantum wavefunction contains position information and momentum information simultaneously. Fourier Transform extracts the momentum information hidden inside this position-space wavefunction.

Given:

ψ(x)\psi(x)

the Fourier Transform reveals the wave-number k spectrum hidden inside the wavefunction.

Since:

p=ℏkp = \hbar k

the Fourier spectrum is also the momentum spectrum.

As we try to localize and determine the position of the particle, the probability of a particle of my finger not being present near a distant star is not zero.