This note is written after seeing this video's title: [[@UnderstandFourierTransform2022|To Understand the Fourier Transform]].  Heisenberg's Uncertainty Principle is closely related to and expressible in terms of the Fourier Transform. This relationship stems from the fundamental properties of Fourier transforms in the context of wave functions, which describe quantum states in quantum mechanics. ### Heisenberg's Uncertainty Principle Heisenberg's Uncertainty Principle is a fundamental theory in quantum mechanics which states that it is impossible to simultaneously know the exact position and momentum of a particle. In more technical terms, the principle sets a limit on the precision with which position $x$ and momentum $p$ can be simultaneously known: $ΔxΔp≥2ℏ​$ where $Δx$ and $Δp$ are the standard deviations of position and momentum, respectively, and $ℏ$ is the [[Planck's constant#Reduced Planck's constant|reduced Planck constant]]. ### Fourier Transform and the Uncertainty Principle The Fourier Transform provides a mathematical framework for transforming a function (or signal) from its original domain (often time or space) into the frequency domain. In quantum mechanics, the wave function $ψ(x)$ of a particle, which provides information about its position, can be transformed into a wave function $ϕ(p)$ in the momentum space through the Fourier Transform: $\phi(p) = \frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-\frac{ipx}{\hbar}} \, dx$ - $\phi(p)$: Represents the wave function in the momentum space. - $\frac{1}{\sqrt{2\pi \hbar}}$: The normalization factor necessary for the Fourier Transform in quantum mechanics. - $\int_{-\infty}^{\infty}$: The integral sign with limits from negative infinity to positive infinity, indicating that the transformation integrates over all space. - $\psi(x)$: The wave function in the position space. - $e^{-\frac{ipx}{\hbar}}$: The exponential function representing the kernel of the Fourier Transform. $i$ is the imaginary unit, $p$ is the momentum variable, $x$ is the position variable, and $ℏ$ (h-bar) is the [[Planck's constant#Reduced Planck's constant|reduced Planck's constant]]. - $dx$: The differential element of the integral in the position space. This transformation reflects the duality between position and momentum. The more localized the wave function is in position space (meaning the particle's position is well-defined), the more spread out it will be in momentum space, and vice versa. This spread is quantified by the standard deviations $Δx$ and $Δp$, which are inversely related through the Fourier Transform properties. ### Mathematical Expression The mathematical underpinning of this relationship comes from the properties of the Fourier Transform applied to the wave functions. The Fourier Transform of a sharply peaked function in position space will be broadly spread in momentum space, demonstrating a fundamental property of the Fourier Transform: a function and its Fourier transform cannot both be sharply localized. This broadening in the transform domain underlies the uncertainty principle. If one tries to confine a particle to a very small region of space (making $Δx$ very small), the Fourier transform (momentum space representation) becomes very wide, increasing $Δp$. ### Physical Implications This relationship between position and momentum via the Fourier Transform is more than a mathematical curiosity—it fundamentally limits our ability to measure and predict the behavior of quantum systems. It implies that the quantum world is inherently probabilistic, rather than deterministic, and that this probabilistic nature arises from the core mathematical structures (like the Fourier Transform) used to describe quantum states. # Conclusion In summary, [[Uncertainty principle|Heisenberg's Uncertainty Principle]] and the [[Fourier Transform]] are fundamentally interconnected in quantum mechanics. This relationship offers profound mathematical insights into the limitations of quantum measurements and illustrates how the **expressed properties** of particles in quantum mechanics differ fundamentally from those in classical mechanics. Unlike classical mechanics, which treats properties like location and momentum as deterministic and precisely measurable, quantum mechanics describes these properties in statistical terms, reflecting the inherent uncertainties and probabilistic nature of quantum states. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Uncertainty") or contains(subject, "Fourier Transform") sort modified desc, authors, title ```